NetworkX Reference Release 1.7 Aric Hagberg, Dan Schult, Pieter Swart July 04, 2012 CONTENTS 1 Overview 1 1.1 Who uses NetworkX? .......................................... 1 1.2 Goals ................................................... 1 1.3 The Python programming language ................................... 1 1.4 Free software ............................................... 2 1.5 History .................................................. 2 2 Introduction 3 2.1 NetworkX Basics ............................................. 3 2.2 Nodes and Edges ............................................. 4 3 Graph types 9 3.1 Which graph class should I use? ..................................... 9 3.2 Basic graph types ............................................. 9 4 Algorithms 127 4.1 Approximation .............................................. 127 4.2 Assortativity ............................................... 131 4.3 Bipartite ................................................. 140 4.4 Blockmodeling .............................................. 158 4.5 Boundary ................................................. 159 4.6 Centrality ................................................. 160 4.7 Chordal .................................................. 176 4.8 Clique .................................................. 179 4.9 Clustering ................................................ 181 4.10 Communities ............................................... 185 4.11 Components ............................................... 186 4.12 Cores ................................................... 199 4.13 Cycles .................................................. 203 4.14 Directed Acyclic Graphs ......................................... 204 4.15 Distance Measures ............................................ 206 4.16 Distance-Regular Graphs ......................................... 208 4.17 Eulerian .................................................. 210 4.18 Flows ................................................... 212 4.19 Graphical degree sequence ........................................ 223 4.20 Hierarchy ................................................. 224 4.21 Isolates .................................................. 225 4.22 Isomorphism ............................................... 226 4.23 Link Analysis ............................................... 239 i 4.24 Matching ................................................. 245 4.25 Maximal independent set ......................................... 246 4.26 Minimum Spanning Tree ......................................... 247 4.27 Operators ................................................. 249 4.28 Rich Club ................................................. 256 4.29 Shortest Paths .............................................. 257 4.30 Simple Paths ............................................... 275 4.31 Swap ................................................... 276 4.32 Traversal ................................................. 278 4.33 Vitality .................................................. 279 5 Functions 281 5.1 Graph ................................................... 281 5.2 Nodes ................................................... 283 5.3 Edges ................................................... 284 5.4 Attributes ................................................. 284 5.5 Freezing graph structure ......................................... 286 6 Graph generators 289 6.1 Atlas ................................................... 289 6.2 Classic .................................................. 289 6.3 Small ................................................... 294 6.4 Random Graphs ............................................. 298 6.5 Degree Sequence ............................................. 307 6.6 Random Clustered ............................................ 312 6.7 Directed ................................................. 314 6.8 Geometric ................................................ 317 6.9 Hybrid .................................................. 320 6.10 Bipartite ................................................. 321 6.11 Line Graph ................................................ 325 6.12 Ego Graph ................................................ 326 6.13 Stochastic ................................................. 326 6.14 Intersection ................................................ 327 6.15 Social Networks ............................................. 329 7 Linear algebra 331 7.1 Graph Matrix ............................................... 331 7.2 Laplacian Matrix ............................................. 332 7.3 Spectrum ................................................. 334 7.4 Attribute Matrices ............................................ 335 8 Converting to and from other data formats 341 8.1 To NetworkX Graph ........................................... 341 8.2 Dictionaries ................................................ 342 8.3 Lists ................................................... 343 8.4 Numpy .................................................. 344 8.5 Scipy ................................................... 347 9 Reading and writing graphs 351 9.1 Adjacency List .............................................. 351 9.2 Multiline Adjacency List ......................................... 355 9.3 Edge List ................................................. 359 9.4 GEXF ................................................... 365 9.5 GML ................................................... 367 9.6 Pickle ................................................... 370 ii 9.7 GraphML ................................................. 371 9.8 JSON ................................................... 373 9.9 LEDA ................................................... 379 9.10 YAML .................................................. 380 9.11 SparseGraph6 .............................................. 381 9.12 Pajek ................................................... 382 9.13 GIS Shapeﬁle ............................................... 384 10 Drawing 387 10.1 Matplotlib ................................................ 387 10.2 Graphviz AGraph (dot) .......................................... 395 10.3 Graphviz with pydot ........................................... 398 10.4 Graph Layout ............................................... 400 11 Exceptions 405 12 Utilities 407 12.1 Helper functions ............................................. 407 12.2 Data structures and Algorithms ..................................... 408 12.3 Random sequence generators ...................................... 408 12.4 Decorators ................................................ 411 13 License 415 14 Citing 417 15 Credits 419 16 Glossary 421 Bibliography 423 Python Module Index 431 Index 433 iii iv CHAPTER ONE OVERVIEW NetworkX is a Python language software package for the creation, manipulation, and study of the structure, dynamics, and function of complex networks. With NetworkX you can load and store networks in standard and nonstandard data formats, generate many types of random and classic networks, analyze network structure, build network models, design new network algorithms, draw networks, and much more. 1.1 Who uses NetworkX? The potential audience for NetworkX includes mathematicians, physicists, biologists, computer scientists, and social scientists. Good reviews of the state-of-the-art in the science of complex networks are presented in Albert and Barabási [BA02], Newman [Newman03], and Dorogovtsev and Mendes [DM03]. See also the classic texts [Bollobas01], [Diestel97] and [West01] for graph theoretic results and terminology. For basic graph algorithms, we recommend the texts of Sedgewick, e.g. [Sedgewick01] and [Sedgewick02] and the survey of Brandes and Erlebach [BE05]. 1.2 Goals NetworkX is intended to provide • tools for the study the structure and dynamics of social, biological, and infrastructure networks, • a standard programming interface and graph implementation that is suitable for many applications, • a rapid development environment for collaborative, multidisciplinary projects, • an interface to existing numerical algorithms and code written in C, C++, and FORTRAN, • the ability to painlessly slurp in large nonstandard data sets. 1.3 The Python programming language Python is a powerful programming language that allows simple and ﬂexible representations of networks, and clear and concise expressions of network algorithms (and other algorithms too). Python has a vibrant and growing ecosystem of packages that NetworkX uses to provide more features such as numerical linear algebra and drawing. In addition Python is also an excellent “glue” language for putting together pieces of software from other languages which allows reuse of legacy code and engineering of high-performance algorithms [Langtangen04]. Equally important, Python is free, well-supported, and a joy to use. 1 NetworkX Reference, Release 1.7 In order to make the most out of NetworkX you will want to know how to write basic programs in Python. Among the many guides to Python, we recommend the documentation at http://www.python.org and the text by Alex Martelli [Martelli03]. 1.4 Free software NetworkX is free software; you can redistribute it and/or modify it under the terms of the BSD License. We welcome contributions from the community. Information on NetworkX development is found at the NetworkX Developer Zone https://networkx.lanl.gov/trac. 1.5 History NetworkX was born in May 2002. The original version was designed and written by Aric Hagberg, Dan Schult, and Pieter Swart in 2002 and 2003. The ﬁrst public release was in April 2005. Many people have contributed to the success of NetworkX. Some of the contributors are listed in the credits. 1.5.1 What Next • A Brief Tour • Installing • Reference • Examples 2 Chapter 1. Overview CHAPTER TWO INTRODUCTION The structure of NetworkX can be seen by the organization of its source code. The package provides classes for graph objects, generators to create standard graphs, IO routines for reading in existing datasets, algorithms to analyse the resulting networks and some basic drawing tools. Most of the NetworkX API is provided by functions which take a graph object as an argument. Methods of the graph object are limited to basic manipulation and reporting. This provides modularity of code and documentation. It also makes it easier for newcomers to learn about the package in stages. The source code for each module is meant to be easy to read and reading this Python code is actually a good way to learn more about network algorithms, but we have put a lot of effort into making the documentation sufﬁcient and friendly. If you have suggestions or questions please contact us by joining the NetworkX Google group. Classes are named using CamelCase (capital letters at the start of each word). functions, methods and variable names are lower_case_underscore (lowercase with an underscore representing a space between words). 2.1 NetworkX Basics After starting Python, import the networkx module with (the recommended way) >>> import networkx as nx To save repetition, in the documentation we assume that NetworkX has been imported this way. If importing networkx fails, it means that Python cannot ﬁnd the installed module. Check your installation and your PYTHONPATH. The following basic graph types are provided as Python classes: Graph This class implements an undirected graph. It ignores multiple edges between two nodes. It does allow self-loop edges between a node and itself. DiGraph Directed graphs, that is, graphs with directed edges. Operations common to directed graphs, (a subclass of Graph). MultiGraph A ﬂexible graph class that allows multiple undirected edges between pairs of nodes. The additional ﬂexibility leads to some degradation in performance, though usually not signiﬁcant. MultiDiGraph A directed version of a MultiGraph. Empty graph-like objects are created with >>> G=nx.Graph() >>> G=nx.DiGraph() >>> G=nx.MultiGraph() >>> G=nx.MultiDiGraph() 3 NetworkX Reference, Release 1.7 All graph classes allow any hashable object as a node. Hashable objects include strings, tuples, integers, and more. Arbitrary edge attributes such as weights and labels can be associated with an edge. The graph internal data structures are based on an adjacency list representation and implemented using Python dic- tionary datastructures. The graph adjaceny structure is implemented as a Python dictionary of dictionaries; the outer dictionary is keyed by nodes to values that are themselves dictionaries keyed by neighboring node to the edge at- tributes associated with that edge. This “dict-of-dicts” structure allows fast addition, deletion, and lookup of nodes and neighbors in large graphs. The underlying datastructure is accessed directly by methods (the programming in- terface “API”) in the class deﬁnitions. All functions, on the other hand, manipulate graph-like objects solely via those API methods and not by acting directly on the datastructure. This design allows for possible replacement of the ‘dicts-of-dicts’-based datastructure with an alternative datastructure that implements the same methods. 2.1.1 Graphs The ﬁrst choice to be made when using NetworkX is what type of graph object to use. A graph (network) is a collection of nodes together with a collection of edges that are pairs of nodes. Attributes are often associated with nodes and/or edges. NetworkX graph objects come in different ﬂavors depending on two main properties of the network: • Directed: Are the edges directed? Does the order of the edge pairs (u,v) matter? A directed graph is speciﬁed by the “Di” preﬁx in the class name, e.g. DiGraph(). We make this distinction because many classical graph properties are deﬁned differently for directed graphs. • Multi-edges: Are multiple edges allowed between each pair of nodes? As you might imagine, multiple edges requires a different data structure, though tricky users could design edge data objects to support this function- ality. We provide a standard data structure and interface for this type of graph using the preﬁx “Multi”, e.g. MultiGraph(). The basic graph classes are named: Graph, DiGraph, MultiGraph, and MultiDiGraph 2.2 Nodes and Edges The next choice you have to make when specifying a graph is what kinds of nodes and edges to use. If the topology of the network is all you care about then using integers or strings as the nodes makes sense and you need not worry about edge data. If you have a data structure already in place to describe nodes you can simply use that structure as your nodes provided it is hashable. If it is not hashable you can use a unique identiﬁer to represent the node and assign the data as a node attribute. Edges often have data associated with them. Arbitrary data can associated with edges as an edge attribute. If the data is numeric and the intent is to represent a weighted graph then use the ‘weight’ keyword for the attribute. Some of the graph algorithms, such as Dijkstra’s shortest path algorithm, use this attribute name to get the weight for each edge. Other attributes can be assigned to an edge by using keyword/value pairs when adding edges. You can use any keyword except ‘weight’ to name your attribute and can then easily query the edge data by that attribute keyword. Once you’ve decided how to encode the nodes and edges, and whether you have an undirected/directed graph with or without multiedges you are ready to build your network. 2.2.1 Graph Creation NetworkX graph objects can be created in one of three ways: • Graph generators – standard algorithms to create network topologies. • Importing data from pre-existing (usually ﬁle) sources. 4 Chapter 2. Introduction NetworkX Reference, Release 1.7 • Adding edges and nodes explicitly. Explicit addition and removal of nodes/edges is the easiest to describe. Each graph object supplies methods to manip- ulate the graph. For example, >>> import networkx as nx >>> G=nx.Graph() >>> G.add_edge(1,2) # default edge data=1 >>> G.add_edge(2,3,weight=0.9) # specify edge data Edge attributes can be anything: >>> import math >>> G.add_edge(’y’,’x’,function=math.cos) >>> G.add_node(math.cos) # any hashable can be a node You can add many edges at one time: >>> elist=[(’a’,’b’,5.0),(’b’,’c’,3.0),(’a’,’c’,1.0),(’c’,’d’,7.3)] >>> G.add_weighted_edges_from(elist) See the /tutorial/index for more examples. Some basic graph operations such as union and intersection are described in the Operators module documentation. Graph generators such as binomial_graph and powerlaw_graph are provided in the Graph generators subpackage. For importing network data from formats such as GML, GraphML, edge list text ﬁles see the Reading and writing graphs subpackage. 2.2.2 Graph Reporting Class methods are used for the basic reporting functions neighbors, edges and degree. Reporting of lists is often needed only to iterate through that list so we supply iterator versions of many property reporting methods. For example edges() and nodes() have corresponding methods edges_iter() and nodes_iter(). Using these methods when you can will save memory and often time as well. The basic graph relationship of an edge can be obtained in two basic ways. One can look for neighbors of a node or one can look for edges incident to a node. We jokingly refer to people who focus on nodes/neighbors as node-centric and people who focus on edges as edge-centric. The designers of NetworkX tend to be node-centric and view edges as a relationship between nodes. You can see this by our avoidance of notation like G[u,v] in favor of G[u][v]. Most data structures for sparse graphs are essentially adjacency lists and so ﬁt this perspective. In the end, of course, it doesn’t really matter which way you examine the graph. G.edges() removes duplicate representations of each edge while G.neighbors(n) or G[n] is slightly faster but doesn’t remove duplicates. Any properties that are more complicated than edges, neighbors and degree are provided by functions. For example nx.triangles(G,n) gives the number of triangles which include node n as a vertex. These functions are grouped in the code and documentation under the term algorithms. 2.2.3 Algorithms A number of graph algorithms are provided with NetworkX. These include shortest path, and breadth ﬁrst search (see traversal), clustering and isomorphism algorithms and others. There are many that we have not developed yet too. If you implement a graph algorithm that might be useful for others please let us know through the NetworkX Google group or the Developer Zone. As an example here is code to use Dijkstra’s algorithm to ﬁnd the shortest weighted path: 2.2. Nodes and Edges 5 NetworkX Reference, Release 1.7 >>> G=nx.Graph() >>> e=[(’a’,’b’,0.3),(’b’,’c’,0.9),(’a’,’c’,0.5),(’c’,’d’,1.2)] >>> G.add_weighted_edges_from(e) >>> print(nx.dijkstra_path(G,’a’,’d’)) [’a’, ’c’, ’d’] 2.2.4 Drawing While NetworkX is not designed as a network layout tool, we provide a simple interface to drawing packages and some simple layout algorithms. We interface to the excellent Graphviz layout tools like dot and neato with the (suggested) pygraphviz package or the pydot interface. Drawing can be done using external programs or the Matplotlib Python package. Interactive GUI interfaces are possible though not provided. The drawing tools are provided in the module drawing. The basic drawing functions essentially place the nodes on a scatterplot using the positions in a dictionary or computed with a layout function. The edges are then lines between those dots. >>> G=nx.cubical_graph() >>> nx.draw(G) # default spring_layout >>> nx.draw(G,pos=nx.spectral_layout(G), nodecolor=’r’,edge_color=’b’) See the examples for more ideas. 2.2.5 Data Structure NetworkX uses a “dictionary of dictionaries of dictionaries” as the basic network data structure. This allows fast lookup with reasonable storage for large sparse networks. The keys are nodes so G[u] returns an adjacency dictionary keyed by neighbor to the edge attribute dictionary. The expression G[u][v] returns the edge attribute dictionary itself. A dictionary of lists would have also been possible, but not allowed fast edge detection nor convenient storage of edge data. Advantages of dict-of-dicts-of-dicts data structure: • Find edges and remove edges with two dictionary look-ups. • Prefer to “lists” because of fast lookup with sparse storage. • Prefer to “sets” since data can be attached to edge. • G[u][v] returns the edge attribute dictionary. • n in G tests if node n is in graph G. • for n in G: iterates through the graph. • for nbr in G[n]: iterates through neighbors. As an example, here is a representation of an undirected graph with the edges (‘A’,’B’), (‘B’,’C’) >>> G=nx.Graph() >>> G.add_edge(’A’,’B’) >>> G.add_edge(’B’,’C’) >>> print(G.adj) {’A’: {’B’: {}}, ’C’: {’B’: {}}, ’B’: {’A’: {}, ’C’: {}}} The data structure gets morphed slightly for each base graph class. For DiGraph two dict-of-dicts-of-dicts structures are provided, one for successors and one for predecessors. For MultiGraph/MultiDiGraph we use a dict-of-dicts-of- 6 Chapter 2. Introduction NetworkX Reference, Release 1.7 dicts-of-dicts 1 where the third dictionary is keyed by an edge key identiﬁer to the fourth dictionary which contains the edge attributes for that edge between the two nodes. Graphs use a dictionary of attributes for each edge. We use a dict-of-dicts-of-dicts data structure with the inner dictionary storing “name-value” relationships for that edge. >>> G=nx.Graph() >>> G.add_edge(1,2,color=’red’,weight=0.84,size=300) >>> print(G[1][2][’size’]) 300 1 “It’s dictionaries all the way down.” 2.2. Nodes and Edges 7 NetworkX Reference, Release 1.7 8 Chapter 2. Introduction CHAPTER THREE GRAPH TYPES NetworkX provides data structures and methods for storing graphs. All NetworkX graph classes allow (hashable) Python objects as nodes. and any Python object can be assigned as an edge attribute. The choice of graph class depends on the structure of the graph you want to represent. 3.1 Which graph class should I use? Graph Type NetworkX Class Undirected Simple Graph Directed Simple DiGraph With Self-loops Graph, DiGraph With Parallel edges MultiGraph, MultiDiGraph 3.2 Basic graph types 3.2.1 Graph – Undirected graphs with self loops Overview Graph(data=None, **attr) Base class for undirected graphs. A Graph stores nodes and edges with optional data, or attributes. Graphs hold undirected edges. Self loops are allowed but multiple (parallel) edges are not. Nodes can be arbitrary (hashable) Python objects with optional key/value attributes. Edges are represented as links between nodes with optional key/value attributes. Parameters data : input graph Data to initialize graph. If data=None (default) an empty graph is created. The data can be an edge list, or any NetworkX graph object. If the corresponding optional Python packages are installed the data can also be a NumPy matrix or 2d ndarray, a SciPy sparse matrix, or a PyGraphviz graph. attr : keyword arguments, optional (default= no attributes) 9 NetworkX Reference, Release 1.7 Attributes to add to graph as key=value pairs. See Also: DiGraph, MultiGraph, MultiDiGraph Examples Create an empty graph structure (a “null graph”) with no nodes and no edges. >>> G = nx.Graph() G can be grown in several ways. Nodes: Add one node at a time: >>> G.add_node(1) Add the nodes from any container (a list, dict, set or even the lines from a ﬁle or the nodes from another graph). >>> G.add_nodes_from([2,3]) >>> G.add_nodes_from(range(100,110)) >>> H=nx.Graph() >>> H.add_path([0,1,2,3,4,5,6,7,8,9]) >>> G.add_nodes_from(H) In addition to strings and integers any hashable Python object (except None) can represent a node, e.g. a customized node object, or even another Graph. >>> G.add_node(H) Edges: G can also be grown by adding edges. Add one edge, >>> G.add_edge(1, 2) a list of edges, >>> G.add_edges_from([(1,2),(1,3)]) or a collection of edges, >>> G.add_edges_from(H.edges()) If some edges connect nodes not yet in the graph, the nodes are added automatically. There are no errors when adding nodes or edges that already exist. Attributes: Each graph, node, and edge can hold key/value attribute pairs in an associated attribute dictionary (the keys must be hashable). By default these are empty, but can be added or changed using add_edge, add_node or direct manipulation of the attribute dictionaries named graph, node and edge respectively. >>> G = nx.Graph(day="Friday") >>> G.graph {’day’: ’Friday’} 10 Chapter 3. Graph types NetworkX Reference, Release 1.7 Add node attributes using add_node(), add_nodes_from() or G.node >>> G.add_node(1, time=’5pm’) >>> G.add_nodes_from([3], time=’2pm’) >>> G.node[1] {’time’: ’5pm’} >>> G.node[1][’room’] = 714 >>> del G.node[1][’room’] # remove attribute >>> G.nodes(data=True) [(1, {’time’: ’5pm’}), (3, {’time’: ’2pm’})] Warning: adding a node to G.node does not add it to the graph. Add edge attributes using add_edge(), add_edges_from(), subscript notation, or G.edge. >>> G.add_edge(1, 2, weight=4.7 ) >>> G.add_edges_from([(3,4),(4,5)], color=’red’) >>> G.add_edges_from([(1,2,{’color’:’blue’}), (2,3,{’weight’:8})]) >>> G[1][2][’weight’] = 4.7 >>> G.edge[1][2][’weight’] = 4 Shortcuts: Many common graph features allow python syntax to speed reporting. >>> 1 in G # check if node in graph True >>> [n for n in G if n<3] # iterate through nodes [1, 2] >>> len(G) # number of nodes in graph 5 >>> G[1] # adjacency dict keyed by neighbor to edge attributes ... # Note: you should not change this dict manually! {2: {’color’: ’blue’, ’weight’: 4}} The fastest way to traverse all edges of a graph is via adjacency_iter(), but the edges() method is often more convenient. >>> for n,nbrsdict in G.adjacency_iter(): ... for nbr,eattr in nbrsdict.items(): ... if ’weight’ in eattr: ... (n,nbr,eattr[’weight’]) (1, 2, 4) (2, 1, 4) (2, 3, 8) (3, 2, 8) >>> [ (u,v,edata[’weight’]) for u,v,edata in G.edges(data=True) if ’weight’ in edata ] [(1, 2, 4), (2, 3, 8)] Reporting: Simple graph information is obtained using methods. Iterator versions of many reporting methods exist for efﬁciency. Methods exist for reporting nodes(), edges(), neighbors() and degree() as well as the number of nodes and edges. For details on these and other miscellaneous methods, see below. Adding and removing nodes and edges 3.2. Basic graph types 11 NetworkX Reference, Release 1.7 Graph.__init__([data]) Initialize a graph with edges, name, graph attributes. Graph.add_node(n[, attr_dict]) Add a single node n and update node attributes. Graph.add_nodes_from(nodes, **attr) Add multiple nodes. Graph.remove_node(n) Remove node n. Graph.remove_nodes_from(nodes) Remove multiple nodes. Graph.add_edge(u, v[, attr_dict]) Add an edge between u and v. Graph.add_edges_from(ebunch[, attr_dict]) Add all the edges in ebunch. Graph.add_weighted_edges_from(ebunch[, weight]) Add all the edges in ebunch as weighted edges with speciﬁed weights. Graph.remove_edge(u, v) Remove the edge between u and v. Graph.remove_edges_from(ebunch) Remove all edges speciﬁed in ebunch. Graph.add_star(nodes, **attr) Add a star. Graph.add_path(nodes, **attr) Add a path. Graph.add_cycle(nodes, **attr) Add a cycle. Graph.clear() Remove all nodes and edges from the graph. __init__ Graph.__init__(data=None, **attr) Initialize a graph with edges, name, graph attributes. Parameters data : input graph Data to initialize graph. If data=None (default) an empty graph is created. The data can be an edge list, or any NetworkX graph object. If the corresponding optional Python packages are installed the data can also be a NumPy matrix or 2d ndarray, a SciPy sparse matrix, or a PyGraphviz graph. name : string, optional (default=’‘) An optional name for the graph. attr : keyword arguments, optional (default= no attributes) Attributes to add to graph as key=value pairs. See Also: convert Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G = nx.Graph(name=’my graph’) >>> e = [(1,2),(2,3),(3,4)] # list of edges >>> G = nx.Graph(e) Arbitrary graph attribute pairs (key=value) may be assigned >>> G=nx.Graph(e, day="Friday") >>> G.graph {’day’: ’Friday’} 12 Chapter 3. Graph types NetworkX Reference, Release 1.7 add_node Graph.add_node(n, attr_dict=None, **attr) Add a single node n and update node attributes. Parameters n : node A node can be any hashable Python object except None. attr_dict : dictionary, optional (default= no attributes) Dictionary of node attributes. Key/value pairs will update existing data associated with the node. attr : keyword arguments, optional Set or change attributes using key=value. See Also: add_nodes_from Notes A hashable object is one that can be used as a key in a Python dictionary. This includes strings, numbers, tuples of strings and numbers, etc. On many platforms hashable items also include mutables such as NetworkX Graphs, though one should be careful that the hash doesn’t change on mutables. Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_node(1) >>> G.add_node(’Hello’) >>> K3 = nx.Graph([(0,1),(1,2),(2,0)]) >>> G.add_node(K3) >>> G.number_of_nodes() 3 Use keywords set/change node attributes: >>> G.add_node(1,size=10) >>> G.add_node(3,weight=0.4,UTM=(’13S’,382871,3972649)) add_nodes_from Graph.add_nodes_from(nodes, **attr) Add multiple nodes. Parameters nodes : iterable container A container of nodes (list, dict, set, etc.). OR A container of (node, attribute dict) tuples. Node attributes are updated using the attribute dict. attr : keyword arguments, optional (default= no attributes) 3.2. Basic graph types 13 NetworkX Reference, Release 1.7 Update attributes for all nodes in nodes. Node attributes speciﬁed in nodes as a tuple take precedence over attributes speciﬁed generally. See Also: add_node Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_nodes_from(’Hello’) >>> K3 = nx.Graph([(0,1),(1,2),(2,0)]) >>> G.add_nodes_from(K3) >>> sorted(G.nodes(),key=str) [0, 1, 2, ’H’, ’e’, ’l’, ’o’] Use keywords to update speciﬁc node attributes for every node. >>> G.add_nodes_from([1,2], size=10) >>> G.add_nodes_from([3,4], weight=0.4) Use (node, attrdict) tuples to update attributes for speciﬁc nodes. >>> G.add_nodes_from([(1,dict(size=11)), (2,{’color’:’blue’})]) >>> G.node[1][’size’] 11 >>> H = nx.Graph() >>> H.add_nodes_from(G.nodes(data=True)) >>> H.node[1][’size’] 11 remove_node Graph.remove_node(n) Remove node n. Removes the node n and all adjacent edges. Attempting to remove a non-existent node will raise an exception. Parameters n : node A node in the graph Raises NetworkXError : If n is not in the graph. See Also: remove_nodes_from Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2]) >>> G.edges() [(0, 1), (1, 2)] >>> G.remove_node(1) 14 Chapter 3. Graph types NetworkX Reference, Release 1.7 >>> G.edges() [] remove_nodes_from Graph.remove_nodes_from(nodes) Remove multiple nodes. Parameters nodes : iterable container A container of nodes (list, dict, set, etc.). If a node in the container is not in the graph it is silently ignored. See Also: remove_node Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2]) >>> e = G.nodes() >>> e [0, 1, 2] >>> G.remove_nodes_from(e) >>> G.nodes() [] add_edge Graph.add_edge(u, v, attr_dict=None, **attr) Add an edge between u and v. The nodes u and v will be automatically added if they are not already in the graph. Edge attributes can be speciﬁed with keywords or by providing a dictionary with key/value pairs. See examples below. Parameters u,v : nodes Nodes can be, for example, strings or numbers. Nodes must be hashable (and not None) Python objects. attr_dict : dictionary, optional (default= no attributes) Dictionary of edge attributes. Key/value pairs will update existing data associated with the edge. attr : keyword arguments, optional Edge data (or labels or objects) can be assigned using keyword arguments. See Also: add_edges_from add a collection of edges 3.2. Basic graph types 15 NetworkX Reference, Release 1.7 Notes Adding an edge that already exists updates the edge data. Many NetworkX algorithms designed for weighted graphs use as the edge weight a numerical value assigned to a keyword which by default is ‘weight’. Examples The following all add the edge e=(1,2) to graph G: >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> e = (1,2) >>> G.add_edge(1, 2) # explicit two-node form >>> G.add_edge(*e) # single edge as tuple of two nodes >>> G.add_edges_from( [(1,2)] ) # add edges from iterable container Associate data to edges using keywords: >>> G.add_edge(1, 2, weight=3) >>> G.add_edge(1, 3, weight=7, capacity=15, length=342.7) add_edges_from Graph.add_edges_from(ebunch, attr_dict=None, **attr) Add all the edges in ebunch. Parameters ebunch : container of edges Each edge given in the container will be added to the graph. The edges must be given as as 2-tuples (u,v) or 3-tuples (u,v,d) where d is a dictionary containing edge data. attr_dict : dictionary, optional (default= no attributes) Dictionary of edge attributes. Key/value pairs will update existing data associated with each edge. attr : keyword arguments, optional Edge data (or labels or objects) can be assigned using keyword arguments. See Also: add_edge add a single edge add_weighted_edges_from convenient way to add weighted edges Notes Adding the same edge twice has no effect but any edge data will be updated when each duplicate edge is added. Examples 16 Chapter 3. Graph types NetworkX Reference, Release 1.7 >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_edges_from([(0,1),(1,2)]) # using a list of edge tuples >>> e = zip(range(0,3),range(1,4)) >>> G.add_edges_from(e) # Add the path graph 0-1-2-3 Associate data to edges >>> G.add_edges_from([(1,2),(2,3)], weight=3) >>> G.add_edges_from([(3,4),(1,4)], label=’WN2898’) add_weighted_edges_from Graph.add_weighted_edges_from(ebunch, weight=’weight’, **attr) Add all the edges in ebunch as weighted edges with speciﬁed weights. Parameters ebunch : container of edges Each edge given in the list or container will be added to the graph. The edges must be given as 3-tuples (u,v,w) where w is a number. weight : string, optional (default= ‘weight’) The attribute name for the edge weights to be added. attr : keyword arguments, optional (default= no attributes) Edge attributes to add/update for all edges. See Also: add_edge add a single edge add_edges_from add multiple edges Notes Adding the same edge twice for Graph/DiGraph simply updates the edge data. For MultiGraph/MultiDiGraph, duplicate edges are stored. Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_weighted_edges_from([(0,1,3.0),(1,2,7.5)]) remove_edge Graph.remove_edge(u, v) Remove the edge between u and v. Parameters u,v: nodes : Remove the edge between nodes u and v. Raises NetworkXError : If there is not an edge between u and v. 3.2. Basic graph types 17 NetworkX Reference, Release 1.7 See Also: remove_edges_from remove a collection of edges Examples >>> G = nx.Graph() # or DiGraph, etc >>> G.add_path([0,1,2,3]) >>> G.remove_edge(0,1) >>> e = (1,2) >>> G.remove_edge(*e) # unpacks e from an edge tuple >>> e = (2,3,{’weight’:7}) # an edge with attribute data >>> G.remove_edge(*e[:2]) # select first part of edge tuple remove_edges_from Graph.remove_edges_from(ebunch) Remove all edges speciﬁed in ebunch. Parameters ebunch: list or container of edge tuples : Each edge given in the list or container will be removed from the graph. The edges can be: • 2-tuples (u,v) edge between u and v. • 3-tuples (u,v,k) where k is ignored. See Also: remove_edge remove a single edge Notes Will fail silently if an edge in ebunch is not in the graph. Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> ebunch=[(1,2),(2,3)] >>> G.remove_edges_from(ebunch) add_star Graph.add_star(nodes, **attr) Add a star. The ﬁrst node in nodes is the middle of the star. It is connected to all other nodes. Parameters nodes : iterable container A container of nodes. 18 Chapter 3. Graph types NetworkX Reference, Release 1.7 attr : keyword arguments, optional (default= no attributes) Attributes to add to every edge in star. See Also: add_path, add_cycle Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_star([0,1,2,3]) >>> G.add_star([10,11,12],weight=2) add_path Graph.add_path(nodes, **attr) Add a path. Parameters nodes : iterable container A container of nodes. A path will be constructed from the nodes (in order) and added to the graph. attr : keyword arguments, optional (default= no attributes) Attributes to add to every edge in path. See Also: add_star, add_cycle Examples >>> G=nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G.add_path([10,11,12],weight=7) add_cycle Graph.add_cycle(nodes, **attr) Add a cycle. Parameters nodes: iterable container : A container of nodes. A cycle will be constructed from the nodes (in order) and added to the graph. attr : keyword arguments, optional (default= no attributes) Attributes to add to every edge in cycle. See Also: add_path, add_star 3.2. Basic graph types 19 NetworkX Reference, Release 1.7 Examples >>> G=nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_cycle([0,1,2,3]) >>> G.add_cycle([10,11,12],weight=7) clear Graph.clear() Remove all nodes and edges from the graph. This also removes the name, and all graph, node, and edge attributes. Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G.clear() >>> G.nodes() [] >>> G.edges() [] Iterating over nodes and edges Graph.nodes([data]) Return a list of the nodes in the graph. Graph.nodes_iter([data]) Return an iterator over the nodes. Graph.__iter__() Iterate over the nodes. Graph.edges([nbunch, data]) Return a list of edges. Graph.edges_iter([nbunch, data]) Return an iterator over the edges. Graph.get_edge_data(u, v[, default]) Return the attribute dictionary associated with edge (u,v). Graph.neighbors(n) Return a list of the nodes connected to the node n. Graph.neighbors_iter(n) Return an iterator over all neighbors of node n. Graph.__getitem__(n) Return a dict of neighbors of node n. Graph.adjacency_list() Return an adjacency list representation of the graph. Graph.adjacency_iter() Return an iterator of (node, adjacency dict) tuples for all nodes. Graph.nbunch_iter([nbunch]) Return an iterator of nodes contained in nbunch that are also in the graph. nodes Graph.nodes(data=False) Return a list of the nodes in the graph. Parameters data : boolean, optional (default=False) If False return a list of nodes. If True return a two-tuple of node and node data dictionary Returns nlist : list A list of nodes. If data=True a list of two-tuples containing (node, node data dictionary). 20 Chapter 3. Graph types NetworkX Reference, Release 1.7 Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2]) >>> G.nodes() [0, 1, 2] >>> G.add_node(1, time=’5pm’) >>> G.nodes(data=True) [(0, {}), (1, {’time’: ’5pm’}), (2, {})] nodes_iter Graph.nodes_iter(data=False) Return an iterator over the nodes. Parameters data : boolean, optional (default=False) If False the iterator returns nodes. If True return a two-tuple of node and node data dictionary Returns niter : iterator An iterator over nodes. If data=True the iterator gives two-tuples containing (node, node data, dictionary) Notes If the node data is not required it is simpler and equivalent to use the expression ‘for n in G’. >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2]) Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2]) >>> [d for n,d in G.nodes_iter(data=True)] [{}, {}, {}] __iter__ Graph.__iter__() Iterate over the nodes. Use the expression ‘for n in G’. Returns niter : iterator An iterator over all nodes in the graph. 3.2. Basic graph types 21 NetworkX Reference, Release 1.7 Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) edges Graph.edges(nbunch=None, data=False) Return a list of edges. Edges are returned as tuples with optional data in the order (node, neighbor, data). Parameters nbunch : iterable container, optional (default= all nodes) A container of nodes. The container will be iterated through once. data : bool, optional (default=False) Return two tuples (u,v) (False) or three-tuples (u,v,data) (True). Returns edge_list: list of edge tuples : Edges that are adjacent to any node in nbunch, or a list of all edges if nbunch is not speciﬁed. See Also: edges_iter return an iterator over the edges Notes Nodes in nbunch that are not in the graph will be (quietly) ignored. For directed graphs this returns the out-edges. Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G.edges() [(0, 1), (1, 2), (2, 3)] >>> G.edges(data=True) # default edge data is {} (empty dictionary) [(0, 1, {}), (1, 2, {}), (2, 3, {})] >>> G.edges([0,3]) [(0, 1), (3, 2)] >>> G.edges(0) [(0, 1)] edges_iter Graph.edges_iter(nbunch=None, data=False) Return an iterator over the edges. Edges are returned as tuples with optional data in the order (node, neighbor, data). Parameters nbunch : iterable container, optional (default= all nodes) A container of nodes. The container will be iterated through once. 22 Chapter 3. Graph types NetworkX Reference, Release 1.7 data : bool, optional (default=False) If True, return edge attribute dict in 3-tuple (u,v,data). Returns edge_iter : iterator An iterator of (u,v) or (u,v,d) tuples of edges. See Also: edges return a list of edges Notes Nodes in nbunch that are not in the graph will be (quietly) ignored. For directed graphs this returns the out-edges. Examples >>> G = nx.Graph() # or MultiGraph, etc >>> G.add_path([0,1,2,3]) >>> [e for e in G.edges_iter()] [(0, 1), (1, 2), (2, 3)] >>> list(G.edges_iter(data=True)) # default data is {} (empty dict) [(0, 1, {}), (1, 2, {}), (2, 3, {})] >>> list(G.edges_iter([0,3])) [(0, 1), (3, 2)] >>> list(G.edges_iter(0)) [(0, 1)] get_edge_data Graph.get_edge_data(u, v, default=None) Return the attribute dictionary associated with edge (u,v). Parameters u,v : nodes default: any Python object (default=None) : Value to return if the edge (u,v) is not found. Returns edge_dict : dictionary The edge attribute dictionary. Notes It is faster to use G[u][v]. >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G[0][1] {} Warning: Assigning G[u][v] corrupts the graph data structure. But it is safe to assign attributes to that dictionary, 3.2. Basic graph types 23 NetworkX Reference, Release 1.7 >>> G[0][1][’weight’] = 7 >>> G[0][1][’weight’] 7 >>> G[1][0][’weight’] 7 Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G.get_edge_data(0,1) # default edge data is {} {} >>> e = (0,1) >>> G.get_edge_data(*e) # tuple form {} >>> G.get_edge_data(’a’,’b’,default=0) # edge not in graph, return 0 0 neighbors Graph.neighbors(n) Return a list of the nodes connected to the node n. Parameters n : node A node in the graph Returns nlist : list A list of nodes that are adjacent to n. Raises NetworkXError : If the node n is not in the graph. Notes It is usually more convenient (and faster) to access the adjacency dictionary as G[n]: >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_edge(’a’,’b’,weight=7) >>> G[’a’] {’b’: {’weight’: 7}} Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G.neighbors(0) [1] 24 Chapter 3. Graph types NetworkX Reference, Release 1.7 neighbors_iter Graph.neighbors_iter(n) Return an iterator over all neighbors of node n. Notes It is faster to use the idiom “in G[0]”, e.g. >>> G = nx.path_graph(4) >>> [n for n in G[0]] [1] Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> [n for n in G.neighbors_iter(0)] [1] __getitem__ Graph.__getitem__(n) Return a dict of neighbors of node n. Use the expression ‘G[n]’. Parameters n : node A node in the graph. Returns adj_dict : dictionary The adjacency dictionary for nodes connected to n. Notes G[n] is similar to G.neighbors(n) but the internal data dictionary is returned instead of a list. Assigning G[n] will corrupt the internal graph data structure. Use G[n] for reading data only. Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G[0] {1: {}} adjacency_list Graph.adjacency_list() Return an adjacency list representation of the graph. 3.2. Basic graph types 25 NetworkX Reference, Release 1.7 The output adjacency list is in the order of G.nodes(). For directed graphs, only outgoing adjacencies are included. Returns adj_list : lists of lists The adjacency structure of the graph as a list of lists. See Also: adjacency_iter Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G.adjacency_list() # in order given by G.nodes() [[1], [0, 2], [1, 3], [2]] adjacency_iter Graph.adjacency_iter() Return an iterator of (node, adjacency dict) tuples for all nodes. This is the fastest way to look at every edge. For directed graphs, only outgoing adjacencies are included. Returns adj_iter : iterator An iterator of (node, adjacency dictionary) for all nodes in the graph. See Also: adjacency_list Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> [(n,nbrdict) for n,nbrdict in G.adjacency_iter()] [(0, {1: {}}), (1, {0: {}, 2: {}}), (2, {1: {}, 3: {}}), (3, {2: {}})] nbunch_iter Graph.nbunch_iter(nbunch=None) Return an iterator of nodes contained in nbunch that are also in the graph. The nodes in nbunch are checked for membership in the graph and if not are silently ignored. Parameters nbunch : iterable container, optional (default=all nodes) A container of nodes. The container will be iterated through once. Returns niter : iterator An iterator over nodes in nbunch that are also in the graph. If nbunch is None, iterate over all nodes in the graph. Raises NetworkXError : 26 Chapter 3. Graph types NetworkX Reference, Release 1.7 If nbunch is not a node or or sequence of nodes. If a node in nbunch is not hashable. See Also: Graph.__iter__ Notes When nbunch is an iterator, the returned iterator yields values directly from nbunch, becoming exhausted when nbunch is exhausted. To test whether nbunch is a single node, one can use “if nbunch in self:”, even after processing with this routine. If nbunch is not a node or a (possibly empty) sequence/iterator or None, a NetworkXError is raised. Also, if any object in nbunch is not hashable, a NetworkXError is raised. Information about graph structure Graph.has_node(n) Return True if the graph contains the node n. Graph.__contains__(n) Return True if n is a node, False otherwise. Use the expression Graph.has_edge(u, v) Return True if the edge (u,v) is in the graph. Graph.order() Return the number of nodes in the graph. Graph.number_of_nodes() Return the number of nodes in the graph. Graph.__len__() Return the number of nodes. Graph.degree([nbunch, weight]) Return the degree of a node or nodes. Graph.degree_iter([nbunch, weight]) Return an iterator for (node, degree). Graph.size([weight]) Return the number of edges. Graph.number_of_edges([u, v]) Return the number of edges between two nodes. Graph.nodes_with_selfloops() Return a list of nodes with self loops. Graph.selfloop_edges([data]) Return a list of selﬂoop edges. Graph.number_of_selfloops() Return the number of selﬂoop edges. has_node Graph.has_node(n) Return True if the graph contains the node n. Parameters n : node Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2]) >>> G.has_node(0) True It is more readable and simpler to use >>> 0 in G True 3.2. Basic graph types 27 NetworkX Reference, Release 1.7 __contains__ Graph.__contains__(n) Return True if n is a node, False otherwise. Use the expression ‘n in G’. Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> 1 in G True has_edge Graph.has_edge(u, v) Return True if the edge (u,v) is in the graph. Parameters u,v : nodes Nodes can be, for example, strings or numbers. Nodes must be hashable (and not None) Python objects. Returns edge_ind : bool True if edge is in the graph, False otherwise. Examples Can be called either using two nodes u,v or edge tuple (u,v) >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G.has_edge(0,1) # using two nodes True >>> e = (0,1) >>> G.has_edge(*e) # e is a 2-tuple (u,v) True >>> e = (0,1,{’weight’:7}) >>> G.has_edge(*e[:2]) # e is a 3-tuple (u,v,data_dictionary) True The following syntax are all equivalent: >>> G.has_edge(0,1) True >>> 1 in G[0] # though this gives KeyError if 0 not in G True order Graph.order() Return the number of nodes in the graph. Returns nnodes : int 28 Chapter 3. Graph types NetworkX Reference, Release 1.7 The number of nodes in the graph. See Also: number_of_nodes, __len__ number_of_nodes Graph.number_of_nodes() Return the number of nodes in the graph. Returns nnodes : int The number of nodes in the graph. See Also: order, __len__ Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2]) >>> len(G) 3 __len__ Graph.__len__() Return the number of nodes. Use the expression ‘len(G)’. Returns nnodes : int The number of nodes in the graph. Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> len(G) 4 degree Graph.degree(nbunch=None, weight=None) Return the degree of a node or nodes. The node degree is the number of edges adjacent to that node. Parameters nbunch : iterable container, optional (default=all nodes) A container of nodes. The container will be iterated through once. weight : string or None, optional (default=None) 3.2. Basic graph types 29 NetworkX Reference, Release 1.7 The edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1. The degree is the sum of the edge weights adjacent to the node. Returns nd : dictionary, or number A dictionary with nodes as keys and degree as values or a number if a single node is speciﬁed. Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G.degree(0) 1 >>> G.degree([0,1]) {0: 1, 1: 2} >>> list(G.degree([0,1]).values()) [1, 2] degree_iter Graph.degree_iter(nbunch=None, weight=None) Return an iterator for (node, degree). The node degree is the number of edges adjacent to the node. Parameters nbunch : iterable container, optional (default=all nodes) A container of nodes. The container will be iterated through once. weight : string or None, optional (default=None) The edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1. The degree is the sum of the edge weights adjacent to the node. Returns nd_iter : an iterator The iterator returns two-tuples of (node, degree). See Also: degree Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> list(G.degree_iter(0)) # node 0 with degree 1 [(0, 1)] >>> list(G.degree_iter([0,1])) [(0, 1), (1, 2)] size Graph.size(weight=None) Return the number of edges. 30 Chapter 3. Graph types NetworkX Reference, Release 1.7 Parameters weight : string or None, optional (default=None) The edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1. Returns nedges : int The number of edges of sum of edge weights in the graph. See Also: number_of_edges Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G.size() 3 >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_edge(’a’,’b’,weight=2) >>> G.add_edge(’b’,’c’,weight=4) >>> G.size() 2 >>> G.size(weight=’weight’) 6.0 number_of_edges Graph.number_of_edges(u=None, v=None) Return the number of edges between two nodes. Parameters u,v : nodes, optional (default=all edges) If u and v are speciﬁed, return the number of edges between u and v. Otherwise return the total number of all edges. Returns nedges : int The number of edges in the graph. If nodes u and v are speciﬁed return the number of edges between those nodes. See Also: size Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G.number_of_edges() 3 >>> G.number_of_edges(0,1) 1 >>> e = (0,1) >>> G.number_of_edges(*e) 1 3.2. Basic graph types 31 NetworkX Reference, Release 1.7 nodes_with_selﬂoops Graph.nodes_with_selfloops() Return a list of nodes with self loops. A node with a self loop has an edge with both ends adjacent to that node. Returns nodelist : list A list of nodes with self loops. See Also: selfloop_edges, number_of_selfloops Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_edge(1,1) >>> G.add_edge(1,2) >>> G.nodes_with_selfloops() [1] selﬂoop_edges Graph.selfloop_edges(data=False) Return a list of selﬂoop edges. A selﬂoop edge has the same node at both ends. Parameters data : bool, optional (default=False) Return selﬂoop edges as two tuples (u,v) (data=False) or three-tuples (u,v,data) (data=True) Returns edgelist : list of edge tuples A list of all selﬂoop edges. See Also: nodes_with_selfloops, number_of_selfloops Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_edge(1,1) >>> G.add_edge(1,2) >>> G.selfloop_edges() [(1, 1)] >>> G.selfloop_edges(data=True) [(1, 1, {})] 32 Chapter 3. Graph types NetworkX Reference, Release 1.7 number_of_selﬂoops Graph.number_of_selfloops() Return the number of selﬂoop edges. A selﬂoop edge has the same node at both ends. Returns nloops : int The number of selﬂoops. See Also: nodes_with_selfloops, selfloop_edges Examples >>> G=nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_edge(1,1) >>> G.add_edge(1,2) >>> G.number_of_selfloops() 1 Making copies and subgraphs Graph.copy() Return a copy of the graph. Graph.to_undirected() Return an undirected copy of the graph. Graph.to_directed() Return a directed representation of the graph. Graph.subgraph(nbunch) Return the subgraph induced on nodes in nbunch. copy Graph.copy() Return a copy of the graph. Returns G : Graph A copy of the graph. See Also: to_directed return a directed copy of the graph. Notes This makes a complete copy of the graph including all of the node or edge attributes. Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> H = G.copy() 3.2. Basic graph types 33 NetworkX Reference, Release 1.7 to_undirected Graph.to_undirected() Return an undirected copy of the graph. Returns G : Graph/MultiGraph A deepcopy of the graph. See Also: copy, add_edge, add_edges_from Notes This returns a “deepcopy” of the edge, node, and graph attributes which attempts to completely copy all of the data and references. This is in contrast to the similar G=DiGraph(D) which returns a shallow copy of the data. See the Python copy module for more information on shallow and deep copies, http://docs.python.org/library/copy.html. Examples >>> G = nx.Graph() # or MultiGraph, etc >>> G.add_path([0,1]) >>> H = G.to_directed() >>> H.edges() [(0, 1), (1, 0)] >>> G2 = H.to_undirected() >>> G2.edges() [(0, 1)] to_directed Graph.to_directed() Return a directed representation of the graph. Returns G : DiGraph A directed graph with the same name, same nodes, and with each edge (u,v,data) re- placed by two directed edges (u,v,data) and (v,u,data). Notes This returns a “deepcopy” of the edge, node, and graph attributes which attempts to completely copy all of the data and references. This is in contrast to the similar D=DiGraph(G) which returns a shallow copy of the data. See the Python copy module for more information on shallow and deep copies, http://docs.python.org/library/copy.html. 34 Chapter 3. Graph types NetworkX Reference, Release 1.7 Examples >>> G = nx.Graph() # or MultiGraph, etc >>> G.add_path([0,1]) >>> H = G.to_directed() >>> H.edges() [(0, 1), (1, 0)] If already directed, return a (deep) copy >>> G = nx.DiGraph() # or MultiDiGraph, etc >>> G.add_path([0,1]) >>> H = G.to_directed() >>> H.edges() [(0, 1)] subgraph Graph.subgraph(nbunch) Return the subgraph induced on nodes in nbunch. The induced subgraph of the graph contains the nodes in nbunch and the edges between those nodes. Parameters nbunch : list, iterable A container of nodes which will be iterated through once. Returns G : Graph A subgraph of the graph with the same edge attributes. Notes The graph, edge or node attributes just point to the original graph. So changes to the node or edge structure will not be reﬂected in the original graph while changes to the attributes will. To create a subgraph with its own copy of the edge/node attributes use: nx.Graph(G.subgraph(nbunch)) If edge attributes are containers, a deep copy can be obtained using: G.subgraph(nbunch).copy() For an inplace reduction of a graph to a subgraph you can remove nodes: G.remove_nodes_from([ n in G if n not in set(nbunch)]) Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> H = G.subgraph([0,1,2]) >>> H.edges() [(0, 1), (1, 2)] 3.2. Basic graph types 35 NetworkX Reference, Release 1.7 3.2.2 DiGraph - Directed graphs with self loops Overview DiGraph(data=None, **attr) Base class for directed graphs. A DiGraph stores nodes and edges with optional data, or attributes. DiGraphs hold directed edges. Self loops are allowed but multiple (parallel) edges are not. Nodes can be arbitrary (hashable) Python objects with optional key/value attributes. Edges are represented as links between nodes with optional key/value attributes. Parameters data : input graph Data to initialize graph. If data=None (default) an empty graph is created. The data can be an edge list, or any NetworkX graph object. If the corresponding optional Python packages are installed the data can also be a NumPy matrix or 2d ndarray, a SciPy sparse matrix, or a PyGraphviz graph. attr : keyword arguments, optional (default= no attributes) Attributes to add to graph as key=value pairs. See Also: Graph, MultiGraph, MultiDiGraph Examples Create an empty graph structure (a “null graph”) with no nodes and no edges. >>> G = nx.DiGraph() G can be grown in several ways. Nodes: Add one node at a time: >>> G.add_node(1) Add the nodes from any container (a list, dict, set or even the lines from a ﬁle or the nodes from another graph). >>> G.add_nodes_from([2,3]) >>> G.add_nodes_from(range(100,110)) >>> H=nx.Graph() >>> H.add_path([0,1,2,3,4,5,6,7,8,9]) >>> G.add_nodes_from(H) In addition to strings and integers any hashable Python object (except None) can represent a node, e.g. a customized node object, or even another Graph. >>> G.add_node(H) Edges: G can also be grown by adding edges. Add one edge, 36 Chapter 3. Graph types NetworkX Reference, Release 1.7 >>> G.add_edge(1, 2) a list of edges, >>> G.add_edges_from([(1,2),(1,3)]) or a collection of edges, >>> G.add_edges_from(H.edges()) If some edges connect nodes not yet in the graph, the nodes are added automatically. There are no errors when adding nodes or edges that already exist. Attributes: Each graph, node, and edge can hold key/value attribute pairs in an associated attribute dictionary (the keys must be hashable). By default these are empty, but can be added or changed using add_edge, add_node or direct manipulation of the attribute dictionaries named graph, node and edge respectively. >>> G = nx.DiGraph(day="Friday") >>> G.graph {’day’: ’Friday’} Add node attributes using add_node(), add_nodes_from() or G.node >>> G.add_node(1, time=’5pm’) >>> G.add_nodes_from([3], time=’2pm’) >>> G.node[1] {’time’: ’5pm’} >>> G.node[1][’room’] = 714 >>> del G.node[1][’room’] # remove attribute >>> G.nodes(data=True) [(1, {’time’: ’5pm’}), (3, {’time’: ’2pm’})] Warning: adding a node to G.node does not add it to the graph. Add edge attributes using add_edge(), add_edges_from(), subscript notation, or G.edge. >>> G.add_edge(1, 2, weight=4.7 ) >>> G.add_edges_from([(3,4),(4,5)], color=’red’) >>> G.add_edges_from([(1,2,{’color’:’blue’}), (2,3,{’weight’:8})]) >>> G[1][2][’weight’] = 4.7 >>> G.edge[1][2][’weight’] = 4 Shortcuts: Many common graph features allow python syntax to speed reporting. >>> 1 in G # check if node in graph True >>> [n for n in G if n<3] # iterate through nodes [1, 2] >>> len(G) # number of nodes in graph 5 >>> G[1] # adjacency dict keyed by neighbor to edge attributes ... # Note: you should not change this dict manually! {2: {’color’: ’blue’, ’weight’: 4}} The fastest way to traverse all edges of a graph is via adjacency_iter(), but the edges() method is often more convenient. 3.2. Basic graph types 37 NetworkX Reference, Release 1.7 >>> for n,nbrsdict in G.adjacency_iter(): ... for nbr,eattr in nbrsdict.items(): ... if ’weight’ in eattr: ... (n,nbr,eattr[’weight’]) (1, 2, 4) (2, 3, 8) >>> [ (u,v,edata[’weight’]) for u,v,edata in G.edges(data=True) if ’weight’ in edata ] [(1, 2, 4), (2, 3, 8)] Reporting: Simple graph information is obtained using methods. Iterator versions of many reporting methods exist for efﬁciency. Methods exist for reporting nodes(), edges(), neighbors() and degree() as well as the number of nodes and edges. For details on these and other miscellaneous methods, see below. Adding and removing nodes and edges DiGraph.__init__([data]) Initialize a graph with edges, name, graph attributes. DiGraph.add_node(n[, attr_dict]) Add a single node n and update node attributes. DiGraph.add_nodes_from(nodes, **attr) Add multiple nodes. DiGraph.remove_node(n) Remove node n. DiGraph.remove_nodes_from(nbunch) Remove multiple nodes. DiGraph.add_edge(u, v[, attr_dict]) Add an edge between u and v. DiGraph.add_edges_from(ebunch[, attr_dict]) Add all the edges in ebunch. DiGraph.add_weighted_edges_from(ebunch[, weight]) Add all the edges in ebunch as weighted edges with speciﬁed weights. DiGraph.remove_edge(u, v) Remove the edge between u and v. DiGraph.remove_edges_from(ebunch) Remove all edges speciﬁed in ebunch. DiGraph.add_star(nodes, **attr) Add a star. DiGraph.add_path(nodes, **attr) Add a path. DiGraph.add_cycle(nodes, **attr) Add a cycle. DiGraph.clear() Remove all nodes and edges from the graph. __init__ DiGraph.__init__(data=None, **attr) Initialize a graph with edges, name, graph attributes. Parameters data : input graph Data to initialize graph. If data=None (default) an empty graph is created. The data can be an edge list, or any NetworkX graph object. If the corresponding optional Python packages are installed the data can also be a NumPy matrix or 2d ndarray, a SciPy sparse matrix, or a PyGraphviz graph. name : string, optional (default=’‘) An optional name for the graph. attr : keyword arguments, optional (default= no attributes) Attributes to add to graph as key=value pairs. See Also: convert 38 Chapter 3. Graph types NetworkX Reference, Release 1.7 Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G = nx.Graph(name=’my graph’) >>> e = [(1,2),(2,3),(3,4)] # list of edges >>> G = nx.Graph(e) Arbitrary graph attribute pairs (key=value) may be assigned >>> G=nx.Graph(e, day="Friday") >>> G.graph {’day’: ’Friday’} add_node DiGraph.add_node(n, attr_dict=None, **attr) Add a single node n and update node attributes. Parameters n : node A node can be any hashable Python object except None. attr_dict : dictionary, optional (default= no attributes) Dictionary of node attributes. Key/value pairs will update existing data associated with the node. attr : keyword arguments, optional Set or change attributes using key=value. See Also: add_nodes_from Notes A hashable object is one that can be used as a key in a Python dictionary. This includes strings, numbers, tuples of strings and numbers, etc. On many platforms hashable items also include mutables such as NetworkX Graphs, though one should be careful that the hash doesn’t change on mutables. Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_node(1) >>> G.add_node(’Hello’) >>> K3 = nx.Graph([(0,1),(1,2),(2,0)]) >>> G.add_node(K3) >>> G.number_of_nodes() 3 Use keywords set/change node attributes: >>> G.add_node(1,size=10) >>> G.add_node(3,weight=0.4,UTM=(’13S’,382871,3972649)) 3.2. Basic graph types 39 NetworkX Reference, Release 1.7 add_nodes_from DiGraph.add_nodes_from(nodes, **attr) Add multiple nodes. Parameters nodes : iterable container A container of nodes (list, dict, set, etc.). OR A container of (node, attribute dict) tuples. Node attributes are updated using the attribute dict. attr : keyword arguments, optional (default= no attributes) Update attributes for all nodes in nodes. Node attributes speciﬁed in nodes as a tuple take precedence over attributes speciﬁed generally. See Also: add_node Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_nodes_from(’Hello’) >>> K3 = nx.Graph([(0,1),(1,2),(2,0)]) >>> G.add_nodes_from(K3) >>> sorted(G.nodes(),key=str) [0, 1, 2, ’H’, ’e’, ’l’, ’o’] Use keywords to update speciﬁc node attributes for every node. >>> G.add_nodes_from([1,2], size=10) >>> G.add_nodes_from([3,4], weight=0.4) Use (node, attrdict) tuples to update attributes for speciﬁc nodes. >>> G.add_nodes_from([(1,dict(size=11)), (2,{’color’:’blue’})]) >>> G.node[1][’size’] 11 >>> H = nx.Graph() >>> H.add_nodes_from(G.nodes(data=True)) >>> H.node[1][’size’] 11 remove_node DiGraph.remove_node(n) Remove node n. Removes the node n and all adjacent edges. Attempting to remove a non-existent node will raise an exception. Parameters n : node A node in the graph Raises NetworkXError : If n is not in the graph. See Also: remove_nodes_from 40 Chapter 3. Graph types NetworkX Reference, Release 1.7 Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2]) >>> G.edges() [(0, 1), (1, 2)] >>> G.remove_node(1) >>> G.edges() [] remove_nodes_from DiGraph.remove_nodes_from(nbunch) Remove multiple nodes. Parameters nodes : iterable container A container of nodes (list, dict, set, etc.). If a node in the container is not in the graph it is silently ignored. See Also: remove_node Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2]) >>> e = G.nodes() >>> e [0, 1, 2] >>> G.remove_nodes_from(e) >>> G.nodes() [] add_edge DiGraph.add_edge(u, v, attr_dict=None, **attr) Add an edge between u and v. The nodes u and v will be automatically added if they are not already in the graph. Edge attributes can be speciﬁed with keywords or by providing a dictionary with key/value pairs. See examples below. Parameters u,v : nodes Nodes can be, for example, strings or numbers. Nodes must be hashable (and not None) Python objects. attr_dict : dictionary, optional (default= no attributes) Dictionary of edge attributes. Key/value pairs will update existing data associated with the edge. attr : keyword arguments, optional 3.2. Basic graph types 41 NetworkX Reference, Release 1.7 Edge data (or labels or objects) can be assigned using keyword arguments. See Also: add_edges_from add a collection of edges Notes Adding an edge that already exists updates the edge data. Many NetworkX algorithms designed for weighted graphs use as the edge weight a numerical value assigned to a keyword which by default is ‘weight’. Examples The following all add the edge e=(1,2) to graph G: >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> e = (1,2) >>> G.add_edge(1, 2) # explicit two-node form >>> G.add_edge(*e) # single edge as tuple of two nodes >>> G.add_edges_from( [(1,2)] ) # add edges from iterable container Associate data to edges using keywords: >>> G.add_edge(1, 2, weight=3) >>> G.add_edge(1, 3, weight=7, capacity=15, length=342.7) add_edges_from DiGraph.add_edges_from(ebunch, attr_dict=None, **attr) Add all the edges in ebunch. Parameters ebunch : container of edges Each edge given in the container will be added to the graph. The edges must be given as as 2-tuples (u,v) or 3-tuples (u,v,d) where d is a dictionary containing edge data. attr_dict : dictionary, optional (default= no attributes) Dictionary of edge attributes. Key/value pairs will update existing data associated with each edge. attr : keyword arguments, optional Edge data (or labels or objects) can be assigned using keyword arguments. See Also: add_edge add a single edge add_weighted_edges_from convenient way to add weighted edges Notes Adding the same edge twice has no effect but any edge data will be updated when each duplicate edge is added. 42 Chapter 3. Graph types NetworkX Reference, Release 1.7 Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_edges_from([(0,1),(1,2)]) # using a list of edge tuples >>> e = zip(range(0,3),range(1,4)) >>> G.add_edges_from(e) # Add the path graph 0-1-2-3 Associate data to edges >>> G.add_edges_from([(1,2),(2,3)], weight=3) >>> G.add_edges_from([(3,4),(1,4)], label=’WN2898’) add_weighted_edges_from DiGraph.add_weighted_edges_from(ebunch, weight=’weight’, **attr) Add all the edges in ebunch as weighted edges with speciﬁed weights. Parameters ebunch : container of edges Each edge given in the list or container will be added to the graph. The edges must be given as 3-tuples (u,v,w) where w is a number. weight : string, optional (default= ‘weight’) The attribute name for the edge weights to be added. attr : keyword arguments, optional (default= no attributes) Edge attributes to add/update for all edges. See Also: add_edge add a single edge add_edges_from add multiple edges Notes Adding the same edge twice for Graph/DiGraph simply updates the edge data. For MultiGraph/MultiDiGraph, duplicate edges are stored. Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_weighted_edges_from([(0,1,3.0),(1,2,7.5)]) remove_edge DiGraph.remove_edge(u, v) Remove the edge between u and v. Parameters u,v: nodes : Remove the edge between nodes u and v. Raises NetworkXError : 3.2. Basic graph types 43 NetworkX Reference, Release 1.7 If there is not an edge between u and v. See Also: remove_edges_from remove a collection of edges Examples >>> G = nx.Graph() # or DiGraph, etc >>> G.add_path([0,1,2,3]) >>> G.remove_edge(0,1) >>> e = (1,2) >>> G.remove_edge(*e) # unpacks e from an edge tuple >>> e = (2,3,{’weight’:7}) # an edge with attribute data >>> G.remove_edge(*e[:2]) # select first part of edge tuple remove_edges_from DiGraph.remove_edges_from(ebunch) Remove all edges speciﬁed in ebunch. Parameters ebunch: list or container of edge tuples : Each edge given in the list or container will be removed from the graph. The edges can be: • 2-tuples (u,v) edge between u and v. • 3-tuples (u,v,k) where k is ignored. See Also: remove_edge remove a single edge Notes Will fail silently if an edge in ebunch is not in the graph. Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> ebunch=[(1,2),(2,3)] >>> G.remove_edges_from(ebunch) add_star DiGraph.add_star(nodes, **attr) Add a star. The ﬁrst node in nodes is the middle of the star. It is connected to all other nodes. Parameters nodes : iterable container 44 Chapter 3. Graph types NetworkX Reference, Release 1.7 A container of nodes. attr : keyword arguments, optional (default= no attributes) Attributes to add to every edge in star. See Also: add_path, add_cycle Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_star([0,1,2,3]) >>> G.add_star([10,11,12],weight=2) add_path DiGraph.add_path(nodes, **attr) Add a path. Parameters nodes : iterable container A container of nodes. A path will be constructed from the nodes (in order) and added to the graph. attr : keyword arguments, optional (default= no attributes) Attributes to add to every edge in path. See Also: add_star, add_cycle Examples >>> G=nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G.add_path([10,11,12],weight=7) add_cycle DiGraph.add_cycle(nodes, **attr) Add a cycle. Parameters nodes: iterable container : A container of nodes. A cycle will be constructed from the nodes (in order) and added to the graph. attr : keyword arguments, optional (default= no attributes) Attributes to add to every edge in cycle. See Also: add_path, add_star 3.2. Basic graph types 45 NetworkX Reference, Release 1.7 Examples >>> G=nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_cycle([0,1,2,3]) >>> G.add_cycle([10,11,12],weight=7) clear DiGraph.clear() Remove all nodes and edges from the graph. This also removes the name, and all graph, node, and edge attributes. Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G.clear() >>> G.nodes() [] >>> G.edges() [] Iterating over nodes and edges DiGraph.nodes([data]) Return a list of the nodes in the graph. DiGraph.nodes_iter([data]) Return an iterator over the nodes. DiGraph.__iter__() Iterate over the nodes. DiGraph.edges([nbunch, data]) Return a list of edges. DiGraph.edges_iter([nbunch, data]) Return an iterator over the edges. DiGraph.out_edges([nbunch, data]) Return a list of edges. DiGraph.out_edges_iter([nbunch, data]) Return an iterator over the edges. DiGraph.in_edges([nbunch, data]) Return a list of the incoming edges. DiGraph.in_edges_iter([nbunch, data]) Return an iterator over the incoming edges. DiGraph.get_edge_data(u, v[, default]) Return the attribute dictionary associated with edge (u,v). DiGraph.neighbors(n) Return a list of successor nodes of n. DiGraph.neighbors_iter(n) Return an iterator over successor nodes of n. DiGraph.__getitem__(n) Return a dict of neighbors of node n. DiGraph.successors(n) Return a list of successor nodes of n. DiGraph.successors_iter(n) Return an iterator over successor nodes of n. DiGraph.predecessors(n) Return a list of predecessor nodes of n. DiGraph.predecessors_iter(n) Return an iterator over predecessor nodes of n. DiGraph.adjacency_list() Return an adjacency list representation of the graph. DiGraph.adjacency_iter() Return an iterator of (node, adjacency dict) tuples for all nodes. DiGraph.nbunch_iter([nbunch]) Return an iterator of nodes contained in nbunch that are also in the graph. 46 Chapter 3. Graph types NetworkX Reference, Release 1.7 nodes DiGraph.nodes(data=False) Return a list of the nodes in the graph. Parameters data : boolean, optional (default=False) If False return a list of nodes. If True return a two-tuple of node and node data dictionary Returns nlist : list A list of nodes. If data=True a list of two-tuples containing (node, node data dictionary). Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2]) >>> G.nodes() [0, 1, 2] >>> G.add_node(1, time=’5pm’) >>> G.nodes(data=True) [(0, {}), (1, {’time’: ’5pm’}), (2, {})] nodes_iter DiGraph.nodes_iter(data=False) Return an iterator over the nodes. Parameters data : boolean, optional (default=False) If False the iterator returns nodes. If True return a two-tuple of node and node data dictionary Returns niter : iterator An iterator over nodes. If data=True the iterator gives two-tuples containing (node, node data, dictionary) Notes If the node data is not required it is simpler and equivalent to use the expression ‘for n in G’. >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2]) Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2]) >>> [d for n,d in G.nodes_iter(data=True)] [{}, {}, {}] 3.2. Basic graph types 47 NetworkX Reference, Release 1.7 __iter__ DiGraph.__iter__() Iterate over the nodes. Use the expression ‘for n in G’. Returns niter : iterator An iterator over all nodes in the graph. Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) edges DiGraph.edges(nbunch=None, data=False) Return a list of edges. Edges are returned as tuples with optional data in the order (node, neighbor, data). Parameters nbunch : iterable container, optional (default= all nodes) A container of nodes. The container will be iterated through once. data : bool, optional (default=False) Return two tuples (u,v) (False) or three-tuples (u,v,data) (True). Returns edge_list: list of edge tuples : Edges that are adjacent to any node in nbunch, or a list of all edges if nbunch is not speciﬁed. See Also: edges_iter return an iterator over the edges Notes Nodes in nbunch that are not in the graph will be (quietly) ignored. For directed graphs this returns the out-edges. Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G.edges() [(0, 1), (1, 2), (2, 3)] >>> G.edges(data=True) # default edge data is {} (empty dictionary) [(0, 1, {}), (1, 2, {}), (2, 3, {})] >>> G.edges([0,3]) [(0, 1), (3, 2)] >>> G.edges(0) [(0, 1)] 48 Chapter 3. Graph types NetworkX Reference, Release 1.7 edges_iter DiGraph.edges_iter(nbunch=None, data=False) Return an iterator over the edges. Edges are returned as tuples with optional data in the order (node, neighbor, data). Parameters nbunch : iterable container, optional (default= all nodes) A container of nodes. The container will be iterated through once. data : bool, optional (default=False) If True, return edge attribute dict in 3-tuple (u,v,data). Returns edge_iter : iterator An iterator of (u,v) or (u,v,d) tuples of edges. See Also: edges return a list of edges Notes Nodes in nbunch that are not in the graph will be (quietly) ignored. For directed graphs this returns the out-edges. Examples >>> G = nx.DiGraph() # or MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> [e for e in G.edges_iter()] [(0, 1), (1, 2), (2, 3)] >>> list(G.edges_iter(data=True)) # default data is {} (empty dict) [(0, 1, {}), (1, 2, {}), (2, 3, {})] >>> list(G.edges_iter([0,2])) [(0, 1), (2, 3)] >>> list(G.edges_iter(0)) [(0, 1)] out_edges DiGraph.out_edges(nbunch=None, data=False) Return a list of edges. Edges are returned as tuples with optional data in the order (node, neighbor, data). Parameters nbunch : iterable container, optional (default= all nodes) A container of nodes. The container will be iterated through once. data : bool, optional (default=False) Return two tuples (u,v) (False) or three-tuples (u,v,data) (True). Returns edge_list: list of edge tuples : Edges that are adjacent to any node in nbunch, or a list of all edges if nbunch is not speciﬁed. 3.2. Basic graph types 49 NetworkX Reference, Release 1.7 See Also: edges_iter return an iterator over the edges Notes Nodes in nbunch that are not in the graph will be (quietly) ignored. For directed graphs this returns the out-edges. Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G.edges() [(0, 1), (1, 2), (2, 3)] >>> G.edges(data=True) # default edge data is {} (empty dictionary) [(0, 1, {}), (1, 2, {}), (2, 3, {})] >>> G.edges([0,3]) [(0, 1), (3, 2)] >>> G.edges(0) [(0, 1)] out_edges_iter DiGraph.out_edges_iter(nbunch=None, data=False) Return an iterator over the edges. Edges are returned as tuples with optional data in the order (node, neighbor, data). Parameters nbunch : iterable container, optional (default= all nodes) A container of nodes. The container will be iterated through once. data : bool, optional (default=False) If True, return edge attribute dict in 3-tuple (u,v,data). Returns edge_iter : iterator An iterator of (u,v) or (u,v,d) tuples of edges. See Also: edges return a list of edges Notes Nodes in nbunch that are not in the graph will be (quietly) ignored. For directed graphs this returns the out-edges. Examples 50 Chapter 3. Graph types NetworkX Reference, Release 1.7 >>> G = nx.DiGraph() # or MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> [e for e in G.edges_iter()] [(0, 1), (1, 2), (2, 3)] >>> list(G.edges_iter(data=True)) # default data is {} (empty dict) [(0, 1, {}), (1, 2, {}), (2, 3, {})] >>> list(G.edges_iter([0,2])) [(0, 1), (2, 3)] >>> list(G.edges_iter(0)) [(0, 1)] in_edges DiGraph.in_edges(nbunch=None, data=False) Return a list of the incoming edges. See Also: edges return a list of edges in_edges_iter DiGraph.in_edges_iter(nbunch=None, data=False) Return an iterator over the incoming edges. Parameters nbunch : iterable container, optional (default= all nodes) A container of nodes. The container will be iterated through once. data : bool, optional (default=False) If True, return edge attribute dict in 3-tuple (u,v,data). Returns in_edge_iter : iterator An iterator of (u,v) or (u,v,d) tuples of incoming edges. See Also: edges_iter return an iterator of edges get_edge_data DiGraph.get_edge_data(u, v, default=None) Return the attribute dictionary associated with edge (u,v). Parameters u,v : nodes default: any Python object (default=None) : Value to return if the edge (u,v) is not found. Returns edge_dict : dictionary The edge attribute dictionary. 3.2. Basic graph types 51 NetworkX Reference, Release 1.7 Notes It is faster to use G[u][v]. >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G[0][1] {} Warning: Assigning G[u][v] corrupts the graph data structure. But it is safe to assign attributes to that dictionary, >>> G[0][1][’weight’] = 7 >>> G[0][1][’weight’] 7 >>> G[1][0][’weight’] 7 Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G.get_edge_data(0,1) # default edge data is {} {} >>> e = (0,1) >>> G.get_edge_data(*e) # tuple form {} >>> G.get_edge_data(’a’,’b’,default=0) # edge not in graph, return 0 0 neighbors DiGraph.neighbors(n) Return a list of successor nodes of n. neighbors() and successors() are the same function. neighbors_iter DiGraph.neighbors_iter(n) Return an iterator over successor nodes of n. neighbors_iter() and successors_iter() are the same. __getitem__ DiGraph.__getitem__(n) Return a dict of neighbors of node n. Use the expression ‘G[n]’. Parameters n : node A node in the graph. Returns adj_dict : dictionary The adjacency dictionary for nodes connected to n. 52 Chapter 3. Graph types NetworkX Reference, Release 1.7 Notes G[n] is similar to G.neighbors(n) but the internal data dictionary is returned instead of a list. Assigning G[n] will corrupt the internal graph data structure. Use G[n] for reading data only. Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G[0] {1: {}} successors DiGraph.successors(n) Return a list of successor nodes of n. neighbors() and successors() are the same function. successors_iter DiGraph.successors_iter(n) Return an iterator over successor nodes of n. neighbors_iter() and successors_iter() are the same. predecessors DiGraph.predecessors(n) Return a list of predecessor nodes of n. predecessors_iter DiGraph.predecessors_iter(n) Return an iterator over predecessor nodes of n. adjacency_list DiGraph.adjacency_list() Return an adjacency list representation of the graph. The output adjacency list is in the order of G.nodes(). For directed graphs, only outgoing adjacencies are included. Returns adj_list : lists of lists The adjacency structure of the graph as a list of lists. See Also: adjacency_iter 3.2. Basic graph types 53 NetworkX Reference, Release 1.7 Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G.adjacency_list() # in order given by G.nodes() [[1], [0, 2], [1, 3], [2]] adjacency_iter DiGraph.adjacency_iter() Return an iterator of (node, adjacency dict) tuples for all nodes. This is the fastest way to look at every edge. For directed graphs, only outgoing adjacencies are included. Returns adj_iter : iterator An iterator of (node, adjacency dictionary) for all nodes in the graph. See Also: adjacency_list Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> [(n,nbrdict) for n,nbrdict in G.adjacency_iter()] [(0, {1: {}}), (1, {0: {}, 2: {}}), (2, {1: {}, 3: {}}), (3, {2: {}})] nbunch_iter DiGraph.nbunch_iter(nbunch=None) Return an iterator of nodes contained in nbunch that are also in the graph. The nodes in nbunch are checked for membership in the graph and if not are silently ignored. Parameters nbunch : iterable container, optional (default=all nodes) A container of nodes. The container will be iterated through once. Returns niter : iterator An iterator over nodes in nbunch that are also in the graph. If nbunch is None, iterate over all nodes in the graph. Raises NetworkXError : If nbunch is not a node or or sequence of nodes. If a node in nbunch is not hashable. See Also: Graph.__iter__ 54 Chapter 3. Graph types NetworkX Reference, Release 1.7 Notes When nbunch is an iterator, the returned iterator yields values directly from nbunch, becoming exhausted when nbunch is exhausted. To test whether nbunch is a single node, one can use “if nbunch in self:”, even after processing with this routine. If nbunch is not a node or a (possibly empty) sequence/iterator or None, a NetworkXError is raised. Also, if any object in nbunch is not hashable, a NetworkXError is raised. Information about graph structure DiGraph.has_node(n) Return True if the graph contains the node n. DiGraph.__contains__(n) Return True if n is a node, False otherwise. Use the expression DiGraph.has_edge(u, v) Return True if the edge (u,v) is in the graph. DiGraph.order() Return the number of nodes in the graph. DiGraph.number_of_nodes() Return the number of nodes in the graph. DiGraph.__len__() Return the number of nodes. DiGraph.degree([nbunch, weight]) Return the degree of a node or nodes. DiGraph.degree_iter([nbunch, weight]) Return an iterator for (node, degree). DiGraph.in_degree([nbunch, weight]) Return the in-degree of a node or nodes. DiGraph.in_degree_iter([nbunch, weight]) Return an iterator for (node, in-degree). DiGraph.out_degree([nbunch, weight]) Return the out-degree of a node or nodes. DiGraph.out_degree_iter([nbunch, weight]) Return an iterator for (node, out-degree). DiGraph.size([weight]) Return the number of edges. DiGraph.number_of_edges([u, v]) Return the number of edges between two nodes. DiGraph.nodes_with_selfloops() Return a list of nodes with self loops. DiGraph.selfloop_edges([data]) Return a list of selﬂoop edges. DiGraph.number_of_selfloops() Return the number of selﬂoop edges. has_node DiGraph.has_node(n) Return True if the graph contains the node n. Parameters n : node Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2]) >>> G.has_node(0) True It is more readable and simpler to use >>> 0 in G True 3.2. Basic graph types 55 NetworkX Reference, Release 1.7 __contains__ DiGraph.__contains__(n) Return True if n is a node, False otherwise. Use the expression ‘n in G’. Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> 1 in G True has_edge DiGraph.has_edge(u, v) Return True if the edge (u,v) is in the graph. Parameters u,v : nodes Nodes can be, for example, strings or numbers. Nodes must be hashable (and not None) Python objects. Returns edge_ind : bool True if edge is in the graph, False otherwise. Examples Can be called either using two nodes u,v or edge tuple (u,v) >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G.has_edge(0,1) # using two nodes True >>> e = (0,1) >>> G.has_edge(*e) # e is a 2-tuple (u,v) True >>> e = (0,1,{’weight’:7}) >>> G.has_edge(*e[:2]) # e is a 3-tuple (u,v,data_dictionary) True The following syntax are all equivalent: >>> G.has_edge(0,1) True >>> 1 in G[0] # though this gives KeyError if 0 not in G True order DiGraph.order() Return the number of nodes in the graph. Returns nnodes : int 56 Chapter 3. Graph types NetworkX Reference, Release 1.7 The number of nodes in the graph. See Also: number_of_nodes, __len__ number_of_nodes DiGraph.number_of_nodes() Return the number of nodes in the graph. Returns nnodes : int The number of nodes in the graph. See Also: order, __len__ Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2]) >>> len(G) 3 __len__ DiGraph.__len__() Return the number of nodes. Use the expression ‘len(G)’. Returns nnodes : int The number of nodes in the graph. Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> len(G) 4 degree DiGraph.degree(nbunch=None, weight=None) Return the degree of a node or nodes. The node degree is the number of edges adjacent to that node. Parameters nbunch : iterable container, optional (default=all nodes) A container of nodes. The container will be iterated through once. weight : string or None, optional (default=None) 3.2. Basic graph types 57 NetworkX Reference, Release 1.7 The edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1. The degree is the sum of the edge weights adjacent to the node. Returns nd : dictionary, or number A dictionary with nodes as keys and degree as values or a number if a single node is speciﬁed. Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G.degree(0) 1 >>> G.degree([0,1]) {0: 1, 1: 2} >>> list(G.degree([0,1]).values()) [1, 2] degree_iter DiGraph.degree_iter(nbunch=None, weight=None) Return an iterator for (node, degree). The node degree is the number of edges adjacent to the node. Parameters nbunch : iterable container, optional (default=all nodes) A container of nodes. The container will be iterated through once. weight : string or None, optional (default=None) The edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1. The degree is the sum of the edge weights adjacent to the node. Returns nd_iter : an iterator The iterator returns two-tuples of (node, degree). See Also: degree, in_degree, out_degree, in_degree_iter, out_degree_iter Examples >>> G = nx.DiGraph() # or MultiDiGraph >>> G.add_path([0,1,2,3]) >>> list(G.degree_iter(0)) # node 0 with degree 1 [(0, 1)] >>> list(G.degree_iter([0,1])) [(0, 1), (1, 2)] in_degree DiGraph.in_degree(nbunch=None, weight=None) Return the in-degree of a node or nodes. 58 Chapter 3. Graph types NetworkX Reference, Release 1.7 The node in-degree is the number of edges pointing in to the node. Parameters nbunch : iterable container, optional (default=all nodes) A container of nodes. The container will be iterated through once. weight : string or None, optional (default=None) The edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1. The degree is the sum of the edge weights adjacent to the node. Returns nd : dictionary, or number A dictionary with nodes as keys and in-degree as values or a number if a single node is speciﬁed. See Also: degree, out_degree, in_degree_iter Examples >>> G = nx.DiGraph() # or MultiDiGraph >>> G.add_path([0,1,2,3]) >>> G.in_degree(0) 0 >>> G.in_degree([0,1]) {0: 0, 1: 1} >>> list(G.in_degree([0,1]).values()) [0, 1] in_degree_iter DiGraph.in_degree_iter(nbunch=None, weight=None) Return an iterator for (node, in-degree). The node in-degree is the number of edges pointing in to the node. Parameters nbunch : iterable container, optional (default=all nodes) A container of nodes. The container will be iterated through once. weight : string or None, optional (default=None) The edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1. The degree is the sum of the edge weights adjacent to the node. Returns nd_iter : an iterator The iterator returns two-tuples of (node, in-degree). See Also: degree, in_degree, out_degree, out_degree_iter Examples 3.2. Basic graph types 59 NetworkX Reference, Release 1.7 >>> G = nx.DiGraph() >>> G.add_path([0,1,2,3]) >>> list(G.in_degree_iter(0)) # node 0 with degree 0 [(0, 0)] >>> list(G.in_degree_iter([0,1])) [(0, 0), (1, 1)] out_degree DiGraph.out_degree(nbunch=None, weight=None) Return the out-degree of a node or nodes. The node out-degree is the number of edges pointing out of the node. Parameters nbunch : iterable container, optional (default=all nodes) A container of nodes. The container will be iterated through once. weight : string or None, optional (default=None) The edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1. The degree is the sum of the edge weights adjacent to the node. Returns nd : dictionary, or number A dictionary with nodes as keys and out-degree as values or a number if a single node is speciﬁed. Examples >>> G = nx.DiGraph() # or MultiDiGraph >>> G.add_path([0,1,2,3]) >>> G.out_degree(0) 1 >>> G.out_degree([0,1]) {0: 1, 1: 1} >>> list(G.out_degree([0,1]).values()) [1, 1] out_degree_iter DiGraph.out_degree_iter(nbunch=None, weight=None) Return an iterator for (node, out-degree). The node out-degree is the number of edges pointing out of the node. Parameters nbunch : iterable container, optional (default=all nodes) A container of nodes. The container will be iterated through once. weight : string or None, optional (default=None) The edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1. The degree is the sum of the edge weights adjacent to the node. Returns nd_iter : an iterator The iterator returns two-tuples of (node, out-degree). 60 Chapter 3. Graph types NetworkX Reference, Release 1.7 See Also: degree, in_degree, out_degree, in_degree_iter Examples >>> G = nx.DiGraph() >>> G.add_path([0,1,2,3]) >>> list(G.out_degree_iter(0)) # node 0 with degree 1 [(0, 1)] >>> list(G.out_degree_iter([0,1])) [(0, 1), (1, 1)] size DiGraph.size(weight=None) Return the number of edges. Parameters weight : string or None, optional (default=None) The edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1. Returns nedges : int The number of edges of sum of edge weights in the graph. See Also: number_of_edges Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G.size() 3 >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_edge(’a’,’b’,weight=2) >>> G.add_edge(’b’,’c’,weight=4) >>> G.size() 2 >>> G.size(weight=’weight’) 6.0 number_of_edges DiGraph.number_of_edges(u=None, v=None) Return the number of edges between two nodes. Parameters u,v : nodes, optional (default=all edges) If u and v are speciﬁed, return the number of edges between u and v. Otherwise return the total number of all edges. 3.2. Basic graph types 61 NetworkX Reference, Release 1.7 Returns nedges : int The number of edges in the graph. If nodes u and v are speciﬁed return the number of edges between those nodes. See Also: size Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G.number_of_edges() 3 >>> G.number_of_edges(0,1) 1 >>> e = (0,1) >>> G.number_of_edges(*e) 1 nodes_with_selﬂoops DiGraph.nodes_with_selfloops() Return a list of nodes with self loops. A node with a self loop has an edge with both ends adjacent to that node. Returns nodelist : list A list of nodes with self loops. See Also: selfloop_edges, number_of_selfloops Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_edge(1,1) >>> G.add_edge(1,2) >>> G.nodes_with_selfloops() [1] selﬂoop_edges DiGraph.selfloop_edges(data=False) Return a list of selﬂoop edges. A selﬂoop edge has the same node at both ends. Parameters data : bool, optional (default=False) Return selﬂoop edges as two tuples (u,v) (data=False) or three-tuples (u,v,data) (data=True) Returns edgelist : list of edge tuples 62 Chapter 3. Graph types NetworkX Reference, Release 1.7 A list of all selﬂoop edges. See Also: nodes_with_selfloops, number_of_selfloops Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_edge(1,1) >>> G.add_edge(1,2) >>> G.selfloop_edges() [(1, 1)] >>> G.selfloop_edges(data=True) [(1, 1, {})] number_of_selﬂoops DiGraph.number_of_selfloops() Return the number of selﬂoop edges. A selﬂoop edge has the same node at both ends. Returns nloops : int The number of selﬂoops. See Also: nodes_with_selfloops, selfloop_edges Examples >>> G=nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_edge(1,1) >>> G.add_edge(1,2) >>> G.number_of_selfloops() 1 Making copies and subgraphs DiGraph.copy() Return a copy of the graph. DiGraph.to_undirected([reciprocal]) Return an undirected representation of the digraph. DiGraph.to_directed() Return a directed copy of the graph. DiGraph.subgraph(nbunch) Return the subgraph induced on nodes in nbunch. DiGraph.reverse([copy]) Return the reverse of the graph. copy DiGraph.copy() Return a copy of the graph. Returns G : Graph 3.2. Basic graph types 63 NetworkX Reference, Release 1.7 A copy of the graph. See Also: to_directed return a directed copy of the graph. Notes This makes a complete copy of the graph including all of the node or edge attributes. Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> H = G.copy() to_undirected DiGraph.to_undirected(reciprocal=False) Return an undirected representation of the digraph. Parameters reciprocal : bool (optional) If True only keep edges that appear in both directions in the original digraph. Returns G : Graph An undirected graph with the same name and nodes and with edge (u,v,data) if either (u,v,data) or (v,u,data) is in the digraph. If both edges exist in digraph and their edge data is different, only one edge is created with an arbitrary choice of which edge data to use. You must check and correct for this manually if desired. Notes If edges in both directions (u,v) and (v,u) exist in the graph, attributes for the new undirected edge will be a combination of the attributes of the directed edges. The edge data is updated in the (arbitrary) order that the edges are encountered. For more customized control of the edge attributes use add_edge(). This returns a “deepcopy” of the edge, node, and graph attributes which attempts to completely copy all of the data and references. This is in contrast to the similar G=DiGraph(D) which returns a shallow copy of the data. See the Python copy module for more information on shallow and deep copies, http://docs.python.org/library/copy.html. to_directed DiGraph.to_directed() Return a directed copy of the graph. Returns G : DiGraph A deepcopy of the graph. 64 Chapter 3. Graph types NetworkX Reference, Release 1.7 Notes This returns a “deepcopy” of the edge, node, and graph attributes which attempts to completely copy all of the data and references. This is in contrast to the similar D=DiGraph(G) which returns a shallow copy of the data. See the Python copy module for more information on shallow and deep copies, http://docs.python.org/library/copy.html. Examples >>> G = nx.Graph() # or MultiGraph, etc >>> G.add_path([0,1]) >>> H = G.to_directed() >>> H.edges() [(0, 1), (1, 0)] If already directed, return a (deep) copy >>> G = nx.DiGraph() # or MultiDiGraph, etc >>> G.add_path([0,1]) >>> H = G.to_directed() >>> H.edges() [(0, 1)] subgraph DiGraph.subgraph(nbunch) Return the subgraph induced on nodes in nbunch. The induced subgraph of the graph contains the nodes in nbunch and the edges between those nodes. Parameters nbunch : list, iterable A container of nodes which will be iterated through once. Returns G : Graph A subgraph of the graph with the same edge attributes. Notes The graph, edge or node attributes just point to the original graph. So changes to the node or edge structure will not be reﬂected in the original graph while changes to the attributes will. To create a subgraph with its own copy of the edge/node attributes use: nx.Graph(G.subgraph(nbunch)) If edge attributes are containers, a deep copy can be obtained using: G.subgraph(nbunch).copy() For an inplace reduction of a graph to a subgraph you can remove nodes: G.remove_nodes_from([ n in G if n not in set(nbunch)]) 3.2. Basic graph types 65 NetworkX Reference, Release 1.7 Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> H = G.subgraph([0,1,2]) >>> H.edges() [(0, 1), (1, 2)] reverse DiGraph.reverse(copy=True) Return the reverse of the graph. The reverse is a graph with the same nodes and edges but with the directions of the edges reversed. Parameters copy : bool optional (default=True) If True, return a new DiGraph holding the reversed edges. If False, reverse the reverse graph is created using the original graph (this changes the original graph). 3.2.3 MultiGraph - Undirected graphs with self loops and parallel edges Overview MultiGraph(data=None, **attr) An undirected graph class that can store multiedges. Multiedges are multiple edges between two nodes. Each edge can hold optional data or attributes. A MultiGraph holds undirected edges. Self loops are allowed. Nodes can be arbitrary (hashable) Python objects with optional key/value attributes. Edges are represented as links between nodes with optional key/value attributes. Parameters data : input graph Data to initialize graph. If data=None (default) an empty graph is created. The data can be an edge list, or any NetworkX graph object. If the corresponding optional Python packages are installed the data can also be a NumPy matrix or 2d ndarray, a SciPy sparse matrix, or a PyGraphviz graph. attr : keyword arguments, optional (default= no attributes) Attributes to add to graph as key=value pairs. See Also: Graph, DiGraph, MultiDiGraph Examples Create an empty graph structure (a “null graph”) with no nodes and no edges. >>> G = nx.MultiGraph() 66 Chapter 3. Graph types NetworkX Reference, Release 1.7 G can be grown in several ways. Nodes: Add one node at a time: >>> G.add_node(1) Add the nodes from any container (a list, dict, set or even the lines from a ﬁle or the nodes from another graph). >>> G.add_nodes_from([2,3]) >>> G.add_nodes_from(range(100,110)) >>> H=nx.Graph() >>> H.add_path([0,1,2,3,4,5,6,7,8,9]) >>> G.add_nodes_from(H) In addition to strings and integers any hashable Python object (except None) can represent a node, e.g. a customized node object, or even another Graph. >>> G.add_node(H) Edges: G can also be grown by adding edges. Add one edge, >>> G.add_edge(1, 2) a list of edges, >>> G.add_edges_from([(1,2),(1,3)]) or a collection of edges, >>> G.add_edges_from(H.edges()) If some edges connect nodes not yet in the graph, the nodes are added automatically. If an edge already exists, an additional edge is created and stored using a key to identify the edge. By default the key is the lowest unused integer. >>> G.add_edges_from([(4,5,dict(route=282)), (4,5,dict(route=37))]) >>> G[4] {3: {0: {}}, 5: {0: {}, 1: {’route’: 282}, 2: {’route’: 37}}} Attributes: Each graph, node, and edge can hold key/value attribute pairs in an associated attribute dictionary (the keys must be hashable). By default these are empty, but can be added or changed using add_edge, add_node or direct manipulation of the attribute dictionaries named graph, node and edge respectively. >>> G = nx.MultiGraph(day="Friday") >>> G.graph {’day’: ’Friday’} Add node attributes using add_node(), add_nodes_from() or G.node >>> G.add_node(1, time=’5pm’) >>> G.add_nodes_from([3], time=’2pm’) >>> G.node[1] {’time’: ’5pm’} >>> G.node[1][’room’] = 714 >>> del G.node[1][’room’] # remove attribute 3.2. Basic graph types 67 NetworkX Reference, Release 1.7 >>> G.nodes(data=True) [(1, {’time’: ’5pm’}), (3, {’time’: ’2pm’})] Warning: adding a node to G.node does not add it to the graph. Add edge attributes using add_edge(), add_edges_from(), subscript notation, or G.edge. >>> G.add_edge(1, 2, weight=4.7 ) >>> G.add_edges_from([(3,4),(4,5)], color=’red’) >>> G.add_edges_from([(1,2,{’color’:’blue’}), (2,3,{’weight’:8})]) >>> G[1][2][0][’weight’] = 4.7 >>> G.edge[1][2][0][’weight’] = 4 Shortcuts: Many common graph features allow python syntax to speed reporting. >>> 1 in G # check if node in graph True >>> [n for n in G if n<3] # iterate through nodes [1, 2] >>> len(G) # number of nodes in graph 5 >>> G[1] # adjacency dict keyed by neighbor to edge attributes ... # Note: you should not change this dict manually! {2: {0: {’weight’: 4}, 1: {’color’: ’blue’}}} The fastest way to traverse all edges of a graph is via adjacency_iter(), but the edges() method is often more convenient. >>> for n,nbrsdict in G.adjacency_iter(): ... for nbr,keydict in nbrsdict.items(): ... for key,eattr in keydict.items(): ... if ’weight’ in eattr: ... (n,nbr,eattr[’weight’]) (1, 2, 4) (2, 1, 4) (2, 3, 8) (3, 2, 8) >>> [ (u,v,edata[’weight’]) for u,v,edata in G.edges(data=True) if ’weight’ in edata ] [(1, 2, 4), (2, 3, 8)] Reporting: Simple graph information is obtained using methods. Iterator versions of many reporting methods exist for efﬁciency. Methods exist for reporting nodes(), edges(), neighbors() and degree() as well as the number of nodes and edges. For details on these and other miscellaneous methods, see below. Adding and removing nodes and edges MultiGraph.__init__([data]) Initialize a graph with edges, name, graph attributes. MultiGraph.add_node(n[, attr_dict]) Add a single node n and update node attributes. MultiGraph.add_nodes_from(nodes, **attr) Add multiple nodes. MultiGraph.remove_node(n) Remove node n. MultiGraph.remove_nodes_from(nodes) Remove multiple nodes. Continued on next page 68 Chapter 3. Graph types NetworkX Reference, Release 1.7 Table 3.9 – continued from previous page MultiGraph.add_edge(u, v[, key, attr_dict]) Add an edge between u and v. MultiGraph.add_edges_from(ebunch[, attr_dict]) Add all the edges in ebunch. MultiGraph.add_weighted_edges_from(ebunch[, ...]) Add all the edges in ebunch as weighted edges with speciﬁed weights. MultiGraph.remove_edge(u, v[, key]) Remove an edge between u and v. MultiGraph.remove_edges_from(ebunch) Remove all edges speciﬁed in ebunch. MultiGraph.add_star(nodes, **attr) Add a star. MultiGraph.add_path(nodes, **attr) Add a path. MultiGraph.add_cycle(nodes, **attr) Add a cycle. MultiGraph.clear() Remove all nodes and edges from the graph. __init__ MultiGraph.__init__(data=None, **attr) Initialize a graph with edges, name, graph attributes. Parameters data : input graph Data to initialize graph. If data=None (default) an empty graph is created. The data can be an edge list, or any NetworkX graph object. If the corresponding optional Python packages are installed the data can also be a NumPy matrix or 2d ndarray, a SciPy sparse matrix, or a PyGraphviz graph. name : string, optional (default=’‘) An optional name for the graph. attr : keyword arguments, optional (default= no attributes) Attributes to add to graph as key=value pairs. See Also: convert Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G = nx.Graph(name=’my graph’) >>> e = [(1,2),(2,3),(3,4)] # list of edges >>> G = nx.Graph(e) Arbitrary graph attribute pairs (key=value) may be assigned >>> G=nx.Graph(e, day="Friday") >>> G.graph {’day’: ’Friday’} add_node MultiGraph.add_node(n, attr_dict=None, **attr) Add a single node n and update node attributes. Parameters n : node A node can be any hashable Python object except None. 3.2. Basic graph types 69 NetworkX Reference, Release 1.7 attr_dict : dictionary, optional (default= no attributes) Dictionary of node attributes. Key/value pairs will update existing data associated with the node. attr : keyword arguments, optional Set or change attributes using key=value. See Also: add_nodes_from Notes A hashable object is one that can be used as a key in a Python dictionary. This includes strings, numbers, tuples of strings and numbers, etc. On many platforms hashable items also include mutables such as NetworkX Graphs, though one should be careful that the hash doesn’t change on mutables. Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_node(1) >>> G.add_node(’Hello’) >>> K3 = nx.Graph([(0,1),(1,2),(2,0)]) >>> G.add_node(K3) >>> G.number_of_nodes() 3 Use keywords set/change node attributes: >>> G.add_node(1,size=10) >>> G.add_node(3,weight=0.4,UTM=(’13S’,382871,3972649)) add_nodes_from MultiGraph.add_nodes_from(nodes, **attr) Add multiple nodes. Parameters nodes : iterable container A container of nodes (list, dict, set, etc.). OR A container of (node, attribute dict) tuples. Node attributes are updated using the attribute dict. attr : keyword arguments, optional (default= no attributes) Update attributes for all nodes in nodes. Node attributes speciﬁed in nodes as a tuple take precedence over attributes speciﬁed generally. See Also: add_node 70 Chapter 3. Graph types NetworkX Reference, Release 1.7 Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_nodes_from(’Hello’) >>> K3 = nx.Graph([(0,1),(1,2),(2,0)]) >>> G.add_nodes_from(K3) >>> sorted(G.nodes(),key=str) [0, 1, 2, ’H’, ’e’, ’l’, ’o’] Use keywords to update speciﬁc node attributes for every node. >>> G.add_nodes_from([1,2], size=10) >>> G.add_nodes_from([3,4], weight=0.4) Use (node, attrdict) tuples to update attributes for speciﬁc nodes. >>> G.add_nodes_from([(1,dict(size=11)), (2,{’color’:’blue’})]) >>> G.node[1][’size’] 11 >>> H = nx.Graph() >>> H.add_nodes_from(G.nodes(data=True)) >>> H.node[1][’size’] 11 remove_node MultiGraph.remove_node(n) Remove node n. Removes the node n and all adjacent edges. Attempting to remove a non-existent node will raise an exception. Parameters n : node A node in the graph Raises NetworkXError : If n is not in the graph. See Also: remove_nodes_from Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2]) >>> G.edges() [(0, 1), (1, 2)] >>> G.remove_node(1) >>> G.edges() [] remove_nodes_from MultiGraph.remove_nodes_from(nodes) Remove multiple nodes. 3.2. Basic graph types 71 NetworkX Reference, Release 1.7 Parameters nodes : iterable container A container of nodes (list, dict, set, etc.). If a node in the container is not in the graph it is silently ignored. See Also: remove_node Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2]) >>> e = G.nodes() >>> e [0, 1, 2] >>> G.remove_nodes_from(e) >>> G.nodes() [] add_edge MultiGraph.add_edge(u, v, key=None, attr_dict=None, **attr) Add an edge between u and v. The nodes u and v will be automatically added if they are not already in the graph. Edge attributes can be speciﬁed with keywords or by providing a dictionary with key/value pairs. See examples below. Parameters u,v : nodes Nodes can be, for example, strings or numbers. Nodes must be hashable (and not None) Python objects. key : hashable identiﬁer, optional (default=lowest unused integer) Used to distinguish multiedges between a pair of nodes. attr_dict : dictionary, optional (default= no attributes) Dictionary of edge attributes. Key/value pairs will update existing data associated with the edge. attr : keyword arguments, optional Edge data (or labels or objects) can be assigned using keyword arguments. See Also: add_edges_from add a collection of edges Notes To replace/update edge data, use the optional key argument to identify a unique edge. Otherwise a new edge will be created. 72 Chapter 3. Graph types NetworkX Reference, Release 1.7 NetworkX algorithms designed for weighted graphs cannot use multigraphs directly because it is not clear how to handle multiedge weights. Convert to Graph using edge attribute ‘weight’ to enable weighted graph algorithms. Examples The following all add the edge e=(1,2) to graph G: >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> e = (1,2) >>> G.add_edge(1, 2) # explicit two-node form >>> G.add_edge(*e) # single edge as tuple of two nodes >>> G.add_edges_from( [(1,2)] ) # add edges from iterable container Associate data to edges using keywords: >>> G.add_edge(1, 2, weight=3) >>> G.add_edge(1, 2, key=0, weight=4) # update data for key=0 >>> G.add_edge(1, 3, weight=7, capacity=15, length=342.7) add_edges_from MultiGraph.add_edges_from(ebunch, attr_dict=None, **attr) Add all the edges in ebunch. Parameters ebunch : container of edges Each edge given in the container will be added to the graph. The edges can be: • 2-tuples (u,v) or • 3-tuples (u,v,d) for an edge attribute dict d, or • 4-tuples (u,v,k,d) for an edge identiﬁed by key k attr_dict : dictionary, optional (default= no attributes) Dictionary of edge attributes. Key/value pairs will update existing data associated with each edge. attr : keyword arguments, optional Edge data (or labels or objects) can be assigned using keyword arguments. See Also: add_edge add a single edge add_weighted_edges_from convenient way to add weighted edges Notes Adding the same edge twice has no effect but any edge data will be updated when each duplicate edge is added. 3.2. Basic graph types 73 NetworkX Reference, Release 1.7 Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_edges_from([(0,1),(1,2)]) # using a list of edge tuples >>> e = zip(range(0,3),range(1,4)) >>> G.add_edges_from(e) # Add the path graph 0-1-2-3 Associate data to edges >>> G.add_edges_from([(1,2),(2,3)], weight=3) >>> G.add_edges_from([(3,4),(1,4)], label=’WN2898’) add_weighted_edges_from MultiGraph.add_weighted_edges_from(ebunch, weight=’weight’, **attr) Add all the edges in ebunch as weighted edges with speciﬁed weights. Parameters ebunch : container of edges Each edge given in the list or container will be added to the graph. The edges must be given as 3-tuples (u,v,w) where w is a number. weight : string, optional (default= ‘weight’) The attribute name for the edge weights to be added. attr : keyword arguments, optional (default= no attributes) Edge attributes to add/update for all edges. See Also: add_edge add a single edge add_edges_from add multiple edges Notes Adding the same edge twice for Graph/DiGraph simply updates the edge data. For MultiGraph/MultiDiGraph, duplicate edges are stored. Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_weighted_edges_from([(0,1,3.0),(1,2,7.5)]) remove_edge MultiGraph.remove_edge(u, v, key=None) Remove an edge between u and v. Parameters u,v: nodes : Remove an edge between nodes u and v. key : hashable identiﬁer, optional (default=None) 74 Chapter 3. Graph types NetworkX Reference, Release 1.7 Used to distinguish multiple edges between a pair of nodes. If None remove a single (abritrary) edge between u and v. Raises NetworkXError : If there is not an edge between u and v, or if there is no edge with the speciﬁed key. See Also: remove_edges_from remove a collection of edges Examples >>> G = nx.MultiGraph() >>> G.add_path([0,1,2,3]) >>> G.remove_edge(0,1) >>> e = (1,2) >>> G.remove_edge(*e) # unpacks e from an edge tuple For multiple edges >>> G = nx.MultiGraph() # or MultiDiGraph, etc >>> G.add_edges_from([(1,2),(1,2),(1,2)]) >>> G.remove_edge(1,2) # remove a single (arbitrary) edge For edges with keys >>> G = nx.MultiGraph() # or MultiDiGraph, etc >>> G.add_edge(1,2,key=’first’) >>> G.add_edge(1,2,key=’second’) >>> G.remove_edge(1,2,key=’second’) remove_edges_from MultiGraph.remove_edges_from(ebunch) Remove all edges speciﬁed in ebunch. Parameters ebunch: list or container of edge tuples : Each edge given in the list or container will be removed from the graph. The edges can be: • 2-tuples (u,v) All edges between u and v are removed. • 3-tuples (u,v,key) The edge identiﬁed by key is removed. • 4-tuples (u,v,key,data) where data is ignored. See Also: remove_edge remove a single edge Notes Will fail silently if an edge in ebunch is not in the graph. 3.2. Basic graph types 75 NetworkX Reference, Release 1.7 Examples >>> G = nx.MultiGraph() # or MultiDiGraph >>> G.add_path([0,1,2,3]) >>> ebunch=[(1,2),(2,3)] >>> G.remove_edges_from(ebunch) Removing multiple copies of edges >>> G = nx.MultiGraph() >>> G.add_edges_from([(1,2),(1,2),(1,2)]) >>> G.remove_edges_from([(1,2),(1,2)]) >>> G.edges() [(1, 2)] >>> G.remove_edges_from([(1,2),(1,2)]) # silently ignore extra copy >>> G.edges() # now empty graph [] add_star MultiGraph.add_star(nodes, **attr) Add a star. The ﬁrst node in nodes is the middle of the star. It is connected to all other nodes. Parameters nodes : iterable container A container of nodes. attr : keyword arguments, optional (default= no attributes) Attributes to add to every edge in star. See Also: add_path, add_cycle Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_star([0,1,2,3]) >>> G.add_star([10,11,12],weight=2) add_path MultiGraph.add_path(nodes, **attr) Add a path. Parameters nodes : iterable container A container of nodes. A path will be constructed from the nodes (in order) and added to the graph. attr : keyword arguments, optional (default= no attributes) Attributes to add to every edge in path. 76 Chapter 3. Graph types NetworkX Reference, Release 1.7 See Also: add_star, add_cycle Examples >>> G=nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G.add_path([10,11,12],weight=7) add_cycle MultiGraph.add_cycle(nodes, **attr) Add a cycle. Parameters nodes: iterable container : A container of nodes. A cycle will be constructed from the nodes (in order) and added to the graph. attr : keyword arguments, optional (default= no attributes) Attributes to add to every edge in cycle. See Also: add_path, add_star Examples >>> G=nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_cycle([0,1,2,3]) >>> G.add_cycle([10,11,12],weight=7) clear MultiGraph.clear() Remove all nodes and edges from the graph. This also removes the name, and all graph, node, and edge attributes. Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G.clear() >>> G.nodes() [] >>> G.edges() [] 3.2. Basic graph types 77 NetworkX Reference, Release 1.7 Iterating over nodes and edges 78 Chapter 3. Graph types NetworkX Reference, Release 1.7 MultiGraph.nodes([data]) Return a list of the nodes in the graph. MultiGraph.nodes_iter([data]) Return an iterator over the nodes. MultiGraph.__iter__() Iterate over the nodes. MultiGraph.edges([nbunch, data, keys]) Return a list of edges. MultiGraph.edges_iter([nbunch, data, keys]) Return an iterator over the edges. MultiGraph.get_edge_data(u, v[, key, default]) Return the attribute dictionary associated with edge (u,v). MultiGraph.neighbors(n) Return a list of the nodes connected to the node n. MultiGraph.neighbors_iter(n) Return an iterator over all neighbors of node n. MultiGraph.__getitem__(n) Return a dict of neighbors of node n. MultiGraph.adjacency_list() Return an adjacency list representation of the graph. MultiGraph.adjacency_iter() Return an iterator of (node, adjacency dict) tuples for all nodes. MultiGraph.nbunch_iter([nbunch]) Return an iterator of nodes contained in nbunch that are also in the graph. nodes MultiGraph.nodes(data=False) Return a list of the nodes in the graph. Parameters data : boolean, optional (default=False) If False return a list of nodes. If True return a two-tuple of node and node data dictionary Returns nlist : list A list of nodes. If data=True a list of two-tuples containing (node, node data dictionary). Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2]) >>> G.nodes() [0, 1, 2] >>> G.add_node(1, time=’5pm’) >>> G.nodes(data=True) [(0, {}), (1, {’time’: ’5pm’}), (2, {})] nodes_iter MultiGraph.nodes_iter(data=False) Return an iterator over the nodes. Parameters data : boolean, optional (default=False) If False the iterator returns nodes. If True return a two-tuple of node and node data dictionary Returns niter : iterator An iterator over nodes. If data=True the iterator gives two-tuples containing (node, node data, dictionary) 3.2. Basic graph types 79 NetworkX Reference, Release 1.7 Notes If the node data is not required it is simpler and equivalent to use the expression ‘for n in G’. >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2]) Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2]) >>> [d for n,d in G.nodes_iter(data=True)] [{}, {}, {}] __iter__ MultiGraph.__iter__() Iterate over the nodes. Use the expression ‘for n in G’. Returns niter : iterator An iterator over all nodes in the graph. Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) edges MultiGraph.edges(nbunch=None, data=False, keys=False) Return a list of edges. Edges are returned as tuples with optional data and keys in the order (node, neighbor, key, data). Parameters nbunch : iterable container, optional (default= all nodes) A container of nodes. The container will be iterated through once. data : bool, optional (default=False) Return two tuples (u,v) (False) or three-tuples (u,v,data) (True). keys : bool, optional (default=False) Return two tuples (u,v) (False) or three-tuples (u,v,key) (True). Returns edge_list: list of edge tuples : Edges that are adjacent to any node in nbunch, or a list of all edges if nbunch is not speciﬁed. See Also: edges_iter return an iterator over the edges 80 Chapter 3. Graph types NetworkX Reference, Release 1.7 Notes Nodes in nbunch that are not in the graph will be (quietly) ignored. For directed graphs this returns the out-edges. Examples >>> G = nx.MultiGraph() # or MultiDiGraph >>> G.add_path([0,1,2,3]) >>> G.edges() [(0, 1), (1, 2), (2, 3)] >>> G.edges(data=True) # default edge data is {} (empty dictionary) [(0, 1, {}), (1, 2, {}), (2, 3, {})] >>> G.edges(keys=True) # default keys are integers [(0, 1, 0), (1, 2, 0), (2, 3, 0)] >>> G.edges(data=True,keys=True) # default keys are integers [(0, 1, 0, {}), (1, 2, 0, {}), (2, 3, 0, {})] >>> G.edges([0,3]) [(0, 1), (3, 2)] >>> G.edges(0) [(0, 1)] edges_iter MultiGraph.edges_iter(nbunch=None, data=False, keys=False) Return an iterator over the edges. Edges are returned as tuples with optional data and keys in the order (node, neighbor, key, data). Parameters nbunch : iterable container, optional (default= all nodes) A container of nodes. The container will be iterated through once. data : bool, optional (default=False) If True, return edge attribute dict with each edge. keys : bool, optional (default=False) If True, return edge keys with each edge. Returns edge_iter : iterator An iterator of (u,v), (u,v,d) or (u,v,key,d) tuples of edges. See Also: edges return a list of edges Notes Nodes in nbunch that are not in the graph will be (quietly) ignored. For directed graphs this returns the out-edges. 3.2. Basic graph types 81 NetworkX Reference, Release 1.7 Examples >>> G = nx.MultiGraph() # or MultiDiGraph >>> G.add_path([0,1,2,3]) >>> [e for e in G.edges_iter()] [(0, 1), (1, 2), (2, 3)] >>> list(G.edges_iter(data=True)) # default data is {} (empty dict) [(0, 1, {}), (1, 2, {}), (2, 3, {})] >>> list(G.edges(keys=True)) # default keys are integers [(0, 1, 0), (1, 2, 0), (2, 3, 0)] >>> list(G.edges(data=True,keys=True)) # default keys are integers [(0, 1, 0, {}), (1, 2, 0, {}), (2, 3, 0, {})] >>> list(G.edges_iter([0,3])) [(0, 1), (3, 2)] >>> list(G.edges_iter(0)) [(0, 1)] get_edge_data MultiGraph.get_edge_data(u, v, key=None, default=None) Return the attribute dictionary associated with edge (u,v). Parameters u,v : nodes default: any Python object (default=None) : Value to return if the edge (u,v) is not found. key : hashable identiﬁer, optional (default=None) Return data only for the edge with speciﬁed key. Returns edge_dict : dictionary The edge attribute dictionary. Notes It is faster to use G[u][v][key]. >>> G = nx.MultiGraph() # or MultiDiGraph >>> G.add_edge(0,1,key=’a’,weight=7) >>> G[0][1][’a’] # key=’a’ {’weight’: 7} Warning: Assigning G[u][v][key] corrupts the graph data structure. But it is safe to assign attributes to that dictionary, >>> G[0][1][’a’][’weight’] = 10 >>> G[0][1][’a’][’weight’] 10 >>> G[1][0][’a’][’weight’] 10 82 Chapter 3. Graph types NetworkX Reference, Release 1.7 Examples >>> G = nx.MultiGraph() # or MultiDiGraph >>> G.add_path([0,1,2,3]) >>> G.get_edge_data(0,1) {0: {}} >>> e = (0,1) >>> G.get_edge_data(*e) # tuple form {0: {}} >>> G.get_edge_data(’a’,’b’,default=0) # edge not in graph, return 0 0 neighbors MultiGraph.neighbors(n) Return a list of the nodes connected to the node n. Parameters n : node A node in the graph Returns nlist : list A list of nodes that are adjacent to n. Raises NetworkXError : If the node n is not in the graph. Notes It is usually more convenient (and faster) to access the adjacency dictionary as G[n]: >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_edge(’a’,’b’,weight=7) >>> G[’a’] {’b’: {’weight’: 7}} Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G.neighbors(0) [1] neighbors_iter MultiGraph.neighbors_iter(n) Return an iterator over all neighbors of node n. 3.2. Basic graph types 83 NetworkX Reference, Release 1.7 Notes It is faster to use the idiom “in G[0]”, e.g. >>> G = nx.path_graph(4) >>> [n for n in G[0]] [1] Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> [n for n in G.neighbors_iter(0)] [1] __getitem__ MultiGraph.__getitem__(n) Return a dict of neighbors of node n. Use the expression ‘G[n]’. Parameters n : node A node in the graph. Returns adj_dict : dictionary The adjacency dictionary for nodes connected to n. Notes G[n] is similar to G.neighbors(n) but the internal data dictionary is returned instead of a list. Assigning G[n] will corrupt the internal graph data structure. Use G[n] for reading data only. Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G[0] {1: {}} adjacency_list MultiGraph.adjacency_list() Return an adjacency list representation of the graph. The output adjacency list is in the order of G.nodes(). For directed graphs, only outgoing adjacencies are included. Returns adj_list : lists of lists The adjacency structure of the graph as a list of lists. 84 Chapter 3. Graph types NetworkX Reference, Release 1.7 See Also: adjacency_iter Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G.adjacency_list() # in order given by G.nodes() [[1], [0, 2], [1, 3], [2]] adjacency_iter MultiGraph.adjacency_iter() Return an iterator of (node, adjacency dict) tuples for all nodes. This is the fastest way to look at every edge. For directed graphs, only outgoing adjacencies are included. Returns adj_iter : iterator An iterator of (node, adjacency dictionary) for all nodes in the graph. See Also: adjacency_list Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> [(n,nbrdict) for n,nbrdict in G.adjacency_iter()] [(0, {1: {}}), (1, {0: {}, 2: {}}), (2, {1: {}, 3: {}}), (3, {2: {}})] nbunch_iter MultiGraph.nbunch_iter(nbunch=None) Return an iterator of nodes contained in nbunch that are also in the graph. The nodes in nbunch are checked for membership in the graph and if not are silently ignored. Parameters nbunch : iterable container, optional (default=all nodes) A container of nodes. The container will be iterated through once. Returns niter : iterator An iterator over nodes in nbunch that are also in the graph. If nbunch is None, iterate over all nodes in the graph. Raises NetworkXError : If nbunch is not a node or or sequence of nodes. If a node in nbunch is not hashable. See Also: Graph.__iter__ 3.2. Basic graph types 85 NetworkX Reference, Release 1.7 Notes When nbunch is an iterator, the returned iterator yields values directly from nbunch, becoming exhausted when nbunch is exhausted. To test whether nbunch is a single node, one can use “if nbunch in self:”, even after processing with this routine. If nbunch is not a node or a (possibly empty) sequence/iterator or None, a NetworkXError is raised. Also, if any object in nbunch is not hashable, a NetworkXError is raised. Information about graph structure MultiGraph.has_node(n) Return True if the graph contains the node n. MultiGraph.__contains__(n) Return True if n is a node, False otherwise. Use the expression MultiGraph.has_edge(u, v[, key]) Return True if the graph has an edge between nodes u and v. MultiGraph.order() Return the number of nodes in the graph. MultiGraph.number_of_nodes() Return the number of nodes in the graph. MultiGraph.__len__() Return the number of nodes. MultiGraph.degree([nbunch, weight]) Return the degree of a node or nodes. MultiGraph.degree_iter([nbunch, weight]) Return an iterator for (node, degree). MultiGraph.size([weight]) Return the number of edges. MultiGraph.number_of_edges([u, v]) Return the number of edges between two nodes. MultiGraph.nodes_with_selfloops() Return a list of nodes with self loops. MultiGraph.selfloop_edges([data, keys]) Return a list of selﬂoop edges. MultiGraph.number_of_selfloops() Return the number of selﬂoop edges. has_node MultiGraph.has_node(n) Return True if the graph contains the node n. Parameters n : node Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2]) >>> G.has_node(0) True It is more readable and simpler to use >>> 0 in G True __contains__ MultiGraph.__contains__(n) Return True if n is a node, False otherwise. Use the expression ‘n in G’. 86 Chapter 3. Graph types NetworkX Reference, Release 1.7 Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> 1 in G True has_edge MultiGraph.has_edge(u, v, key=None) Return True if the graph has an edge between nodes u and v. Parameters u,v : nodes Nodes can be, for example, strings or numbers. key : hashable identiﬁer, optional (default=None) If speciﬁed return True only if the edge with key is found. Returns edge_ind : bool True if edge is in the graph, False otherwise. Examples Can be called either using two nodes u,v, an edge tuple (u,v), or an edge tuple (u,v,key). >>> G = nx.MultiGraph() # or MultiDiGraph >>> G.add_path([0,1,2,3]) >>> G.has_edge(0,1) # using two nodes True >>> e = (0,1) >>> G.has_edge(*e) # e is a 2-tuple (u,v) True >>> G.add_edge(0,1,key=’a’) >>> G.has_edge(0,1,key=’a’) # specify key True >>> e=(0,1,’a’) >>> G.has_edge(*e) # e is a 3-tuple (u,v,’a’) True The following syntax are equivalent: >>> G.has_edge(0,1) True >>> 1 in G[0] # though this gives KeyError if 0 not in G True order MultiGraph.order() Return the number of nodes in the graph. Returns nnodes : int The number of nodes in the graph. 3.2. Basic graph types 87 NetworkX Reference, Release 1.7 See Also: number_of_nodes, __len__ number_of_nodes MultiGraph.number_of_nodes() Return the number of nodes in the graph. Returns nnodes : int The number of nodes in the graph. See Also: order, __len__ Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2]) >>> len(G) 3 __len__ MultiGraph.__len__() Return the number of nodes. Use the expression ‘len(G)’. Returns nnodes : int The number of nodes in the graph. Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> len(G) 4 degree MultiGraph.degree(nbunch=None, weight=None) Return the degree of a node or nodes. The node degree is the number of edges adjacent to that node. Parameters nbunch : iterable container, optional (default=all nodes) A container of nodes. The container will be iterated through once. weight : string or None, optional (default=None) The edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1. The degree is the sum of the edge weights adjacent to the node. 88 Chapter 3. Graph types NetworkX Reference, Release 1.7 Returns nd : dictionary, or number A dictionary with nodes as keys and degree as values or a number if a single node is speciﬁed. Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G.degree(0) 1 >>> G.degree([0,1]) {0: 1, 1: 2} >>> list(G.degree([0,1]).values()) [1, 2] degree_iter MultiGraph.degree_iter(nbunch=None, weight=None) Return an iterator for (node, degree). The node degree is the number of edges adjacent to the node. Parameters nbunch : iterable container, optional (default=all nodes) A container of nodes. The container will be iterated through once. weight : string or None, optional (default=None) The edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1. The degree is the sum of the edge weights adjacent to the node. Returns nd_iter : an iterator The iterator returns two-tuples of (node, degree). See Also: degree Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> list(G.degree_iter(0)) # node 0 with degree 1 [(0, 1)] >>> list(G.degree_iter([0,1])) [(0, 1), (1, 2)] size MultiGraph.size(weight=None) Return the number of edges. Parameters weight : string or None, optional (default=None) 3.2. Basic graph types 89 NetworkX Reference, Release 1.7 The edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1. Returns nedges : int The number of edges of sum of edge weights in the graph. See Also: number_of_edges Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G.size() 3 >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_edge(’a’,’b’,weight=2) >>> G.add_edge(’b’,’c’,weight=4) >>> G.size() 2 >>> G.size(weight=’weight’) 6.0 number_of_edges MultiGraph.number_of_edges(u=None, v=None) Return the number of edges between two nodes. Parameters u,v : nodes, optional (default=all edges) If u and v are speciﬁed, return the number of edges between u and v. Otherwise return the total number of all edges. Returns nedges : int The number of edges in the graph. If nodes u and v are speciﬁed return the number of edges between those nodes. See Also: size Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G.number_of_edges() 3 >>> G.number_of_edges(0,1) 1 >>> e = (0,1) >>> G.number_of_edges(*e) 1 90 Chapter 3. Graph types NetworkX Reference, Release 1.7 nodes_with_selﬂoops MultiGraph.nodes_with_selfloops() Return a list of nodes with self loops. A node with a self loop has an edge with both ends adjacent to that node. Returns nodelist : list A list of nodes with self loops. See Also: selfloop_edges, number_of_selfloops Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_edge(1,1) >>> G.add_edge(1,2) >>> G.nodes_with_selfloops() [1] selﬂoop_edges MultiGraph.selfloop_edges(data=False, keys=False) Return a list of selﬂoop edges. A selﬂoop edge has the same node at both ends. Parameters data : bool, optional (default=False) Return selﬂoop edges as two tuples (u,v) (data=False) or three-tuples (u,v,data) (data=True) keys : bool, optional (default=False) If True, return edge keys with each edge. Returns edgelist : list of edge tuples A list of all selﬂoop edges. See Also: nodes_with_selfloops, number_of_selfloops Examples >>> G = nx.MultiGraph() # or MultiDiGraph >>> G.add_edge(1,1) >>> G.add_edge(1,2) >>> G.selfloop_edges() [(1, 1)] >>> G.selfloop_edges(data=True) [(1, 1, {})] >>> G.selfloop_edges(keys=True) [(1, 1, 0)] 3.2. Basic graph types 91 NetworkX Reference, Release 1.7 >>> G.selfloop_edges(keys=True, data=True) [(1, 1, 0, {})] number_of_selﬂoops MultiGraph.number_of_selfloops() Return the number of selﬂoop edges. A selﬂoop edge has the same node at both ends. Returns nloops : int The number of selﬂoops. See Also: nodes_with_selfloops, selfloop_edges Examples >>> G=nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_edge(1,1) >>> G.add_edge(1,2) >>> G.number_of_selfloops() 1 Making copies and subgraphs MultiGraph.copy() Return a copy of the graph. MultiGraph.to_undirected() Return an undirected copy of the graph. MultiGraph.to_directed() Return a directed representation of the graph. MultiGraph.subgraph(nbunch) Return the subgraph induced on nodes in nbunch. copy MultiGraph.copy() Return a copy of the graph. Returns G : Graph A copy of the graph. See Also: to_directed return a directed copy of the graph. Notes This makes a complete copy of the graph including all of the node or edge attributes. 92 Chapter 3. Graph types NetworkX Reference, Release 1.7 Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> H = G.copy() to_undirected MultiGraph.to_undirected() Return an undirected copy of the graph. Returns G : Graph/MultiGraph A deepcopy of the graph. See Also: copy, add_edge, add_edges_from Notes This returns a “deepcopy” of the edge, node, and graph attributes which attempts to completely copy all of the data and references. This is in contrast to the similar G=DiGraph(D) which returns a shallow copy of the data. See the Python copy module for more information on shallow and deep copies, http://docs.python.org/library/copy.html. Examples >>> G = nx.Graph() # or MultiGraph, etc >>> G.add_path([0,1]) >>> H = G.to_directed() >>> H.edges() [(0, 1), (1, 0)] >>> G2 = H.to_undirected() >>> G2.edges() [(0, 1)] to_directed MultiGraph.to_directed() Return a directed representation of the graph. Returns G : MultiDiGraph A directed graph with the same name, same nodes, and with each edge (u,v,data) re- placed by two directed edges (u,v,data) and (v,u,data). 3.2. Basic graph types 93 NetworkX Reference, Release 1.7 Notes This returns a “deepcopy” of the edge, node, and graph attributes which attempts to completely copy all of the data and references. This is in contrast to the similar D=DiGraph(G) which returns a shallow copy of the data. See the Python copy module for more information on shallow and deep copies, http://docs.python.org/library/copy.html. Examples >>> G = nx.Graph() # or MultiGraph, etc >>> G.add_path([0,1]) >>> H = G.to_directed() >>> H.edges() [(0, 1), (1, 0)] If already directed, return a (deep) copy >>> G = nx.DiGraph() # or MultiDiGraph, etc >>> G.add_path([0,1]) >>> H = G.to_directed() >>> H.edges() [(0, 1)] subgraph MultiGraph.subgraph(nbunch) Return the subgraph induced on nodes in nbunch. The induced subgraph of the graph contains the nodes in nbunch and the edges between those nodes. Parameters nbunch : list, iterable A container of nodes which will be iterated through once. Returns G : Graph A subgraph of the graph with the same edge attributes. Notes The graph, edge or node attributes just point to the original graph. So changes to the node or edge structure will not be reﬂected in the original graph while changes to the attributes will. To create a subgraph with its own copy of the edge/node attributes use: nx.Graph(G.subgraph(nbunch)) If edge attributes are containers, a deep copy can be obtained using: G.subgraph(nbunch).copy() For an inplace reduction of a graph to a subgraph you can remove nodes: G.remove_nodes_from([ n in G if n not in set(nbunch)]) 94 Chapter 3. Graph types NetworkX Reference, Release 1.7 Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> H = G.subgraph([0,1,2]) >>> H.edges() [(0, 1), (1, 2)] 3.2.4 MultiDiGraph - Directed graphs with self loops and parallel edges Overview MultiDiGraph(data=None, **attr) A directed graph class that can store multiedges. Multiedges are multiple edges between two nodes. Each edge can hold optional data or attributes. A MultiDiGraph holds directed edges. Self loops are allowed. Nodes can be arbitrary (hashable) Python objects with optional key/value attributes. Edges are represented as links between nodes with optional key/value attributes. Parameters data : input graph Data to initialize graph. If data=None (default) an empty graph is created. The data can be an edge list, or any NetworkX graph object. If the corresponding optional Python packages are installed the data can also be a NumPy matrix or 2d ndarray, a SciPy sparse matrix, or a PyGraphviz graph. attr : keyword arguments, optional (default= no attributes) Attributes to add to graph as key=value pairs. See Also: Graph, DiGraph, MultiGraph Examples Create an empty graph structure (a “null graph”) with no nodes and no edges. >>> G = nx.MultiDiGraph() G can be grown in several ways. Nodes: Add one node at a time: >>> G.add_node(1) Add the nodes from any container (a list, dict, set or even the lines from a ﬁle or the nodes from another graph). >>> G.add_nodes_from([2,3]) >>> G.add_nodes_from(range(100,110)) >>> H=nx.Graph() >>> H.add_path([0,1,2,3,4,5,6,7,8,9]) >>> G.add_nodes_from(H) 3.2. Basic graph types 95 NetworkX Reference, Release 1.7 In addition to strings and integers any hashable Python object (except None) can represent a node, e.g. a customized node object, or even another Graph. >>> G.add_node(H) Edges: G can also be grown by adding edges. Add one edge, >>> G.add_edge(1, 2) a list of edges, >>> G.add_edges_from([(1,2),(1,3)]) or a collection of edges, >>> G.add_edges_from(H.edges()) If some edges connect nodes not yet in the graph, the nodes are added automatically. If an edge already exists, an additional edge is created and stored using a key to identify the edge. By default the key is the lowest unused integer. >>> G.add_edges_from([(4,5,dict(route=282)), (4,5,dict(route=37))]) >>> G[4] {5: {0: {}, 1: {’route’: 282}, 2: {’route’: 37}}} Attributes: Each graph, node, and edge can hold key/value attribute pairs in an associated attribute dictionary (the keys must be hashable). By default these are empty, but can be added or changed using add_edge, add_node or direct manipulation of the attribute dictionaries named graph, node and edge respectively. >>> G = nx.MultiDiGraph(day="Friday") >>> G.graph {’day’: ’Friday’} Add node attributes using add_node(), add_nodes_from() or G.node >>> G.add_node(1, time=’5pm’) >>> G.add_nodes_from([3], time=’2pm’) >>> G.node[1] {’time’: ’5pm’} >>> G.node[1][’room’] = 714 >>> del G.node[1][’room’] # remove attribute >>> G.nodes(data=True) [(1, {’time’: ’5pm’}), (3, {’time’: ’2pm’})] Warning: adding a node to G.node does not add it to the graph. Add edge attributes using add_edge(), add_edges_from(), subscript notation, or G.edge. >>> G.add_edge(1, 2, weight=4.7 ) >>> G.add_edges_from([(3,4),(4,5)], color=’red’) >>> G.add_edges_from([(1,2,{’color’:’blue’}), (2,3,{’weight’:8})]) >>> G[1][2][0][’weight’] = 4.7 >>> G.edge[1][2][0][’weight’] = 4 Shortcuts: Many common graph features allow python syntax to speed reporting. 96 Chapter 3. Graph types NetworkX Reference, Release 1.7 >>> 1 in G # check if node in graph True >>> [n for n in G if n<3] # iterate through nodes [1, 2] >>> len(G) # number of nodes in graph 5 >>> G[1] # adjacency dict keyed by neighbor to edge attributes ... # Note: you should not change this dict manually! {2: {0: {’weight’: 4}, 1: {’color’: ’blue’}}} The fastest way to traverse all edges of a graph is via adjacency_iter(), but the edges() method is often more convenient. >>> for n,nbrsdict in G.adjacency_iter(): ... for nbr,keydict in nbrsdict.items(): ... for key,eattr in keydict.items(): ... if ’weight’ in eattr: ... (n,nbr,eattr[’weight’]) (1, 2, 4) (2, 3, 8) >>> [ (u,v,edata[’weight’]) for u,v,edata in G.edges(data=True) if ’weight’ in edata ] [(1, 2, 4), (2, 3, 8)] Reporting: Simple graph information is obtained using methods. Iterator versions of many reporting methods exist for efﬁciency. Methods exist for reporting nodes(), edges(), neighbors() and degree() as well as the number of nodes and edges. For details on these and other miscellaneous methods, see below. Adding and Removing Nodes and Edges MultiDiGraph.__init__([data]) Initialize a graph with edges, name, graph attributes. MultiDiGraph.add_node(n[, attr_dict]) Add a single node n and update node attributes. MultiDiGraph.add_nodes_from(nodes, **attr) Add multiple nodes. MultiDiGraph.remove_node(n) Remove node n. MultiDiGraph.remove_nodes_from(nbunch) Remove multiple nodes. MultiDiGraph.add_edge(u, v[, key, attr_dict]) Add an edge between u and v. MultiDiGraph.add_edges_from(ebunch[, attr_dict]) Add all the edges in ebunch. MultiDiGraph.add_weighted_edges_from(ebunch) Add all the edges in ebunch as weighted edges with speciﬁed weights. MultiDiGraph.remove_edge(u, v[, key]) Remove an edge between u and v. MultiDiGraph.remove_edges_from(ebunch) Remove all edges speciﬁed in ebunch. MultiDiGraph.add_star(nodes, **attr) Add a star. MultiDiGraph.add_path(nodes, **attr) Add a path. MultiDiGraph.add_cycle(nodes, **attr) Add a cycle. MultiDiGraph.clear() Remove all nodes and edges from the graph. __init__ MultiDiGraph.__init__(data=None, **attr) Initialize a graph with edges, name, graph attributes. Parameters data : input graph 3.2. Basic graph types 97 NetworkX Reference, Release 1.7 Data to initialize graph. If data=None (default) an empty graph is created. The data can be an edge list, or any NetworkX graph object. If the corresponding optional Python packages are installed the data can also be a NumPy matrix or 2d ndarray, a SciPy sparse matrix, or a PyGraphviz graph. name : string, optional (default=’‘) An optional name for the graph. attr : keyword arguments, optional (default= no attributes) Attributes to add to graph as key=value pairs. See Also: convert Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G = nx.Graph(name=’my graph’) >>> e = [(1,2),(2,3),(3,4)] # list of edges >>> G = nx.Graph(e) Arbitrary graph attribute pairs (key=value) may be assigned >>> G=nx.Graph(e, day="Friday") >>> G.graph {’day’: ’Friday’} add_node MultiDiGraph.add_node(n, attr_dict=None, **attr) Add a single node n and update node attributes. Parameters n : node A node can be any hashable Python object except None. attr_dict : dictionary, optional (default= no attributes) Dictionary of node attributes. Key/value pairs will update existing data associated with the node. attr : keyword arguments, optional Set or change attributes using key=value. See Also: add_nodes_from Notes A hashable object is one that can be used as a key in a Python dictionary. This includes strings, numbers, tuples of strings and numbers, etc. On many platforms hashable items also include mutables such as NetworkX Graphs, though one should be careful that the hash doesn’t change on mutables. 98 Chapter 3. Graph types NetworkX Reference, Release 1.7 Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_node(1) >>> G.add_node(’Hello’) >>> K3 = nx.Graph([(0,1),(1,2),(2,0)]) >>> G.add_node(K3) >>> G.number_of_nodes() 3 Use keywords set/change node attributes: >>> G.add_node(1,size=10) >>> G.add_node(3,weight=0.4,UTM=(’13S’,382871,3972649)) add_nodes_from MultiDiGraph.add_nodes_from(nodes, **attr) Add multiple nodes. Parameters nodes : iterable container A container of nodes (list, dict, set, etc.). OR A container of (node, attribute dict) tuples. Node attributes are updated using the attribute dict. attr : keyword arguments, optional (default= no attributes) Update attributes for all nodes in nodes. Node attributes speciﬁed in nodes as a tuple take precedence over attributes speciﬁed generally. See Also: add_node Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_nodes_from(’Hello’) >>> K3 = nx.Graph([(0,1),(1,2),(2,0)]) >>> G.add_nodes_from(K3) >>> sorted(G.nodes(),key=str) [0, 1, 2, ’H’, ’e’, ’l’, ’o’] Use keywords to update speciﬁc node attributes for every node. >>> G.add_nodes_from([1,2], size=10) >>> G.add_nodes_from([3,4], weight=0.4) Use (node, attrdict) tuples to update attributes for speciﬁc nodes. >>> G.add_nodes_from([(1,dict(size=11)), (2,{’color’:’blue’})]) >>> G.node[1][’size’] 11 >>> H = nx.Graph() >>> H.add_nodes_from(G.nodes(data=True)) >>> H.node[1][’size’] 11 3.2. Basic graph types 99 NetworkX Reference, Release 1.7 remove_node MultiDiGraph.remove_node(n) Remove node n. Removes the node n and all adjacent edges. Attempting to remove a non-existent node will raise an exception. Parameters n : node A node in the graph Raises NetworkXError : If n is not in the graph. See Also: remove_nodes_from Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2]) >>> G.edges() [(0, 1), (1, 2)] >>> G.remove_node(1) >>> G.edges() [] remove_nodes_from MultiDiGraph.remove_nodes_from(nbunch) Remove multiple nodes. Parameters nodes : iterable container A container of nodes (list, dict, set, etc.). If a node in the container is not in the graph it is silently ignored. See Also: remove_node Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2]) >>> e = G.nodes() >>> e [0, 1, 2] >>> G.remove_nodes_from(e) >>> G.nodes() [] 100 Chapter 3. Graph types NetworkX Reference, Release 1.7 add_edge MultiDiGraph.add_edge(u, v, key=None, attr_dict=None, **attr) Add an edge between u and v. The nodes u and v will be automatically added if they are not already in the graph. Edge attributes can be speciﬁed with keywords or by providing a dictionary with key/value pairs. See examples below. Parameters u,v : nodes Nodes can be, for example, strings or numbers. Nodes must be hashable (and not None) Python objects. key : hashable identiﬁer, optional (default=lowest unused integer) Used to distinguish multiedges between a pair of nodes. attr_dict : dictionary, optional (default= no attributes) Dictionary of edge attributes. Key/value pairs will update existing data associated with the edge. attr : keyword arguments, optional Edge data (or labels or objects) can be assigned using keyword arguments. See Also: add_edges_from add a collection of edges Notes To replace/update edge data, use the optional key argument to identify a unique edge. Otherwise a new edge will be created. NetworkX algorithms designed for weighted graphs cannot use multigraphs directly because it is not clear how to handle multiedge weights. Convert to Graph using edge attribute ‘weight’ to enable weighted graph algorithms. Examples The following all add the edge e=(1,2) to graph G: >>> G = nx.MultiDiGraph() >>> e = (1,2) >>> G.add_edge(1, 2) # explicit two-node form >>> G.add_edge(*e) # single edge as tuple of two nodes >>> G.add_edges_from( [(1,2)] ) # add edges from iterable container Associate data to edges using keywords: >>> G.add_edge(1, 2, weight=3) >>> G.add_edge(1, 2, key=0, weight=4) # update data for key=0 >>> G.add_edge(1, 3, weight=7, capacity=15, length=342.7) 3.2. Basic graph types 101 NetworkX Reference, Release 1.7 add_edges_from MultiDiGraph.add_edges_from(ebunch, attr_dict=None, **attr) Add all the edges in ebunch. Parameters ebunch : container of edges Each edge given in the container will be added to the graph. The edges can be: • 2-tuples (u,v) or • 3-tuples (u,v,d) for an edge attribute dict d, or • 4-tuples (u,v,k,d) for an edge identiﬁed by key k attr_dict : dictionary, optional (default= no attributes) Dictionary of edge attributes. Key/value pairs will update existing data associated with each edge. attr : keyword arguments, optional Edge data (or labels or objects) can be assigned using keyword arguments. See Also: add_edge add a single edge add_weighted_edges_from convenient way to add weighted edges Notes Adding the same edge twice has no effect but any edge data will be updated when each duplicate edge is added. Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_edges_from([(0,1),(1,2)]) # using a list of edge tuples >>> e = zip(range(0,3),range(1,4)) >>> G.add_edges_from(e) # Add the path graph 0-1-2-3 Associate data to edges >>> G.add_edges_from([(1,2),(2,3)], weight=3) >>> G.add_edges_from([(3,4),(1,4)], label=’WN2898’) add_weighted_edges_from MultiDiGraph.add_weighted_edges_from(ebunch, weight=’weight’, **attr) Add all the edges in ebunch as weighted edges with speciﬁed weights. Parameters ebunch : container of edges Each edge given in the list or container will be added to the graph. The edges must be given as 3-tuples (u,v,w) where w is a number. weight : string, optional (default= ‘weight’) The attribute name for the edge weights to be added. 102 Chapter 3. Graph types NetworkX Reference, Release 1.7 attr : keyword arguments, optional (default= no attributes) Edge attributes to add/update for all edges. See Also: add_edge add a single edge add_edges_from add multiple edges Notes Adding the same edge twice for Graph/DiGraph simply updates the edge data. For MultiGraph/MultiDiGraph, duplicate edges are stored. Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_weighted_edges_from([(0,1,3.0),(1,2,7.5)]) remove_edge MultiDiGraph.remove_edge(u, v, key=None) Remove an edge between u and v. Parameters u,v: nodes : Remove an edge between nodes u and v. key : hashable identiﬁer, optional (default=None) Used to distinguish multiple edges between a pair of nodes. If None remove a single (abritrary) edge between u and v. Raises NetworkXError : If there is not an edge between u and v, or if there is no edge with the speciﬁed key. See Also: remove_edges_from remove a collection of edges Examples >>> G = nx.MultiDiGraph() >>> G.add_path([0,1,2,3]) >>> G.remove_edge(0,1) >>> e = (1,2) >>> G.remove_edge(*e) # unpacks e from an edge tuple For multiple edges >>> G = nx.MultiDiGraph() >>> G.add_edges_from([(1,2),(1,2),(1,2)]) >>> G.remove_edge(1,2) # remove a single (arbitrary) edge 3.2. Basic graph types 103 NetworkX Reference, Release 1.7 For edges with keys >>> G = nx.MultiDiGraph() >>> G.add_edge(1,2,key=’first’) >>> G.add_edge(1,2,key=’second’) >>> G.remove_edge(1,2,key=’second’) remove_edges_from MultiDiGraph.remove_edges_from(ebunch) Remove all edges speciﬁed in ebunch. Parameters ebunch: list or container of edge tuples : Each edge given in the list or container will be removed from the graph. The edges can be: • 2-tuples (u,v) All edges between u and v are removed. • 3-tuples (u,v,key) The edge identiﬁed by key is removed. • 4-tuples (u,v,key,data) where data is ignored. See Also: remove_edge remove a single edge Notes Will fail silently if an edge in ebunch is not in the graph. Examples >>> G = nx.MultiGraph() # or MultiDiGraph >>> G.add_path([0,1,2,3]) >>> ebunch=[(1,2),(2,3)] >>> G.remove_edges_from(ebunch) Removing multiple copies of edges >>> G = nx.MultiGraph() >>> G.add_edges_from([(1,2),(1,2),(1,2)]) >>> G.remove_edges_from([(1,2),(1,2)]) >>> G.edges() [(1, 2)] >>> G.remove_edges_from([(1,2),(1,2)]) # silently ignore extra copy >>> G.edges() # now empty graph [] add_star MultiDiGraph.add_star(nodes, **attr) Add a star. The ﬁrst node in nodes is the middle of the star. It is connected to all other nodes. 104 Chapter 3. Graph types NetworkX Reference, Release 1.7 Parameters nodes : iterable container A container of nodes. attr : keyword arguments, optional (default= no attributes) Attributes to add to every edge in star. See Also: add_path, add_cycle Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_star([0,1,2,3]) >>> G.add_star([10,11,12],weight=2) add_path MultiDiGraph.add_path(nodes, **attr) Add a path. Parameters nodes : iterable container A container of nodes. A path will be constructed from the nodes (in order) and added to the graph. attr : keyword arguments, optional (default= no attributes) Attributes to add to every edge in path. See Also: add_star, add_cycle Examples >>> G=nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G.add_path([10,11,12],weight=7) add_cycle MultiDiGraph.add_cycle(nodes, **attr) Add a cycle. Parameters nodes: iterable container : A container of nodes. A cycle will be constructed from the nodes (in order) and added to the graph. attr : keyword arguments, optional (default= no attributes) Attributes to add to every edge in cycle. See Also: add_path, add_star 3.2. Basic graph types 105 NetworkX Reference, Release 1.7 Examples >>> G=nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_cycle([0,1,2,3]) >>> G.add_cycle([10,11,12],weight=7) clear MultiDiGraph.clear() Remove all nodes and edges from the graph. This also removes the name, and all graph, node, and edge attributes. Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G.clear() >>> G.nodes() [] >>> G.edges() [] Iterating over nodes and edges MultiDiGraph.nodes([data]) Return a list of the nodes in the graph. MultiDiGraph.nodes_iter([data]) Return an iterator over the nodes. MultiDiGraph.__iter__() Iterate over the nodes. MultiDiGraph.edges([nbunch, data, keys]) Return a list of edges. MultiDiGraph.edges_iter([nbunch, data, keys]) Return an iterator over the edges. MultiDiGraph.out_edges([nbunch, keys, data]) Return a list of the outgoing edges. MultiDiGraph.out_edges_iter([nbunch, data, keys]) Return an iterator over the edges. MultiDiGraph.in_edges([nbunch, keys, data]) Return a list of the incoming edges. MultiDiGraph.in_edges_iter([nbunch, data, keys]) Return an iterator over the incoming edges. MultiDiGraph.get_edge_data(u, v[, key, default]) Return the attribute dictionary associated with edge (u,v). MultiDiGraph.neighbors(n) Return a list of successor nodes of n. MultiDiGraph.neighbors_iter(n) Return an iterator over successor nodes of n. MultiDiGraph.__getitem__(n) Return a dict of neighbors of node n. MultiDiGraph.successors(n) Return a list of successor nodes of n. MultiDiGraph.successors_iter(n) Return an iterator over successor nodes of n. MultiDiGraph.predecessors(n) Return a list of predecessor nodes of n. MultiDiGraph.predecessors_iter(n) Return an iterator over predecessor nodes of n. MultiDiGraph.adjacency_list() Return an adjacency list representation of the graph. MultiDiGraph.adjacency_iter() Return an iterator of (node, adjacency dict) tuples for all nodes. MultiDiGraph.nbunch_iter([nbunch]) Return an iterator of nodes contained in nbunch that are also in the graph. 106 Chapter 3. Graph types NetworkX Reference, Release 1.7 nodes MultiDiGraph.nodes(data=False) Return a list of the nodes in the graph. Parameters data : boolean, optional (default=False) If False return a list of nodes. If True return a two-tuple of node and node data dictionary Returns nlist : list A list of nodes. If data=True a list of two-tuples containing (node, node data dictionary). Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2]) >>> G.nodes() [0, 1, 2] >>> G.add_node(1, time=’5pm’) >>> G.nodes(data=True) [(0, {}), (1, {’time’: ’5pm’}), (2, {})] nodes_iter MultiDiGraph.nodes_iter(data=False) Return an iterator over the nodes. Parameters data : boolean, optional (default=False) If False the iterator returns nodes. If True return a two-tuple of node and node data dictionary Returns niter : iterator An iterator over nodes. If data=True the iterator gives two-tuples containing (node, node data, dictionary) Notes If the node data is not required it is simpler and equivalent to use the expression ‘for n in G’. >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2]) Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2]) >>> [d for n,d in G.nodes_iter(data=True)] [{}, {}, {}] 3.2. Basic graph types 107 NetworkX Reference, Release 1.7 __iter__ MultiDiGraph.__iter__() Iterate over the nodes. Use the expression ‘for n in G’. Returns niter : iterator An iterator over all nodes in the graph. Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) edges MultiDiGraph.edges(nbunch=None, data=False, keys=False) Return a list of edges. Edges are returned as tuples with optional data and keys in the order (node, neighbor, key, data). Parameters nbunch : iterable container, optional (default= all nodes) A container of nodes. The container will be iterated through once. data : bool, optional (default=False) Return two tuples (u,v) (False) or three-tuples (u,v,data) (True). keys : bool, optional (default=False) Return two tuples (u,v) (False) or three-tuples (u,v,key) (True). Returns edge_list: list of edge tuples : Edges that are adjacent to any node in nbunch, or a list of all edges if nbunch is not speciﬁed. See Also: edges_iter return an iterator over the edges Notes Nodes in nbunch that are not in the graph will be (quietly) ignored. For directed graphs this returns the out-edges. Examples >>> G = nx.MultiGraph() # or MultiDiGraph >>> G.add_path([0,1,2,3]) >>> G.edges() [(0, 1), (1, 2), (2, 3)] >>> G.edges(data=True) # default edge data is {} (empty dictionary) [(0, 1, {}), (1, 2, {}), (2, 3, {})] >>> G.edges(keys=True) # default keys are integers [(0, 1, 0), (1, 2, 0), (2, 3, 0)] 108 Chapter 3. Graph types NetworkX Reference, Release 1.7 >>> G.edges(data=True,keys=True) # default keys are integers [(0, 1, 0, {}), (1, 2, 0, {}), (2, 3, 0, {})] >>> G.edges([0,3]) [(0, 1), (3, 2)] >>> G.edges(0) [(0, 1)] edges_iter MultiDiGraph.edges_iter(nbunch=None, data=False, keys=False) Return an iterator over the edges. Edges are returned as tuples with optional data and keys in the order (node, neighbor, key, data). Parameters nbunch : iterable container, optional (default= all nodes) A container of nodes. The container will be iterated through once. data : bool, optional (default=False) If True, return edge attribute dict with each edge. keys : bool, optional (default=False) If True, return edge keys with each edge. Returns edge_iter : iterator An iterator of (u,v), (u,v,d) or (u,v,key,d) tuples of edges. See Also: edges return a list of edges Notes Nodes in nbunch that are not in the graph will be (quietly) ignored. For directed graphs this returns the out-edges. Examples >>> G = nx.MultiDiGraph() >>> G.add_path([0,1,2,3]) >>> [e for e in G.edges_iter()] [(0, 1), (1, 2), (2, 3)] >>> list(G.edges_iter(data=True)) # default data is {} (empty dict) [(0, 1, {}), (1, 2, {}), (2, 3, {})] >>> list(G.edges_iter([0,2])) [(0, 1), (2, 3)] >>> list(G.edges_iter(0)) [(0, 1)] out_edges MultiDiGraph.out_edges(nbunch=None, keys=False, data=False) Return a list of the outgoing edges. 3.2. Basic graph types 109 NetworkX Reference, Release 1.7 Edges are returned as tuples with optional data and keys in the order (node, neighbor, key, data). Parameters nbunch : iterable container, optional (default= all nodes) A container of nodes. The container will be iterated through once. data : bool, optional (default=False) If True, return edge attribute dict with each edge. keys : bool, optional (default=False) If True, return edge keys with each edge. Returns out_edges : list An listr of (u,v), (u,v,d) or (u,v,key,d) tuples of edges. See Also: in_edges return a list of incoming edges Notes Nodes in nbunch that are not in the graph will be (quietly) ignored. For directed graphs edges() is the same as out_edges(). out_edges_iter MultiDiGraph.out_edges_iter(nbunch=None, data=False, keys=False) Return an iterator over the edges. Edges are returned as tuples with optional data and keys in the order (node, neighbor, key, data). Parameters nbunch : iterable container, optional (default= all nodes) A container of nodes. The container will be iterated through once. data : bool, optional (default=False) If True, return edge attribute dict with each edge. keys : bool, optional (default=False) If True, return edge keys with each edge. Returns edge_iter : iterator An iterator of (u,v), (u,v,d) or (u,v,key,d) tuples of edges. See Also: edges return a list of edges Notes Nodes in nbunch that are not in the graph will be (quietly) ignored. For directed graphs this returns the out-edges. 110 Chapter 3. Graph types NetworkX Reference, Release 1.7 Examples >>> G = nx.MultiDiGraph() >>> G.add_path([0,1,2,3]) >>> [e for e in G.edges_iter()] [(0, 1), (1, 2), (2, 3)] >>> list(G.edges_iter(data=True)) # default data is {} (empty dict) [(0, 1, {}), (1, 2, {}), (2, 3, {})] >>> list(G.edges_iter([0,2])) [(0, 1), (2, 3)] >>> list(G.edges_iter(0)) [(0, 1)] in_edges MultiDiGraph.in_edges(nbunch=None, keys=False, data=False) Return a list of the incoming edges. Parameters nbunch : iterable container, optional (default= all nodes) A container of nodes. The container will be iterated through once. data : bool, optional (default=False) If True, return edge attribute dict with each edge. keys : bool, optional (default=False) If True, return edge keys with each edge. Returns in_edges : list A list of (u,v), (u,v,d) or (u,v,key,d) tuples of edges. See Also: out_edges return a list of outgoing edges in_edges_iter MultiDiGraph.in_edges_iter(nbunch=None, data=False, keys=False) Return an iterator over the incoming edges. Parameters nbunch : iterable container, optional (default= all nodes) A container of nodes. The container will be iterated through once. data : bool, optional (default=False) If True, return edge attribute dict with each edge. keys : bool, optional (default=False) If True, return edge keys with each edge. Returns in_edge_iter : iterator An iterator of (u,v), (u,v,d) or (u,v,key,d) tuples of edges. See Also: 3.2. Basic graph types 111 NetworkX Reference, Release 1.7 edges_iter return an iterator of edges get_edge_data MultiDiGraph.get_edge_data(u, v, key=None, default=None) Return the attribute dictionary associated with edge (u,v). Parameters u,v : nodes default: any Python object (default=None) : Value to return if the edge (u,v) is not found. key : hashable identiﬁer, optional (default=None) Return data only for the edge with speciﬁed key. Returns edge_dict : dictionary The edge attribute dictionary. Notes It is faster to use G[u][v][key]. >>> G = nx.MultiGraph() # or MultiDiGraph >>> G.add_edge(0,1,key=’a’,weight=7) >>> G[0][1][’a’] # key=’a’ {’weight’: 7} Warning: Assigning G[u][v][key] corrupts the graph data structure. But it is safe to assign attributes to that dictionary, >>> G[0][1][’a’][’weight’] = 10 >>> G[0][1][’a’][’weight’] 10 >>> G[1][0][’a’][’weight’] 10 Examples >>> G = nx.MultiGraph() # or MultiDiGraph >>> G.add_path([0,1,2,3]) >>> G.get_edge_data(0,1) {0: {}} >>> e = (0,1) >>> G.get_edge_data(*e) # tuple form {0: {}} >>> G.get_edge_data(’a’,’b’,default=0) # edge not in graph, return 0 0 neighbors MultiDiGraph.neighbors(n) Return a list of successor nodes of n. 112 Chapter 3. Graph types NetworkX Reference, Release 1.7 neighbors() and successors() are the same function. neighbors_iter MultiDiGraph.neighbors_iter(n) Return an iterator over successor nodes of n. neighbors_iter() and successors_iter() are the same. __getitem__ MultiDiGraph.__getitem__(n) Return a dict of neighbors of node n. Use the expression ‘G[n]’. Parameters n : node A node in the graph. Returns adj_dict : dictionary The adjacency dictionary for nodes connected to n. Notes G[n] is similar to G.neighbors(n) but the internal data dictionary is returned instead of a list. Assigning G[n] will corrupt the internal graph data structure. Use G[n] for reading data only. Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G[0] {1: {}} successors MultiDiGraph.successors(n) Return a list of successor nodes of n. neighbors() and successors() are the same function. successors_iter MultiDiGraph.successors_iter(n) Return an iterator over successor nodes of n. neighbors_iter() and successors_iter() are the same. 3.2. Basic graph types 113 NetworkX Reference, Release 1.7 predecessors MultiDiGraph.predecessors(n) Return a list of predecessor nodes of n. predecessors_iter MultiDiGraph.predecessors_iter(n) Return an iterator over predecessor nodes of n. adjacency_list MultiDiGraph.adjacency_list() Return an adjacency list representation of the graph. The output adjacency list is in the order of G.nodes(). For directed graphs, only outgoing adjacencies are included. Returns adj_list : lists of lists The adjacency structure of the graph as a list of lists. See Also: adjacency_iter Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G.adjacency_list() # in order given by G.nodes() [[1], [0, 2], [1, 3], [2]] adjacency_iter MultiDiGraph.adjacency_iter() Return an iterator of (node, adjacency dict) tuples for all nodes. This is the fastest way to look at every edge. For directed graphs, only outgoing adjacencies are included. Returns adj_iter : iterator An iterator of (node, adjacency dictionary) for all nodes in the graph. See Also: adjacency_list Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> [(n,nbrdict) for n,nbrdict in G.adjacency_iter()] [(0, {1: {}}), (1, {0: {}, 2: {}}), (2, {1: {}, 3: {}}), (3, {2: {}})] 114 Chapter 3. Graph types NetworkX Reference, Release 1.7 nbunch_iter MultiDiGraph.nbunch_iter(nbunch=None) Return an iterator of nodes contained in nbunch that are also in the graph. The nodes in nbunch are checked for membership in the graph and if not are silently ignored. Parameters nbunch : iterable container, optional (default=all nodes) A container of nodes. The container will be iterated through once. Returns niter : iterator An iterator over nodes in nbunch that are also in the graph. If nbunch is None, iterate over all nodes in the graph. Raises NetworkXError : If nbunch is not a node or or sequence of nodes. If a node in nbunch is not hashable. See Also: Graph.__iter__ Notes When nbunch is an iterator, the returned iterator yields values directly from nbunch, becoming exhausted when nbunch is exhausted. To test whether nbunch is a single node, one can use “if nbunch in self:”, even after processing with this routine. If nbunch is not a node or a (possibly empty) sequence/iterator or None, a NetworkXError is raised. Also, if any object in nbunch is not hashable, a NetworkXError is raised. Information about graph structure MultiDiGraph.has_node(n) Return True if the graph contains the node n. MultiDiGraph.__contains__(n) Return True if n is a node, False otherwise. Use the expression MultiDiGraph.has_edge(u, v[, key]) Return True if the graph has an edge between nodes u and v. MultiDiGraph.order() Return the number of nodes in the graph. MultiDiGraph.number_of_nodes() Return the number of nodes in the graph. MultiDiGraph.__len__() Return the number of nodes. MultiDiGraph.degree([nbunch, weight]) Return the degree of a node or nodes. MultiDiGraph.degree_iter([nbunch, weight]) Return an iterator for (node, degree). MultiDiGraph.in_degree([nbunch, weight]) Return the in-degree of a node or nodes. MultiDiGraph.in_degree_iter([nbunch, weight]) Return an iterator for (node, in-degree). MultiDiGraph.out_degree([nbunch, weight]) Return the out-degree of a node or nodes. MultiDiGraph.out_degree_iter([nbunch, weight]) Return an iterator for (node, out-degree). MultiDiGraph.size([weight]) Return the number of edges. MultiDiGraph.number_of_edges([u, v]) Return the number of edges between two nodes. MultiDiGraph.nodes_with_selfloops() Return a list of nodes with self loops. MultiDiGraph.selfloop_edges([data, keys]) Return a list of selﬂoop edges. MultiDiGraph.number_of_selfloops() Return the number of selﬂoop edges. 3.2. Basic graph types 115 NetworkX Reference, Release 1.7 has_node MultiDiGraph.has_node(n) Return True if the graph contains the node n. Parameters n : node Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2]) >>> G.has_node(0) True It is more readable and simpler to use >>> 0 in G True __contains__ MultiDiGraph.__contains__(n) Return True if n is a node, False otherwise. Use the expression ‘n in G’. Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> 1 in G True has_edge MultiDiGraph.has_edge(u, v, key=None) Return True if the graph has an edge between nodes u and v. Parameters u,v : nodes Nodes can be, for example, strings or numbers. key : hashable identiﬁer, optional (default=None) If speciﬁed return True only if the edge with key is found. Returns edge_ind : bool True if edge is in the graph, False otherwise. Examples Can be called either using two nodes u,v, an edge tuple (u,v), or an edge tuple (u,v,key). 116 Chapter 3. Graph types NetworkX Reference, Release 1.7 >>> G = nx.MultiGraph() # or MultiDiGraph >>> G.add_path([0,1,2,3]) >>> G.has_edge(0,1) # using two nodes True >>> e = (0,1) >>> G.has_edge(*e) # e is a 2-tuple (u,v) True >>> G.add_edge(0,1,key=’a’) >>> G.has_edge(0,1,key=’a’) # specify key True >>> e=(0,1,’a’) >>> G.has_edge(*e) # e is a 3-tuple (u,v,’a’) True The following syntax are equivalent: >>> G.has_edge(0,1) True >>> 1 in G[0] # though this gives KeyError if 0 not in G True order MultiDiGraph.order() Return the number of nodes in the graph. Returns nnodes : int The number of nodes in the graph. See Also: number_of_nodes, __len__ number_of_nodes MultiDiGraph.number_of_nodes() Return the number of nodes in the graph. Returns nnodes : int The number of nodes in the graph. See Also: order, __len__ Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2]) >>> len(G) 3 3.2. Basic graph types 117 NetworkX Reference, Release 1.7 __len__ MultiDiGraph.__len__() Return the number of nodes. Use the expression ‘len(G)’. Returns nnodes : int The number of nodes in the graph. Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> len(G) 4 degree MultiDiGraph.degree(nbunch=None, weight=None) Return the degree of a node or nodes. The node degree is the number of edges adjacent to that node. Parameters nbunch : iterable container, optional (default=all nodes) A container of nodes. The container will be iterated through once. weight : string or None, optional (default=None) The edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1. The degree is the sum of the edge weights adjacent to the node. Returns nd : dictionary, or number A dictionary with nodes as keys and degree as values or a number if a single node is speciﬁed. Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G.degree(0) 1 >>> G.degree([0,1]) {0: 1, 1: 2} >>> list(G.degree([0,1]).values()) [1, 2] degree_iter MultiDiGraph.degree_iter(nbunch=None, weight=None) Return an iterator for (node, degree). The node degree is the number of edges adjacent to the node. Parameters nbunch : iterable container, optional (default=all nodes) 118 Chapter 3. Graph types NetworkX Reference, Release 1.7 A container of nodes. The container will be iterated through once. weight : string or None, optional (default=None) The edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1. The degree is the sum of the edge weights. Returns nd_iter : an iterator The iterator returns two-tuples of (node, degree). See Also: degree Examples >>> G = nx.MultiDiGraph() >>> G.add_path([0,1,2,3]) >>> list(G.degree_iter(0)) # node 0 with degree 1 [(0, 1)] >>> list(G.degree_iter([0,1])) [(0, 1), (1, 2)] in_degree MultiDiGraph.in_degree(nbunch=None, weight=None) Return the in-degree of a node or nodes. The node in-degree is the number of edges pointing in to the node. Parameters nbunch : iterable container, optional (default=all nodes) A container of nodes. The container will be iterated through once. weight : string or None, optional (default=None) The edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1. The degree is the sum of the edge weights adjacent to the node. Returns nd : dictionary, or number A dictionary with nodes as keys and in-degree as values or a number if a single node is speciﬁed. See Also: degree, out_degree, in_degree_iter Examples >>> G = nx.DiGraph() # or MultiDiGraph >>> G.add_path([0,1,2,3]) >>> G.in_degree(0) 0 >>> G.in_degree([0,1]) {0: 0, 1: 1} >>> list(G.in_degree([0,1]).values()) [0, 1] 3.2. Basic graph types 119 NetworkX Reference, Release 1.7 in_degree_iter MultiDiGraph.in_degree_iter(nbunch=None, weight=None) Return an iterator for (node, in-degree). The node in-degree is the number of edges pointing in to the node. Parameters nbunch : iterable container, optional (default=all nodes) A container of nodes. The container will be iterated through once. weight : string or None, optional (default=None) The edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1. The degree is the sum of the edge weights adjacent to the node. Returns nd_iter : an iterator The iterator returns two-tuples of (node, in-degree). See Also: degree, in_degree, out_degree, out_degree_iter Examples >>> G = nx.MultiDiGraph() >>> G.add_path([0,1,2,3]) >>> list(G.in_degree_iter(0)) # node 0 with degree 0 [(0, 0)] >>> list(G.in_degree_iter([0,1])) [(0, 0), (1, 1)] out_degree MultiDiGraph.out_degree(nbunch=None, weight=None) Return the out-degree of a node or nodes. The node out-degree is the number of edges pointing out of the node. Parameters nbunch : iterable container, optional (default=all nodes) A container of nodes. The container will be iterated through once. weight : string or None, optional (default=None) The edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1. The degree is the sum of the edge weights adjacent to the node. Returns nd : dictionary, or number A dictionary with nodes as keys and out-degree as values or a number if a single node is speciﬁed. Examples 120 Chapter 3. Graph types NetworkX Reference, Release 1.7 >>> G = nx.DiGraph() # or MultiDiGraph >>> G.add_path([0,1,2,3]) >>> G.out_degree(0) 1 >>> G.out_degree([0,1]) {0: 1, 1: 1} >>> list(G.out_degree([0,1]).values()) [1, 1] out_degree_iter MultiDiGraph.out_degree_iter(nbunch=None, weight=None) Return an iterator for (node, out-degree). The node out-degree is the number of edges pointing out of the node. Parameters nbunch : iterable container, optional (default=all nodes) A container of nodes. The container will be iterated through once. weight : string or None, optional (default=None) The edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1. The degree is the sum of the edge weights. Returns nd_iter : an iterator The iterator returns two-tuples of (node, out-degree). See Also: degree, in_degree, out_degree, in_degree_iter Examples >>> G = nx.MultiDiGraph() >>> G.add_path([0,1,2,3]) >>> list(G.out_degree_iter(0)) # node 0 with degree 1 [(0, 1)] >>> list(G.out_degree_iter([0,1])) [(0, 1), (1, 1)] size MultiDiGraph.size(weight=None) Return the number of edges. Parameters weight : string or None, optional (default=None) The edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1. Returns nedges : int The number of edges of sum of edge weights in the graph. See Also: number_of_edges 3.2. Basic graph types 121 NetworkX Reference, Release 1.7 Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G.size() 3 >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_edge(’a’,’b’,weight=2) >>> G.add_edge(’b’,’c’,weight=4) >>> G.size() 2 >>> G.size(weight=’weight’) 6.0 number_of_edges MultiDiGraph.number_of_edges(u=None, v=None) Return the number of edges between two nodes. Parameters u,v : nodes, optional (default=all edges) If u and v are speciﬁed, return the number of edges between u and v. Otherwise return the total number of all edges. Returns nedges : int The number of edges in the graph. If nodes u and v are speciﬁed return the number of edges between those nodes. See Also: size Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> G.number_of_edges() 3 >>> G.number_of_edges(0,1) 1 >>> e = (0,1) >>> G.number_of_edges(*e) 1 nodes_with_selﬂoops MultiDiGraph.nodes_with_selfloops() Return a list of nodes with self loops. A node with a self loop has an edge with both ends adjacent to that node. Returns nodelist : list A list of nodes with self loops. 122 Chapter 3. Graph types NetworkX Reference, Release 1.7 See Also: selfloop_edges, number_of_selfloops Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_edge(1,1) >>> G.add_edge(1,2) >>> G.nodes_with_selfloops() [1] selﬂoop_edges MultiDiGraph.selfloop_edges(data=False, keys=False) Return a list of selﬂoop edges. A selﬂoop edge has the same node at both ends. Parameters data : bool, optional (default=False) Return selﬂoop edges as two tuples (u,v) (data=False) or three-tuples (u,v,data) (data=True) keys : bool, optional (default=False) If True, return edge keys with each edge. Returns edgelist : list of edge tuples A list of all selﬂoop edges. See Also: nodes_with_selfloops, number_of_selfloops Examples >>> G = nx.MultiGraph() # or MultiDiGraph >>> G.add_edge(1,1) >>> G.add_edge(1,2) >>> G.selfloop_edges() [(1, 1)] >>> G.selfloop_edges(data=True) [(1, 1, {})] >>> G.selfloop_edges(keys=True) [(1, 1, 0)] >>> G.selfloop_edges(keys=True, data=True) [(1, 1, 0, {})] number_of_selﬂoops MultiDiGraph.number_of_selfloops() Return the number of selﬂoop edges. A selﬂoop edge has the same node at both ends. 3.2. Basic graph types 123 NetworkX Reference, Release 1.7 Returns nloops : int The number of selﬂoops. See Also: nodes_with_selfloops, selfloop_edges Examples >>> G=nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_edge(1,1) >>> G.add_edge(1,2) >>> G.number_of_selfloops() 1 Making copies and subgraphs MultiDiGraph.copy() Return a copy of the graph. MultiDiGraph.to_undirected([reciprocal]) Return an undirected representation of the digraph. MultiDiGraph.to_directed() Return a directed copy of the graph. MultiDiGraph.subgraph(nbunch) Return the subgraph induced on nodes in nbunch. MultiDiGraph.reverse([copy]) Return the reverse of the graph. copy MultiDiGraph.copy() Return a copy of the graph. Returns G : Graph A copy of the graph. See Also: to_directed return a directed copy of the graph. Notes This makes a complete copy of the graph including all of the node or edge attributes. Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> H = G.copy() to_undirected MultiDiGraph.to_undirected(reciprocal=False) Return an undirected representation of the digraph. 124 Chapter 3. Graph types NetworkX Reference, Release 1.7 Parameters reciprocal : bool (optional) If True only keep edges that appear in both directions in the original digraph. Returns G : MultiGraph An undirected graph with the same name and nodes and with edge (u,v,data) if either (u,v,data) or (v,u,data) is in the digraph. If both edges exist in digraph and their edge data is different, only one edge is created with an arbitrary choice of which edge data to use. You must check and correct for this manually if desired. Notes This returns a “deepcopy” of the edge, node, and graph attributes which attempts to completely copy all of the data and references. This is in contrast to the similar D=DiGraph(G) which returns a shallow copy of the data. See the Python copy module for more information on shallow and deep copies, http://docs.python.org/library/copy.html. to_directed MultiDiGraph.to_directed() Return a directed copy of the graph. Returns G : MultiDiGraph A deepcopy of the graph. Notes If edges in both directions (u,v) and (v,u) exist in the graph, attributes for the new undirected edge will be a combination of the attributes of the directed edges. The edge data is updated in the (arbitrary) order that the edges are encountered. For more customized control of the edge attributes use add_edge(). This returns a “deepcopy” of the edge, node, and graph attributes which attempts to completely copy all of the data and references. This is in contrast to the similar G=DiGraph(D) which returns a shallow copy of the data. See the Python copy module for more information on shallow and deep copies, http://docs.python.org/library/copy.html. Examples >>> G = nx.Graph() # or MultiGraph, etc >>> G.add_path([0,1]) >>> H = G.to_directed() >>> H.edges() [(0, 1), (1, 0)] If already directed, return a (deep) copy 3.2. Basic graph types 125 NetworkX Reference, Release 1.7 >>> G = nx.MultiDiGraph() >>> G.add_path([0,1]) >>> H = G.to_directed() >>> H.edges() [(0, 1)] subgraph MultiDiGraph.subgraph(nbunch) Return the subgraph induced on nodes in nbunch. The induced subgraph of the graph contains the nodes in nbunch and the edges between those nodes. Parameters nbunch : list, iterable A container of nodes which will be iterated through once. Returns G : Graph A subgraph of the graph with the same edge attributes. Notes The graph, edge or node attributes just point to the original graph. So changes to the node or edge structure will not be reﬂected in the original graph while changes to the attributes will. To create a subgraph with its own copy of the edge/node attributes use: nx.Graph(G.subgraph(nbunch)) If edge attributes are containers, a deep copy can be obtained using: G.subgraph(nbunch).copy() For an inplace reduction of a graph to a subgraph you can remove nodes: G.remove_nodes_from([ n in G if n not in set(nbunch)]) Examples >>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc >>> G.add_path([0,1,2,3]) >>> H = G.subgraph([0,1,2]) >>> H.edges() [(0, 1), (1, 2)] reverse MultiDiGraph.reverse(copy=True) Return the reverse of the graph. The reverse is a graph with the same nodes and edges but with the directions of the edges reversed. Parameters copy : bool optional (default=True) If True, return a new DiGraph holding the reversed edges. If False, reverse the reverse graph is created using the original graph (this changes the original graph). 126 Chapter 3. Graph types CHAPTER FOUR ALGORITHMS 4.1 Approximation 4.1.1 Clique Cliques. max_clique(graph) Find the Maximum Clique clique_removal(graph) Repeatedly remove cliques from the graph. max_clique max_clique(graph) Find the Maximum Clique Finds the O(|V |/(log|V |)2) apx of maximum clique/independent set in the worst case. Parameters graph : NetworkX graph Undirected graph Returns clique : set The apx-maximum clique of the graph Notes A clique in an undirected graph G = (V, E) is a subset of the vertex set C ✓ V , such that for every two vertices in C, there exists an edge connecting the two. This is equivalent to saying that the subgraph induced by C is complete (in some cases, the term clique may also refer to the subgraph). A maximum clique is a clique of the largest possible size in a given graph. The clique number !(G) of a graph G is the number of vertices in a maximum clique in G. The intersection number of G is the smallest number of cliques that together cover all edges of G. http://en.wikipedia.org/wiki/Maximum_clique References [R104] 127 NetworkX Reference, Release 1.7 clique_removal clique_removal(graph) Repeatedly remove cliques from the graph. Results in a O(|V |/(log |V |)2) approximation of maximum clique & independent set. Returns the largest inde- pendent set found, along with found maximal cliques. Parameters graph : NetworkX graph Undirected graph Returns max_ind_cliques : (set, list) tuple Maximal independent set and list of maximal cliques (sets) in the graph. References [R103] 4.1.2 Dominating Set A dominating set for a graph G = (V, E) is a subset D of V such that every vertex not in D is joined to at least one member of D by some edge. The domination number gamma(G) is the number of vertices in a smallest dominating set for G. Given a graph G = (V, E) ﬁnd a minimum weight dominating set V’. http://en.wikipedia.org/wiki/Dominating_set This is reducible to the minimum set dom_set problem. min_weighted_dominating_set(graph[, weight]) Return minimum weight dominating set. min_edge_dominating_set(graph) Return minimum weight dominating edge set. min_weighted_dominating_set min_weighted_dominating_set(graph, weight=None) Return minimum weight dominating set. Parameters graph : NetworkX graph Undirected graph weight : None or string, optional (default = None) If None, every edge has weight/distance/weight 1. If a string, use this edge attribute as the edge weight. Any edge attribute not present defaults to 1. Returns min_weight_dominating_set : set Returns a set of vertices whose weight sum is no more than 1 + log w(V) References [R105] 128 Chapter 4. Algorithms NetworkX Reference, Release 1.7 min_edge_dominating_set min_edge_dominating_set(graph) Return minimum weight dominating edge set. Parameters graph : NetworkX graph Undirected graph Returns min_edge_dominating_set : set Returns a set of dominating edges whose size is no more than 2 * OPT. 4.1.3 Independent Set Independent Set Independent set or stable set is a set of vertices in a graph, no two of which are adjacent. That is, it is a set I of vertices such that for every two vertices in I, there is no edge connecting the two. Equivalently, each edge in the graph has at most one endpoint in I. The size of an independent set is the number of vertices it contains. A maximum independent set is a largest independent set for a given graph G and its size is denoted ↵(G). The problem of ﬁnding such a set is called the maximum independent set problem and is an NP-hard optimization problem. As such, it is unlikely that there exists an efﬁcient algorithm for ﬁnding a maximum independent set of a graph. http://en.wikipedia.org/wiki/Independent_set_(graph_theory) Independent set algorithm is based on the following paper: O(|V |/(log|V |)2) apx of maximum clique/independent set. Boppana, R., & Halldórsson, M. M. (1992). Approximating maximum independent sets by excluding subgraphs. BIT Numerical Mathematics, 32(2), 180–196. Springer. doi:10.1007/BF01994876 maximum_independent_set(graph) Return an approximate maximum independent set. maximum_independent_set maximum_independent_set(graph) Return an approximate maximum independent set. Parameters graph : NetworkX graph Undirected graph Returns iset : Set The apx-maximum independent set Notes Finds the O(|V |/(log|V |)2) apx of independent set in the worst case. References [R106] 4.1. Approximation 129 NetworkX Reference, Release 1.7 4.1.4 Matching Given a graph G = (V,E), a matching M in G is a set of pairwise non-adjacent edges; that is, no two edges share a common vertex. http://en.wikipedia.org/wiki/Matching_(graph_theory) min_maximal_matching(graph) Returns a set of edges such that no two edges share a common endpoint and every edge not in the set shares some common endpoint in the set. min_maximal_matching min_maximal_matching(graph) Returns a set of edges such that no two edges share a common endpoint and every edge not in the set shares some common endpoint in the set. Parameters graph : NetworkX graph Undirected graph Returns min_maximal_matching : set Returns a set of edges such that no two edges share a common endpoint and every edge not in the set shares some common endpoint in the set. Cardinality will be 2*OPT in the worst case. References [R107] 4.1.5 Ramsey Ramsey numbers. ramsey_R2(graph) Approximately computes the Ramsey number R(2; s, t) for graph. ramsey_R2 ramsey_R2(graph) Approximately computes the Ramsey number R(2; s, t) for graph. Parameters graph : NetworkX graph Undirected graph Returns max_pair : (set, set) tuple Maximum clique, Maximum independent set. 4.1.6 Vertex Cover Given an undirected graph G =(V,E) and a function w assigning nonnegative weights to its vertices, ﬁnd a minimum weight subset of V such that each edge in E is incident to at least one vertex in the subset. http://en.wikipedia.org/wiki/Vertex_cover 130 Chapter 4. Algorithms NetworkX Reference, Release 1.7 min_weighted_vertex_cover(graph[, weight]) 2-OPT Local Ratio for Minimum Weighted Vertex Cover min_weighted_vertex_cover min_weighted_vertex_cover(graph, weight=None) 2-OPT Local Ratio for Minimum Weighted Vertex Cover Find an approximate minimum weighted vertex cover of a graph. Parameters graph : NetworkX graph Undirected graph weight : None or string, optional (default = None) If None, every edge has weight/distance/cost 1. If a string, use this edge attribute as the edge weight. Any edge attribute not present defaults to 1. Returns min_weighted_cover : set Returns a set of vertices whose weight sum is no more than 2 * OPT. References [R108] 4.2 Assortativity 4.2.1 Assortativity degree_assortativity_coefficient(G[, x, y, ...]) Compute degree assortativity of graph. attribute_assortativity_coefficient(G, attribute) Compute assortativity for node attributes. numeric_assortativity_coefficient(G, attribute) Compute assortativity for numerical node attributes. degree_pearson_correlation_coefficient(G[, ...]) Compute degree assortativity of graph. degree_assortativity_coefﬁcient degree_assortativity_coefficient(G, x=’out’, y=’in’, weight=None, nodes=None) Compute degree assortativity of graph. Assortativity measures the similarity of connections in the graph with respect to the node degree. Parameters G : NetworkX graph x: string (‘in’,’out’) : The degree type for source node (directed graphs only). y: string (‘in’,’out’) : The degree type for target node (directed graphs only). 4.2. Assortativity 131 NetworkX Reference, Release 1.7 weight: string or None, optional (default=None) : The edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1. The degree is the sum of the edge weights adjacent to the node. nodes: list or iterable (optional) : Compute degree assortativity only for nodes in container. The default is all nodes. Returns r : ﬂoat Assortativity of graph by degree. See Also: attribute_assortativity_coefficient, numeric_assortativity_coefficient, neighbor_connectivity, degree_mixing_dict, degree_mixing_matrix Notes This computes Eq. (21) in Ref. [R112] , where e is the joint probability distribution (mixing matrix) of the degrees. If G is directed than the matrix e is the joint probability of the user-speciﬁed degree type for the source and target. References [R112], [R113] Examples >>> G=nx.path_graph(4) >>> r=nx.degree_assortativity_coefficient(G) >>> print("%3.1f"%r) -0.5 attribute_assortativity_coefﬁcient attribute_assortativity_coefficient(G, attribute, nodes=None) Compute assortativity for node attributes. Assortativity measures the similarity of connections in the graph with respect to the given attribute. Parameters G : NetworkX graph attribute : string Node attribute key nodes: list or iterable (optional) : Compute attribute assortativity for nodes in container. The default is all nodes. Returns r: ﬂoat : Assortativity of graph for given attribute 132 Chapter 4. Algorithms NetworkX Reference, Release 1.7 Notes This computes Eq. (2) in Ref. [R109] , trace(M)-sum(M))/(1-sum(M), where M is the joint probability distribu- tion (mixing matrix) of the speciﬁed attribute. References [R109] Examples >>> G=nx.Graph() >>> G.add_nodes_from([0,1],color=’red’) >>> G.add_nodes_from([2,3],color=’blue’) >>> G.add_edges_from([(0,1),(2,3)]) >>> print(nx.attribute_assortativity_coefficient(G,’color’)) 1.0 numeric_assortativity_coefﬁcient numeric_assortativity_coefficient(G, attribute, nodes=None) Compute assortativity for numerical node attributes. Assortativity measures the similarity of connections in the graph with respect to the given numeric attribute. Parameters G : NetworkX graph attribute : string Node attribute key nodes: list or iterable (optional) : Compute numeric assortativity only for attributes of nodes in container. The default is all nodes. Returns r: ﬂoat : Assortativity of graph for given attribute Notes This computes Eq. (21) in Ref. [R117] , for the mixing matrix of of the speciﬁed attribute. References [R117] Examples 4.2. Assortativity 133 NetworkX Reference, Release 1.7 >>> G=nx.Graph() >>> G.add_nodes_from([0,1],size=2) >>> G.add_nodes_from([2,3],size=3) >>> G.add_edges_from([(0,1),(2,3)]) >>> print(nx.numeric_assortativity_coefficient(G,’size’)) 1.0 degree_pearson_correlation_coefﬁcient degree_pearson_correlation_coefficient(G, x=’out’, y=’in’, weight=None, nodes=None) Compute degree assortativity of graph. Assortativity measures the similarity of connections in the graph with respect to the node degree. This is the same as degree_assortativity_coefﬁcient but uses the potentially faster scipy.stats.pearsonr function. Parameters G : NetworkX graph x: string (‘in’,’out’) : The degree type for source node (directed graphs only). y: string (‘in’,’out’) : The degree type for target node (directed graphs only). weight: string or None, optional (default=None) : The edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1. The degree is the sum of the edge weights adjacent to the node. nodes: list or iterable (optional) : Compute pearson correlation of degrees only for speciﬁed nodes. The default is all nodes. Returns r : ﬂoat Assortativity of graph by degree. Notes This calls scipy.stats.pearsonr. References [R114], [R115] Examples >>> G=nx.path_graph(4) >>> r=nx.degree_pearson_correlation_coefficient(G) >>> r -0.5 134 Chapter 4. Algorithms NetworkX Reference, Release 1.7 4.2.2 Average neighbor degree average_neighbor_degree(G[, source, target, ...]) Returns the average degree of the neighborhood of each node. average_neighbor_degree average_neighbor_degree(G, source=’out’, target=’out’, nodes=None, weight=None) Returns the average degree of the neighborhood of each node. The average degree of a node i is knn,i = 1 |N(i)| Xj2N(i) kj where N(i) are the neighbors of node i and kj is the degree of node j which belongs to N(i). For weighted graphs, an analogous measure can be deﬁned [R111], kw nn,i = 1 si Xj2N(i) wijkj where si is the weighted degree of node i, wij is the weight of the edge that links i and j and N(i) are the neighbors of node i. Parameters G : NetworkX graph source : string (“in”|”out”) Directed graphs only. Use “in”- or “out”-degree for source node. target : string (“in”|”out”) Directed graphs only. Use “in”- or “out”-degree for target node. nodes : list or iterable, optional Compute neighbor degree for speciﬁed nodes. The default is all nodes in the graph. weight : string or None, optional (default=None) The edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1. Returns d: dict : A dictionary keyed by node with average neighbors degree value. See Also: average_degree_connectivity Notes For directed graphs you can also specify in-degree or out-degree by passing keyword arguments. References [R111] 4.2. Assortativity 135 NetworkX Reference, Release 1.7 Examples >>> G=nx.path_graph(4) >>> G.edge[0][1][’weight’] = 5 >>> G.edge[2][3][’weight’] = 3 >>> nx.average_neighbor_degree(G) {0: 2.0, 1: 1.5, 2: 1.5, 3: 2.0} >>> nx.average_neighbor_degree(G, weight=’weight’) {0: 2.0, 1: 1.1666666666666667, 2: 1.25, 3: 2.0} >>> G=nx.DiGraph() >>> G.add_path([0,1,2,3]) >>> nx.average_neighbor_degree(G, source=’in’, target=’in’) {0: 1.0, 1: 1.0, 2: 1.0, 3: 0.0} >>> nx.average_neighbor_degree(G, source=’out’, target=’out’) {0: 1.0, 1: 1.0, 2: 0.0, 3: 0.0} 4.2.3 Average degree connectivity average_degree_connectivity(G[, source, ...]) Compute the average degree connectivity of graph. k_nearest_neighbors(G[, source, target, ...]) Compute the average degree connectivity of graph. average_degree_connectivity average_degree_connectivity(G, source=’in+out’, target=’in+out’, nodes=None, weight=None) Compute the average degree connectivity of graph. The average degree connectivity is the average nearest neighbor degree of nodes with degree k. For weighted graphs, an analogous measure can be computed using the weighted average neighbors degree deﬁned in [R110], for a node i, as: kw nn,i = 1 si Xj2N(i) wijkj where si is the weighted degree of node i, wij is the weight of the edge that links i and j, and N(i) are the neighbors of node i. Parameters G : NetworkX graph source : “in”|”out”|”in+out” (default:”in+out”) Directed graphs only. Use “in”- or “out”-degree for source node. target : “in”|”out”|”in+out” (default:”in+out” Directed graphs only. Use “in”- or “out”-degree for target node. nodes: list or iterable (optional) : Compute neighbor connectivity for these nodes. The default is all nodes. weight : string or None, optional (default=None) The edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1. 136 Chapter 4. Algorithms NetworkX Reference, Release 1.7 Returns d: dict : A dictionary keyed by degree k with the value of average connectivity. See Also: neighbors_average_degree Notes This algorithm is sometimes called “k nearest neighbors’. References [R110] Examples >>> G=nx.path_graph(4) >>> G.edge[1][2][’weight’] = 3 >>> nx.k_nearest_neighbors(G) {1: 2.0, 2: 1.5} >>> nx.k_nearest_neighbors(G, weight=’weight’) {1: 2.0, 2: 1.75} k_nearest_neighbors k_nearest_neighbors(G, source=’in+out’, target=’in+out’, nodes=None, weight=None) Compute the average degree connectivity of graph. The average degree connectivity is the average nearest neighbor degree of nodes with degree k. For weighted graphs, an analogous measure can be computed using the weighted average neighbors degree deﬁned in [R116], for a node i, as: kw nn,i = 1 si Xj2N(i) wijkj where si is the weighted degree of node i, wij is the weight of the edge that links i and j, and N(i) are the neighbors of node i. Parameters G : NetworkX graph source : “in”|”out”|”in+out” (default:”in+out”) Directed graphs only. Use “in”- or “out”-degree for source node. target : “in”|”out”|”in+out” (default:”in+out” Directed graphs only. Use “in”- or “out”-degree for target node. nodes: list or iterable (optional) : Compute neighbor connectivity for these nodes. The default is all nodes. weight : string or None, optional (default=None) 4.2. Assortativity 137 NetworkX Reference, Release 1.7 The edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1. Returns d: dict : A dictionary keyed by degree k with the value of average connectivity. See Also: neighbors_average_degree Notes This algorithm is sometimes called “k nearest neighbors’. References [R116] Examples >>> G=nx.path_graph(4) >>> G.edge[1][2][’weight’] = 3 >>> nx.k_nearest_neighbors(G) {1: 2.0, 2: 1.5} >>> nx.k_nearest_neighbors(G, weight=’weight’) {1: 2.0, 2: 1.75} 4.2.4 Mixing attribute_mixing_matrix(G, attribute[, ...]) Return mixing matrix for attribute. degree_mixing_matrix(G[, x, y, weight, ...]) Return mixing matrix for attribute. degree_mixing_dict(G[, x, y, weight, nodes, ...]) Return dictionary representation of mixing matrix for degree. attribute_mixing_dict(G, attribute[, nodes, ...]) Return dictionary representation of mixing matrix for attribute. attribute_mixing_matrix attribute_mixing_matrix(G, attribute, nodes=None, mapping=None, normalized=True) Return mixing matrix for attribute. Parameters G : graph NetworkX graph object. attribute : string Node attribute key. nodes: list or iterable (optional) : Use only nodes in container to build the matrix. The default is all nodes. mapping : dictionary, optional 138 Chapter 4. Algorithms NetworkX Reference, Release 1.7 Mapping from node attribute to integer index in matrix. If not speciﬁed, an arbitrary ordering will be used. normalized : bool (default=False) Return counts if False or probabilities if True. Returns m: numpy array : Counts or joint probability of occurrence of attribute pairs. degree_mixing_matrix degree_mixing_matrix(G, x=’out’, y=’in’, weight=None, nodes=None, normalized=True) Return mixing matrix for attribute. Parameters G : graph NetworkX graph object. x: string (‘in’,’out’) : The degree type for source node (directed graphs only). y: string (‘in’,’out’) : The degree type for target node (directed graphs only). nodes: list or iterable (optional) : Build the matrix using only nodes in container. The default is all nodes. weight: string or None, optional (default=None) : The edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1. The degree is the sum of the edge weights adjacent to the node. normalized : bool (default=False) Return counts if False or probabilities if True. Returns m: numpy array : Counts, or joint probability, of occurrence of node degree. degree_mixing_dict degree_mixing_dict(G, x=’out’, y=’in’, weight=None, nodes=None, normalized=False) Return dictionary representation of mixing matrix for degree. Parameters G : graph NetworkX graph object. x: string (‘in’,’out’) : The degree type for source node (directed graphs only). y: string (‘in’,’out’) : The degree type for target node (directed graphs only). weight: string or None, optional (default=None) : The edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1. The degree is the sum of the edge weights adjacent to the node. 4.2. Assortativity 139 NetworkX Reference, Release 1.7 normalized : bool (default=False) Return counts if False or probabilities if True. Returns d: dictionary : Counts or joint probability of occurrence of degree pairs. attribute_mixing_dict attribute_mixing_dict(G, attribute, nodes=None, normalized=False) Return dictionary representation of mixing matrix for attribute. Parameters G : graph NetworkX graph object. attribute : string Node attribute key. nodes: list or iterable (optional) : Unse nodes in container to build the dict. The default is all nodes. normalized : bool (default=False) Return counts if False or probabilities if True. Returns d : dictionary Counts or joint probability of occurrence of attribute pairs. Examples >>> G=nx.Graph() >>> G.add_nodes_from([0,1],color=’red’) >>> G.add_nodes_from([2,3],color=’blue’) >>> G.add_edge(1,3) >>> d=nx.attribute_mixing_dict(G,’color’) >>> print(d[’red’][’blue’]) 1 >>> print(d[’blue’][’red’]) # d symmetric for undirected graphs 1 4.3 Bipartite This module provides functions and operations for bipartite graphs. Bipartite graphs B =(U, V, E) have two node sets U, V and edges in E that only connect nodes from opposite sets. It is common in the literature to use an spatial analogy referring to the two node sets as top and bottom nodes. The bipartite algorithms are not imported into the networkx namespace at the top level so the easiest way to use them is with: >>> import networkx as nx >>> from networkx.algorithms import bipartite 140 Chapter 4. Algorithms NetworkX Reference, Release 1.7 NetworkX does not have a custom bipartite graph class but the Graph() or DiGraph() classes can be used to represent bipartite graphs. However, you have to keep track of which set each node belongs to, and make sure that there is no edge between nodes of the same set. The convention used in NetworkX is to use a node attribute named “bipartite” with values 0 or 1 to identify the sets each node belongs to. For example: >>> B = nx.Graph() >>> B.add_nodes_from([1,2,3,4], bipartite=0) # Add the node attribute "bipartite" >>> B.add_nodes_from([’a’,’b’,’c’], bipartite=1) >>> B.add_edges_from([(1,’a’), (1,’b’), (2,’b’), (2,’c’), (3,’c’), (4,’a’)]) Many algorithms of the bipartite module of NetworkX require, as an argument, a container with all the nodes that belong to one set, in addition to the bipartite graph B. If B is connected, you can ﬁnd the node sets using a two- coloring algorithm: >>> nx.is_connected(B) True >>> bottom_nodes, top_nodes = bipartite.sets(B) >>> list(top_nodes) [1, 2, 3, 4] >>> list(bottom_nodes) [’a’, ’c’, ’b’] However, if the input graph is not connected, there are more than one possible colorations. Thus, the following result is correct: >>> B.remove_edge(2,’c’) >>> nx.is_connected(B) False >>> bottom_nodes, top_nodes = bipartite.sets(B) >>> list(top_nodes) [1, 2, 4, ’c’] >>> list(bottom_nodes) [’a’, 3, ’b’] Using the “bipartite” node attribute, you can easily get the two node sets: >>> top_nodes = set(n for n,d in B.nodes(data=True) if d[’bipartite’]==0) >>> bottom_nodes = set(B) - top_nodes >>> list(top_nodes) [1, 2, 3, 4] >>> list(bottom_nodes) [’a’, ’c’, ’b’] So you can easily use the bipartite algorithms that require, as an argument, a container with all nodes that belong to one node set: >>> print(round(bipartite.density(B, bottom_nodes),2)) 0.42 >>> G = bipartite.projected_graph(B, top_nodes) >>> G.edges() [(1, 2), (1, 4)] All bipartite graph generators in NetworkX build bipartite graphs with the “bipartite” node attribute. Thus, you can use the same approach: >>> RB = nx.bipartite_random_graph(5, 7, 0.2) >>> nx.is_connected(RB) False 4.3. Bipartite 141 NetworkX Reference, Release 1.7 >>> RB_top = set(n for n,d in RB.nodes(data=True) if d[’bipartite’]==0) >>> RB_bottom = set(RB) - RB_top >>> list(RB_top) [0, 1, 2, 3, 4] >>> list(RB_bottom) [5, 6, 7, 8, 9, 10, 11] For other bipartite graph generators see the bipartite section of Graph generators. 4.3.1 Basic functions is_bipartite(G) Returns True if graph G is bipartite, False if not. is_bipartite_node_set(G, nodes) Returns True if nodes and G/nodes are a bipartition of G. sets(G) Returns bipartite node sets of graph G. color(G) Returns a two-coloring of the graph. density(B, nodes) Return density of bipartite graph B. degrees(B, nodes[, weight]) Return the degrees of the two node sets in the bipartite graph B. biadjacency_matrix(G, row_order[, ...]) Return the biadjacency matrix of the bipartite graph G. is_bipartite is_bipartite(G) Returns True if graph G is bipartite, False if not. Parameters G : NetworkX graph See Also: color, is_bipartite_node_set Examples >>> from networkx.algorithms import bipartite >>> G = nx.path_graph(4) >>> print(bipartite.is_bipartite(G)) True is_bipartite_node_set is_bipartite_node_set(G, nodes) Returns True if nodes and G/nodes are a bipartition of G. Parameters G : NetworkX graph nodes: list or container : Check if nodes are a one of a bipartite set. Notes For connected graphs the bipartite sets are unique. This function handles disconnected graphs. 142 Chapter 4. Algorithms NetworkX Reference, Release 1.7 Examples >>> from networkx.algorithms import bipartite >>> G = nx.path_graph(4) >>> X = set([1,3]) >>> bipartite.is_bipartite_node_set(G,X) True sets sets(G) Returns bipartite node sets of graph G. Raises an exception if the graph is not bipartite. Parameters G : NetworkX graph Returns (X,Y) : two-tuple of sets One set of nodes for each part of the bipartite graph. See Also: color Examples >>> from networkx.algorithms import bipartite >>> G = nx.path_graph(4) >>> X, Y = bipartite.sets(G) >>> list(X) [0, 2] >>> list(Y) [1, 3] color color(G) Returns a two-coloring of the graph. Raises an exception if the graph is not bipartite. Parameters G : NetworkX graph Returns color : dictionary A dictionary keyed by node with a 1 or 0 as data for each node color. Raises NetworkXError if the graph is not two-colorable. : Examples >>> from networkx.algorithms import bipartite >>> G = nx.path_graph(4) >>> c = bipartite.color(G) 4.3. Bipartite 143 NetworkX Reference, Release 1.7 >>> print(c) {0: 1, 1: 0, 2: 1, 3: 0} You can use this to set a node attribute indicating the biparite set: >>> nx.set_node_attributes(G, ’bipartite’, c) >>> print(G.node[0][’bipartite’]) 1 >>> print(G.node[1][’bipartite’]) 0 density density(B, nodes) Return density of bipartite graph B. Parameters G : NetworkX graph nodes: list or container : Nodes in one set of the bipartite graph. Returns d : ﬂoat The bipartite density See Also: color Examples >>> from networkx.algorithms import bipartite >>> G = nx.complete_bipartite_graph(3,2) >>> X=set([0,1,2]) >>> bipartite.density(G,X) 1.0 >>> Y=set([3,4]) >>> bipartite.density(G,Y) 1.0 degrees degrees(B, nodes, weight=None) Return the degrees of the two node sets in the bipartite graph B. Parameters G : NetworkX graph nodes: list or container : Nodes in one set of the bipartite graph. weight : string or None, optional (default=None) The edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1. The degree is the sum of the edge weights adjacent to the node. Returns (degX,degY) : tuple of dictionaries 144 Chapter 4. Algorithms NetworkX Reference, Release 1.7 The degrees of the two bipartite sets as dictionaries keyed by node. See Also: color, density Examples >>> from networkx.algorithms import bipartite >>> G = nx.complete_bipartite_graph(3,2) >>> Y=set([3,4]) >>> degX,degY=bipartite.degrees(G,Y) >>> degX {0: 2, 1: 2, 2: 2} biadjacency_matrix biadjacency_matrix(G, row_order, column_order=None, weight=’weight’, dtype=None) Return the biadjacency matrix of the bipartite graph G. Let G =(U, V, E) be a bipartite graph with node sets U = u1,...,ur and V = v1,...,vs. The biadjacency matrix [1] is the r x s matrix B in which bi,j =1if, and only if, (ui,vj) 2 E. If the parameter weight is not Noneand matches the name of an edge attribute, its value is used instead of 1. Parameters G : graph A NetworkX graph row_order : list of nodes The rows of the matrix are ordered according to the list of nodes. column_order : list, optional The columns of the matrix are ordered according to the list of nodes. If column_order is None, then the ordering of columns is arbitrary. weight : string or None, optional (default=’weight’) The edge data key used to provide each value in the matrix. If None, then each edge has weight 1. dtype : NumPy data type, optional A valid single NumPy data type used to initialize the array. This must be a simple type such as int or numpy.ﬂoat64 and not a compound data type (see to_numpy_recarray) If None, then the NumPy default is used. Returns B : numpy matrix Biadjacency matrix representation of the bipartite graph G. See Also: to_numpy_matrix, adjacency_matrix Notes No attempt is made to check that the input graph is bipartite. 4.3. Bipartite 145 NetworkX Reference, Release 1.7 For directed bipartite graphs only successors are considered as neighbors. To obtain an adjacency matrix with ones (or weight values) for both predecessors and successors you have to generate two biadjacency matrices where the rows of one of them are the columns of the other, and then add one to the transpose of the other. References [1] http://en.wikipedia.org/wiki/Adjacency_matrix#Adjacency_matrix_of_a_bipartite_graph 4.3.2 Projections One-mode (unipartite) projections of bipartite graphs. projected_graph(B, nodes[, multigraph]) Returns the projection of B onto one of its node sets. weighted_projected_graph(B, nodes[, ratio]) Returns a weighted projection of B onto one of its node sets. collaboration_weighted_projected_graph(B, nodes) Newman’s weighted projection of B onto one of its node sets. overlap_weighted_projected_graph(B, nodes[, ...]) Overlap weighted projection of B onto one of its node sets. generic_weighted_projected_graph(B, nodes[, ...]) Weighted projection of B with a user-speciﬁed weight function. projected_graph projected_graph(B, nodes, multigraph=False) Returns the projection of B onto one of its node sets. Returns the graph G that is the projection of the bipartite graph B onto the speciﬁed nodes. They retain their attributes and are connected in G if they have a common neighbor in B. Parameters B : NetworkX graph The input graph should be bipartite. nodes : list or iterable Nodes to project onto (the “bottom” nodes). multigraph: bool (default=False) : If True return a multigraph where the multiple edges represent multiple shared neigh- bors. They edge key in the multigraph is assigned to the label of the neighbor. Returns Graph : NetworkX graph or multigraph A graph that is the projection onto the given nodes. See Also: is_bipartite, is_bipartite_node_set, sets, weighted_projected_graph, collaboration_weighted_projected_graph, overlap_weighted_projected_graph, generic_weighted_projected_graph Notes No attempt is made to verify that the input graph B is bipartite. Returns a simple graph that is the projection of the bipartite graph B onto the set of nodes given in list nodes. If multigraph=True then a multigraph is returned with an edge for every shared neighbor. 146 Chapter 4. Algorithms NetworkX Reference, Release 1.7 Directed graphs are allowed as input. The output will also then be a directed graph with edges if there is a directed path between the nodes. The graph and node properties are (shallow) copied to the projected graph. Examples >>> from networkx.algorithms import bipartite >>> B = nx.path_graph(4) >>> G = bipartite.projected_graph(B, [1,3]) >>> print(G.nodes()) [1, 3] >>> print(G.edges()) [(1, 3)] If nodes a, and b are connected through both nodes 1 and 2 then building a multigraph results in two edges in the projection onto [a,‘b‘]: >>> B = nx.Graph() >>> B.add_edges_from([(’a’, 1), (’b’, 1), (’a’, 2), (’b’, 2)]) >>> G = bipartite.projected_graph(B, [’a’, ’b’], multigraph=True) >>> print(G.edges(keys=True)) [(’a’, ’b’, 1), (’a’, ’b’, 2)] weighted_projected_graph weighted_projected_graph(B, nodes, ratio=False) Returns a weighted projection of B onto one of its node sets. The weighted projected graph is the projection of the bipartite network B onto the speciﬁed nodes with weights representing the number of shared neighbors or the ratio between actual shared neighbors and possible shared neighbors if ratio=True [R125]. The nodes retain their attributes and are connected in the resulting graph if they have an edge to a common node in the original graph. Parameters B : NetworkX graph The input graph should be bipartite. nodes : list or iterable Nodes to project onto (the “bottom” nodes). ratio: Bool (default=False) : If True, edge weight is the ratio between actual shared neighbors and possible shared neighbors. If False, edges weight is the number of shared neighbors. Returns Graph : NetworkX graph A graph that is the projection onto the given nodes. See Also: is_bipartite, is_bipartite_node_set, sets, collaboration_weighted_projected_graph, overlap_weighted_projected_graph, generic_weighted_projected_graph, projected_graph 4.3. Bipartite 147 NetworkX Reference, Release 1.7 Notes No attempt is made to verify that the input graph B is bipartite. The graph and node properties are (shallow) copied to the projected graph. References [R125] Examples >>> from networkx.algorithms import bipartite >>> B = nx.path_graph(4) >>> G = bipartite.weighted_projected_graph(B, [1,3]) >>> print(G.nodes()) [1, 3] >>> print(G.edges(data=True)) [(1, 3, {’weight’: 1})] >>> G = bipartite.weighted_projected_graph(B, [1,3], ratio=True) >>> print(G.edges(data=True)) [(1, 3, {’weight’: 0.5})] collaboration_weighted_projected_graph collaboration_weighted_projected_graph(B, nodes) Newman’s weighted projection of B onto one of its node sets. The collaboration weighted projection is the projection of the bipartite network B onto the speciﬁed nodes with weights assigned using Newman’s collaboration model [R123]: wv,u = Xk w v k w kw 1 where v and u are nodes from the same bipartite node set, and w is a node of the opposite node set. The value kw is the degree of node w in the bipartite network and w v is 1 if node v is linked to node w in the original bipartite graph or 0 otherwise. The nodes retain their attributes and are connected in the resulting graph if have an edge to a common node in the original bipartite graph. Parameters B : NetworkX graph The input graph should be bipartite. nodes : list or iterable Nodes to project onto (the “bottom” nodes). Returns Graph : NetworkX graph A graph that is the projection onto the given nodes. See Also: is_bipartite, is_bipartite_node_set, sets, weighted_projected_graph, overlap_weighted_projected_graph, generic_weighted_projected_graph, projected_graph 148 Chapter 4. Algorithms NetworkX Reference, Release 1.7 Notes No attempt is made to verify that the input graph B is bipartite. The graph and node properties are (shallow) copied to the projected graph. References [R123] Examples >>> from networkx.algorithms import bipartite >>> B = nx.path_graph(5) >>> B.add_edge(1,5) >>> G = bipartite.collaboration_weighted_projected_graph(B, [0, 2, 4, 5]) >>> print(G.nodes()) [0, 2, 4, 5] >>> for edge in G.edges(data=True): print(edge) ... (0, 2, {’weight’: 0.5}) (0, 5, {’weight’: 0.5}) (2, 4, {’weight’: 1.0}) (2, 5, {’weight’: 0.5}) overlap_weighted_projected_graph overlap_weighted_projected_graph(B, nodes, jaccard=True) Overlap weighted projection of B onto one of its node sets. The overlap weighted projection is the projection of the bipartite network B onto the speciﬁed nodes with weights representing the Jaccard index between the neighborhoods of the two nodes in the original bipartite network [R124]: wv,u = |N(u) \ N(v)| |N(u) [ N(v)| or if the parameter ‘jaccard’ is False, the fraction of common neighbors by minimum of both nodes degree in the original bipartite graph [R124]: wv,u = |N(u) \ N(v)|min(|N(u)|, |N(v)|) The nodes retain their attributes and are connected in the resulting graph if have an edge to a common node in the original bipartite graph. Parameters B : NetworkX graph The input graph should be bipartite. nodes : list or iterable Nodes to project onto (the “bottom” nodes). jaccard: Bool (default=True) : Returns Graph : NetworkX graph 4.3. Bipartite 149 NetworkX Reference, Release 1.7 A graph that is the projection onto the given nodes. See Also: is_bipartite, is_bipartite_node_set, sets, weighted_projected_graph, collaboration_weighted_projected_graph, generic_weighted_projected_graph, projected_graph Notes No attempt is made to verify that the input graph B is bipartite. The graph and node properties are (shallow) copied to the projected graph. References [R124] Examples >>> from networkx.algorithms import bipartite >>> B = nx.path_graph(5) >>> G = bipartite.overlap_weighted_projected_graph(B, [0, 2, 4]) >>> print(G.nodes()) [0, 2, 4] >>> print(G.edges(data=True)) [(0, 2, {’weight’: 0.5}), (2, 4, {’weight’: 0.5})] >>> G = bipartite.overlap_weighted_projected_graph(B, [0, 2, 4], jaccard=False) >>> print(G.edges(data=True)) [(0, 2, {’weight’: 1.0}), (2, 4, {’weight’: 1.0})] generic_weighted_projected_graph generic_weighted_projected_graph(B, nodes, weight_function=None) Weighted projection of B with a user-speciﬁed weight function. The bipartite network B is projected on to the speciﬁed nodes with weights computed by a user-speciﬁed func- tion. This function must accept as a parameter the neighborhood sets of two nodes and return an integer or a ﬂoat. The nodes retain their attributes and are connected in the resulting graph if they have an edge to a common node in the original graph. Parameters B : NetworkX graph The input graph should be bipartite. nodes : list or iterable Nodes to project onto (the “bottom” nodes). weight_function: function : This function must accept as a parameters two sets, the neighborhoods of two nodes, and return an integer or a ﬂoat. The default function computes the number of shared neighbors. 150 Chapter 4. Algorithms NetworkX Reference, Release 1.7 Returns Graph : NetworkX graph A graph that is the projection onto the given nodes. See Also: is_bipartite, is_bipartite_node_set, sets, weighted_projected_graph, collaboration_weighted_projected_graph, overlap_weighted_projected_graph, projected_graph Notes No attempt is made to verify that the input graph B is bipartite. The graph and node properties are (shallow) copied to the projected graph. Examples >>> from networkx.algorithms import bipartite >>> def jaccard(unbrs, vnbrs): ... return float(len(unbrs & vnbrs)) / len(unbrs | vnbrs) ... >>> def shared(unbrs, vnbrs): ... return len(unbrs & vnbrs) ... >>> B = nx.path_graph(5) >>> G = bipartite.generic_weighted_projected_graph(B, [0, 2, 4], weight_function=jaccard) >>> print(G.nodes()) [0, 2, 4] >>> print(G.edges(data=True)) [(0, 2, {’weight’: 0.5}), (2, 4, {’weight’: 0.5})] >>> G = bipartite.generic_weighted_projected_graph(B, [0, 2, 4], weight_function=shared) >>> print(G.nodes()) [0, 2, 4] >>> print(G.edges(data=True)) [(0, 2, {’weight’: 1}), (2, 4, {’weight’: 1})] 4.3.3 Spectral Spectral bipartivity measure. spectral_bipartivity(G[, nodes, weight]) Returns the spectral bipartivity. spectral_bipartivity spectral_bipartivity(G, nodes=None, weight=’weight’) Returns the spectral bipartivity. Parameters G : NetworkX graph nodes : list or container optional(default is all nodes) Nodes to return value of spectral bipartivity contribution. weight : string or None optional (default = ‘weight’) 4.3. Bipartite 151 NetworkX Reference, Release 1.7 Edge data key to use for edge weights. If None, weights set to 1. Returns sb : ﬂoat or dict A single number if the keyword nodes is not speciﬁed, or a dictionary keyed by node with the spectral bipartivity contribution of that node as the value. See Also: color Notes This implementation uses Numpy (dense) matrices which are not efﬁcient for storing large sparse graphs. References [R127] Examples >>> from networkx.algorithms import bipartite >>> G = nx.path_graph(4) >>> bipartite.spectral_bipartivity(G) 1.0 4.3.4 Clustering clustering(G[, nodes, mode]) Compute a bipartite clustering coefﬁcient for nodes. average_clustering(G[, nodes, mode]) Compute the average bipartite clustering coefﬁcient. clustering clustering(G, nodes=None, mode=’dot’) Compute a bipartite clustering coefﬁcient for nodes. The bipartie clustering coefﬁcient is a measure of local density of connections deﬁned as [R122] cu = P v2N(N(v)) cuv |N(N(u))| where N(N(u)) are the second order neighbors of u in G excluding u, and cuv is the pairwise clustering coefﬁcient between nodes u and v. The mode selects the function for cuv ‘dot’: cuv = |N(u) \ N(v)| |N(u) [ N(v)| ‘min’: cuv = |N(u) \ N(v)|min(|N(u)|, |N(v)|) 152 Chapter 4. Algorithms NetworkX Reference, Release 1.7 ‘max’: cuv = |N(u) \ N(v)|max(|N(u)|, |N(v)|) Parameters G : graph A bipartite graph nodes : list or iterable (optional) Compute bipartite clustering for these nodes. The default is all nodes in G. mode : string The pariwise bipartite clustering method to be used in the computation. It must be “dot”, “max”, or “min”. Returns clustering : dictionary A dictionary keyed by node with the clustering coefﬁcient value. See Also: average_clustering References [R122] Examples >>> from networkx.algorithms import bipartite >>> G=nx.path_graph(4) # path is bipartite >>> c=bipartite.clustering(G) >>> c[0] 0.5 >>> c=bipartite.clustering(G,mode=’min’) >>> c[0] 1.0 average_clustering average_clustering(G, nodes=None, mode=’dot’) Compute the average bipartite clustering coefﬁcient. A clustering coefﬁcient for the whole graph is the average, C = 1 n Xv2G cv, where n is the number of nodes in G. Similar measures for the two bipartite sets can be deﬁned [R121] CX = 1 |X| Xv2X cv, where X is a bipartite set of G. 4.3. Bipartite 153 NetworkX Reference, Release 1.7 Parameters G : graph A bipartite graph nodes : list or iterable, optional A container of nodes to use in computing the average. The nodes should be either the entire graph (the default) or one of the bipartite sets. mode : string The pariwise bipartite clustering method. It must be “dot”, “max”, or “min” Returns clustering : ﬂoat The average bipartite clustering for the given set of nodes or the entire graph if no nodes are speciﬁed. See Also: clustering Notes The container of nodes passed to this function must contain all of the nodes in one of the bipartite sets (“top” or “bottom”) in order to compute the correct average bipartite clustering coefﬁcients. References [R121] Examples >>> from networkx.algorithms import bipartite >>> G=nx.star_graph(3) # path is bipartite >>> bipartite.average_clustering(G) 0.75 >>> X,Y=bipartite.sets(G) >>> bipartite.average_clustering(G,X) 0.0 >>> bipartite.average_clustering(G,Y) 1.0 4.3.5 Redundancy Node redundancy for bipartite graphs. node_redundancy(G[, nodes]) Compute bipartite node redundancy coefﬁcient. node_redundancy node_redundancy(G, nodes=None) Compute bipartite node redundancy coefﬁcient. 154 Chapter 4. Algorithms NetworkX Reference, Release 1.7 The redundancy coefﬁcient of a node v is the fraction of pairs of neighbors of v that are both linked to other nodes. In a one-mode projection these nodes would be linked together even if v were not there. rc(v)=|{{u, w} ✓ N(v), 9v0 6= v, (v0,u) 2 E and (v0,w) 2 E}| |N(v)|(|N(v)|1) 2 where N(v) are the neighbors of v in G. Parameters G : graph A bipartite graph nodes : list or iterable (optional) Compute redundancy for these nodes. The default is all nodes in G. Returns redundancy : dictionary A dictionary keyed by node with the node redundancy value. References [R126] Examples >>> from networkx.algorithms import bipartite >>> G = nx.cycle_graph(4) >>> rc = bipartite.node_redundancy(G) >>> rc[0] 1.0 Compute the average redundancy for the graph: >>> sum(rc.values())/len(G) 1.0 Compute the average redundancy for a set of nodes: >>> nodes = [0, 2] >>> sum(rc[n] for n in nodes)/len(nodes) 1.0 4.3.6 Centrality closeness_centrality(G, nodes[, normalized]) Compute the closeness centrality for nodes in a bipartite network. degree_centrality(G, nodes) Compute the degree centrality for nodes in a bipartite network. betweenness_centrality(G, nodes) Compute betweenness centrality for nodes in a bipartite network. closeness_centrality closeness_centrality(G, nodes, normalized=True) Compute the closeness centrality for nodes in a bipartite network. The closeness of a node is the distance to all other nodes in the graph or in the case that the graph is not connected 4.3. Bipartite 155 NetworkX Reference, Release 1.7 to all other nodes in the connected component containing that node. Parameters G : graph A bipartite network nodes : list or container Container with all nodes in one bipartite node set. normalized : bool, optional If True (default) normalize by connected component size. Returns closeness : dictionary Dictionary keyed by node with bipartite closeness centrality as the value. See Also: betweenness_centrality, degree_centrality, sets, is_bipartite Notes The nodes input parameter must conatin all nodes in one bipartite node set, but the dictionary returned contains all nodes from both node sets. Closeness centrality is normalized by the minimum distance possible. In the bipartite case the minimum distance for a node in one bipartite node set is 1 from all nodes in the other node set and 2 from all other nodes in its own set [R119]. Thus the closeness centrality for node v in the two bipartite sets U with n nodes and V with m nodes is cv = m + 2(n 1) d , forv 2 U, cv = n + 2(m 1) d , forv 2 V, where d is the sum of the distances from v to all other nodes. Higher values of closeness indicate higher centrality. As in the unipartite case, setting normalized=True causes the values to normalized further to n-1 / size(G)-1 where n is the number of nodes in the connected part of graph containing the node. If the graph is not completely connected, this algorithm computes the closeness centrality for each connected part separately. References [R119] degree_centrality degree_centrality(G, nodes) Compute the degree centrality for nodes in a bipartite network. The degree centrality for a node v is the fraction of nodes connected to it. Parameters G : graph A bipartite network nodes : list or container 156 Chapter 4. Algorithms NetworkX Reference, Release 1.7 Container with all nodes in one bipartite node set. Returns centrality : dictionary Dictionary keyed by node with bipartite degree centrality as the value. See Also: betweenness_centrality, closeness_centrality, sets, is_bipartite Notes The nodes input parameter must conatin all nodes in one bipartite node set, but the dictionary returned contains all nodes from both bipartite node sets. For unipartite networks, the degree centrality values are normalized by dividing by the maximum possible degree (which is n 1 where n is the number of nodes in G). In the bipartite case, the maximum possible degree of a node in a bipartite node set is the number of nodes in the opposite node set [R120]. The degree centrality for a node v in the bipartite sets U with n nodes and V with m nodes is dv = deg(v) m , forv 2 U, dv = deg(v) n , forv 2 V, where deg(v) is the degree of node v. References [R120] betweenness_centrality betweenness_centrality(G, nodes) Compute betweenness centrality for nodes in a bipartite network. Betweenness centrality of a node v is the sum of the fraction of all-pairs shortest paths that pass through v. Values of betweenness are normalized by the maximum possible value which for bipartite graphs is limited by the relative size of the two node sets [R118]. Let n be the number of nodes in the node set U and m be the number of nodes in the node set V , then nodes in U are normalized by dividing by 1 2[m2(s + 1)2 + m(s + 1)(2t s 1) t(2s t + 3)], where s =(n 1) ÷ m, t =(n 1) mod m, and nodes in V are normalized by dividing by 1 2[n2(p + 1)2 + n(p + 1)(2r p 1) r(2p r + 3)], where, p =(m 1) ÷ n, r =(m 1) mod n. 4.3. Bipartite 157 NetworkX Reference, Release 1.7 Parameters G : graph A bipartite graph nodes : list or container Container with all nodes in one bipartite node set. Returns betweenness : dictionary Dictionary keyed by node with bipartite betweenness centrality as the value. See Also: degree_centrality, closeness_centrality, sets, is_bipartite Notes The nodes input parameter must contain all nodes in one bipartite node set, but the dictionary returned contains all nodes from both node sets. References [R118] 4.4 Blockmodeling Functions for creating network blockmodels from node partitions. Created by Drew Conway Copyright (c) 2010. All rights reserved. blockmodel(G, partitions[, multigraph]) Returns a reduced graph constructed using the generalized block modeling technique. 4.4.1 blockmodel blockmodel(G, partitions, multigraph=False) Returns a reduced graph constructed using the generalized block modeling technique. The blockmodel technique collapses nodes into blocks based on a given partitioning of the node set. Each partition of nodes (block) is represented as a single node in the reduced graph. Edges between nodes in the block graph are added according to the edges in the original graph. If the parameter multigraph is False (the default) a single edge is added with a weight equal to the sum of the edge weights between nodes in the original graph The default is a weight of 1 if weights are not speciﬁed. If the parameter multigraph is True then multiple edges are added each with the edge data from the original graph. Parameters G : graph A networkx Graph or DiGraph partitions : list of lists, or list of sets The partition of the nodes. Must be non-overlapping. multigraph : bool, optional 158 Chapter 4. Algorithms NetworkX Reference, Release 1.7 If True return a MultiGraph with the edge data of the original graph applied to each corresponding edge in the new graph. If False return a Graph with the sum of the edge weights, or a count of the edges if the original graph is unweighted. Returns blockmodel : a Networkx graph object References [R128] Examples >>> G=nx.path_graph(6) >>> partition=[[0,1],[2,3],[4,5]] >>> M=nx.blockmodel(G,partition) 4.5 Boundary Routines to ﬁnd the boundary of a set of nodes. Edge boundaries are edges that have only one end in the set of nodes. Node boundaries are nodes outside the set of nodes that have an edge to a node in the set. edge_boundary(G, nbunch1[, nbunch2]) Return the edge boundary. node_boundary(G, nbunch1[, nbunch2]) Return the node boundary. 4.5.1 edge_boundary edge_boundary(G, nbunch1, nbunch2=None) Return the edge boundary. Edge boundaries are edges that have only one end in the given set of nodes. Parameters G : graph A networkx graph nbunch1 : list, container Interior node set nbunch2 : list, container Exterior node set. If None then it is set to all of the nodes in G not in nbunch1. Returns elist : list List of edges Notes Nodes in nbunch1 and nbunch2 that are not in G are ignored. 4.5. Boundary 159 NetworkX Reference, Release 1.7 nbunch1 and nbunch2 are usually meant to be disjoint, but in the interest of speed and generality, that is not required here. 4.5.2 node_boundary node_boundary(G, nbunch1, nbunch2=None) Return the node boundary. The node boundary is all nodes in the edge boundary of a given set of nodes that are in the set. Parameters G : graph A networkx graph nbunch1 : list, container Interior node set nbunch2 : list, container Exterior node set. If None then it is set to all of the nodes in G not in nbunch1. Returns nlist : list List of nodes. Notes Nodes in nbunch1 and nbunch2 that are not in G are ignored. nbunch1 and nbunch2 are usually meant to be disjoint, but in the interest of speed and generality, that is not required here. 4.6 Centrality 4.6.1 Degree degree_centrality(G) Compute the degree centrality for nodes. in_degree_centrality(G) Compute the in-degree centrality for nodes. out_degree_centrality(G) Compute the out-degree centrality for nodes. degree_centrality degree_centrality(G) Compute the degree centrality for nodes. The degree centrality for a node v is the fraction of nodes it is connected to. Parameters G : graph A networkx graph Returns nodes : dictionary Dictionary of nodes with degree centrality as the value. See Also: 160 Chapter 4. Algorithms NetworkX Reference, Release 1.7 betweenness_centrality, load_centrality, eigenvector_centrality Notes The degree centrality values are normalized by dividing by the maximum possible degree in a simple graph n-1 where n is the number of nodes in G. For multigraphs or graphs with self loops the maximum degree might be higher than n-1 and values of degree centrality greater than 1 are possible. in_degree_centrality in_degree_centrality(G) Compute the in-degree centrality for nodes. The in-degree centrality for a node v is the fraction of nodes its incoming edges are connected to. Parameters G : graph A NetworkX graph Returns nodes : dictionary Dictionary of nodes with in-degree centrality as values. See Also: degree_centrality, out_degree_centrality Notes The degree centrality values are normalized by dividing by the maximum possible degree in a simple graph n-1 where n is the number of nodes in G. For multigraphs or graphs with self loops the maximum degree might be higher than n-1 and values of degree centrality greater than 1 are possible. out_degree_centrality out_degree_centrality(G) Compute the out-degree centrality for nodes. The out-degree centrality for a node v is the fraction of nodes its outgoing edges are connected to. Parameters G : graph A NetworkX graph Returns nodes : dictionary Dictionary of nodes with out-degree centrality as values. See Also: degree_centrality, in_degree_centrality 4.6. Centrality 161 NetworkX Reference, Release 1.7 Notes The degree centrality values are normalized by dividing by the maximum possible degree in a simple graph n-1 where n is the number of nodes in G. For multigraphs or graphs with self loops the maximum degree might be higher than n-1 and values of degree centrality greater than 1 are possible. 4.6.2 Closeness closeness_centrality(G[, v, distance, ...]) Compute closeness centrality for nodes. closeness_centrality closeness_centrality(G, v=None, distance=None, normalized=True) Compute closeness centrality for nodes. Closeness centrality at a node is 1/average distance to all other nodes. Parameters G : graph A networkx graph v : node, optional Return only the value for node v distance : string key, optional (default=None) Use speciﬁed edge key as edge distance. If True, use ‘weight’ as the edge key. normalized : bool, optional If True (default) normalize by the graph size. Returns nodes : dictionary Dictionary of nodes with closeness centrality as the value. See Also: betweenness_centrality, load_centrality, eigenvector_centrality, degree_centrality Notes The closeness centrality is normalized to to n-1 / size(G)-1 where n is the number of nodes in the connected part of graph containing the node. If the graph is not completely connected, this algorithm computes the closeness centrality for each connected part separately. 4.6.3 Betweenness betweenness_centrality(G[, k, normalized, ...]) Compute the shortest-path betweenness centrality for nodes. edge_betweenness_centrality(G[, normalized, ...]) Compute betweenness centrality for edges. 162 Chapter 4. Algorithms NetworkX Reference, Release 1.7 betweenness_centrality betweenness_centrality(G, k=None, normalized=True, weight=None, endpoints=False, seed=None) Compute the shortest-path betweenness centrality for nodes. Betweenness centrality of a node v is the sum of the fraction of all-pairs shortest paths that pass through v: cB(v)= Xs,t2V (s, t|v) (s, t) where V is the set of nodes, (s, t) is the number of shortest (s, t)-paths, and (s, t|v) is the number of those paths passing through some node v other than s, t. If s = t, (s, t)=1, and if v 2 s, t, (s, t|v)=0[R131]. Parameters G : graph A NetworkX graph k : int, optional (default=None) If k is not None use k node samples to estimate betweenness. The value of k <= n where n is the number of nodes in the graph. Higher values give better approximation. normalized : bool, optional If True the betweenness values are normalized by 2/((n 1)(n 2)) for graphs, and 1/((n 1)(n 2)) for directed graphs where n is the number of nodes in G. weight : None or string, optional If None, all edge weights are considered equal. Otherwise holds the name of the edge attribute used as weight. endpoints : bool, optional If True include the endpoints in the shortest path counts. Returns nodes : dictionary Dictionary of nodes with betweenness centrality as the value. See Also: edge_betweenness_centrality, load_centrality Notes The algorithm is from Ulrik Brandes [R130]. See [R131] for details on algorithms for variations and related metrics. For approximate betweenness calculations set k=#samples to use k nodes (“pivots”) to estimate the betweenness values. For an estimate of the number of pivots needed see [R132]. For weighted graphs the edge weights must be greater than zero. Zero edge weights can produce an inﬁnite number of equal length paths between pairs of nodes. References [R130], [R131], [R132] 4.6. Centrality 163 NetworkX Reference, Release 1.7 edge_betweenness_centrality edge_betweenness_centrality(G, normalized=True, weight=None) Compute betweenness centrality for edges. Betweenness centrality of an edge e is the sum of the fraction of all-pairs shortest paths that pass through e: cB(v)= Xs,t2V (s, t|e) (s, t) where V is the set of nodes,‘sigma(s, t)‘ is the number of shortest (s, t)-paths, and (s, t|e) is the number of those paths passing through edge e [R146]. Parameters G : graph A NetworkX graph normalized : bool, optional If True the betweenness values are normalized by 2/(n(n1)) for graphs, and 1/(n(n1)) for directed graphs where n is the number of nodes in G. weight : None or string, optional If None, all edge weights are considered equal. Otherwise holds the name of the edge attribute used as weight. Returns edges : dictionary Dictionary of edges with betweenness centrality as the value. See Also: betweenness_centrality, edge_load Notes The algorithm is from Ulrik Brandes [R145]. For weighted graphs the edge weights must be greater than zero. Zero edge weights can produce an inﬁnite number of equal length paths between pairs of nodes. References [R145], [R146] 4.6.4 Current Flow Closeness current_flow_closeness_centrality(G[, ...]) Compute current-ﬂow closeness centrality for nodes. current_ﬂow_closeness_centrality current_flow_closeness_centrality(G, normalized=True, weight=’weight’, dtype=, solver=’lu’) Compute current-ﬂow closeness centrality for nodes. 164 Chapter 4. Algorithms NetworkX Reference, Release 1.7 A variant of closeness centrality based on effective resistance between nodes in a network. This metric is also known as information centrality. Parameters G : graph A NetworkX graph normalized : bool, optional If True the values are normalized by 1/(n-1) where n is the number of nodes in G. dtype: data type (ﬂoat) : Default data type for internal matrices. Set to np.ﬂoat32 for lower memory consump- tion. solver: string (default=’lu’) : Type of linear solver to use for computing the ﬂow matrix. Options are “full” (uses most memory), “lu” (recommended), and “cg” (uses least memory). Returns nodes : dictionary Dictionary of nodes with current ﬂow closeness centrality as the value. See Also: closeness_centrality Notes The algorithm is from Brandes [R143]. See also [R144] for the original deﬁnition of information centrality. References [R143], [R144] 4.6.5 Current-Flow Betweenness current_flow_betweenness_centrality(G[, ...]) Compute current-ﬂow betweenness centrality for nodes. edge_current_flow_betweenness_centrality(G) Compute current-ﬂow betweenness centrality for edges. approximate_current_flow_betweenness_centrality(G) Compute the approximate current-ﬂow betweenness centrality for nodes. current_ﬂow_betweenness_centrality current_flow_betweenness_centrality(G, normalized=True, weight=’weight’, dtype=, solver=’lu’) Compute current-ﬂow betweenness centrality for nodes. Current-ﬂow betweenness centrality uses an electrical current model for information spreading in contrast to betweenness centrality which uses shortest paths. Current-ﬂow betweenness centrality is also known as random-walk betweenness centrality [R142]. Parameters G : graph 4.6. Centrality 165 NetworkX Reference, Release 1.7 A NetworkX graph normalized : bool, optional (default=True) If True the betweenness values are normalized by b=b/(n-1)(n-2) where n is the number of nodes in G. weight : string or None, optional (default=’weight’) Key for edge data used as the edge weight. If None, then use 1 as each edge weight. dtype: data type (ﬂoat) : Default data type for internal matrices. Set to np.ﬂoat32 for lower memory consump- tion. solver: string (default=’lu’) : Type of linear solver to use for computing the ﬂow matrix. Options are “full” (uses most memory), “lu” (recommended), and “cg” (uses least memory). Returns nodes : dictionary Dictionary of nodes with betweenness centrality as the value. See Also: approximate_current_flow_betweenness_centrality, betweenness_centrality, edge_betweenness_centrality, edge_current_flow_betweenness_centrality Notes Current-ﬂow betweenness can be computed in O(I(n 1) + mn log n) time [R141], where I(n 1) is the time needed to compute the inverse Laplacian. For a full matrix this is O(n3) but using sparse methods you can achieve O(nmpk) where k is the Laplacian matrix condition number. The space required is O(nw)where‘w is the width of the sparse Laplacian matrix. Worse case is w = n for O(n2). If the edges have a ‘weight’ attribute they will be used as weights in this algorithm. Unspeciﬁed weights are set to 1. References [R141], [R142] edge_current_ﬂow_betweenness_centrality edge_current_flow_betweenness_centrality(G, normalized=True, weight=’weight’, dtype=, solver=’lu’) Compute current-ﬂow betweenness centrality for edges. Current-ﬂow betweenness centrality uses an electrical current model for information spreading in contrast to betweenness centrality which uses shortest paths. Current-ﬂow betweenness centrality is also known as random-walk betweenness centrality [R148]. Parameters G : graph A NetworkX graph 166 Chapter 4. Algorithms NetworkX Reference, Release 1.7 normalized : bool, optional (default=True) If True the betweenness values are normalized by b=b/(n-1)(n-2) where n is the number of nodes in G. weight : string or None, optional (default=’weight’) Key for edge data used as the edge weight. If None, then use 1 as each edge weight. dtype: data type (ﬂoat) : Default data type for internal matrices. Set to np.ﬂoat32 for lower memory consump- tion. solver: string (default=’lu’) : Type of linear solver to use for computing the ﬂow matrix. Options are “full” (uses most memory), “lu” (recommended), and “cg” (uses least memory). Returns nodes : dictionary Dictionary of edge tuples with betweenness centrality as the value. See Also: betweenness_centrality, edge_betweenness_centrality, current_flow_betweenness_centrality Notes Current-ﬂow betweenness can be computed in O(I(n 1) + mn log n) time [R147], where I(n 1) is the time needed to compute the inverse Laplacian. For a full matrix this is O(n3) but using sparse methods you can achieve O(nmpk) where k is the Laplacian matrix condition number. The space required is O(nw)where‘w is the width of the sparse Laplacian matrix. Worse case is w = n for O(n2). If the edges have a ‘weight’ attribute they will be used as weights in this algorithm. Unspeciﬁed weights are set to 1. References [R147], [R148] approximate_current_ﬂow_betweenness_centrality approximate_current_flow_betweenness_centrality(G, normalized=True, weight=’weight’, dtype=, solver=’lu’, ep- silon=0.5, kmax=10000) Compute the approximate current-ﬂow betweenness centrality for nodes. Approximates the current-ﬂow betweenness centrality within absolute error of epsilon with high probability [R129]. Parameters G : graph A NetworkX graph normalized : bool, optional (default=True) If True the betweenness values are normalized by b=b/(n-1)(n-2) where n is the number of nodes in G. 4.6. Centrality 167 NetworkX Reference, Release 1.7 weight : string or None, optional (default=’weight’) Key for edge data used as the edge weight. If None, then use 1 as each edge weight. dtype: data type (ﬂoat) : Default data type for internal matrices. Set to np.ﬂoat32 for lower memory consump- tion. solver: string (default=’lu’) : Type of linear solver to use for computing the ﬂow matrix. Options are “full” (uses most memory), “lu” (recommended), and “cg” (uses least memory). epsilon: ﬂoat : Absolute error tolerance. kmax: int : Maximum number of sample node pairs to use for approximation. Returns nodes : dictionary Dictionary of nodes with betweenness centrality as the value. See Also: current_flow_betweenness_centrality Notes The running time is O((1/✏2)mpk log n) and the space required is O(m) for n nodes and m edges. If the edges have a ‘weight’ attribute they will be used as weights in this algorithm. Unspeciﬁed weights are set to 1. References [R129] 4.6.6 Eigenvector eigenvector_centrality(G[, max_iter, tol, ...]) Compute the eigenvector centrality for the graph G. eigenvector_centrality_numpy(G) Compute the eigenvector centrality for the graph G. eigenvector_centrality eigenvector_centrality(G, max_iter=100, tol=1e-06, nstart=None) Compute the eigenvector centrality for the graph G. Uses the power method to ﬁnd the eigenvector for the largest eigenvalue of the adjacency matrix of G. Parameters G : graph A networkx graph max_iter : interger, optional 168 Chapter 4. Algorithms NetworkX Reference, Release 1.7 Maximum number of iterations in power method. tol : ﬂoat, optional Error tolerance used to check convergence in power method iteration. nstart : dictionary, optional Starting value of eigenvector iteration for each node. Returns nodes : dictionary Dictionary of nodes with eigenvector centrality as the value. See Also: eigenvector_centrality_numpy, pagerank, hits Notes The eigenvector calculation is done by the power iteration method and has no guarantee of convergence. The iteration will stop after max_iter iterations or an error tolerance of number_of_nodes(G)*tol has been reached. For directed graphs this is “right” eigevector centrality. For “left” eigenvector centrality, ﬁrst reverse the graph with G.reverse(). Examples >>> G=nx.path_graph(4) >>> centrality=nx.eigenvector_centrality(G) >>> print([’%s %0.2f’%(node,centrality[node]) for node in centrality]) [’0 0.37’, ’1 0.60’, ’2 0.60’, ’3 0.37’] eigenvector_centrality_numpy eigenvector_centrality_numpy(G) Compute the eigenvector centrality for the graph G. Parameters G : graph A networkx graph Returns nodes : dictionary Dictionary of nodes with eigenvector centrality as the value. See Also: eigenvector_centrality, pagerank, hits Notes This algorithm uses the NumPy eigenvalue solver. For directed graphs this is “right” eigevector centrality. For “left” eigenvector centrality, ﬁrst reverse the graph with G.reverse(). 4.6. Centrality 169 NetworkX Reference, Release 1.7 Examples >>> G=nx.path_graph(4) >>> centrality=nx.eigenvector_centrality_numpy(G) >>> print([’%s %0.2f’%(node,centrality[node]) for node in centrality]) [’0 0.37’, ’1 0.60’, ’2 0.60’, ’3 0.37’] 4.6.7 Communicability communicability(G) Return communicability between all pairs of nodes in G. communicability_exp(G) Return communicability between all pairs of nodes in G. communicability_centrality(G) Return communicability centrality for each node in G. communicability_centrality_exp(G) Return the communicability centrality for each node of G communicability_betweenness_centrality(G[, ...]) Return communicability betweenness for all pairs of nodes in G. estrada_index(G) Return the Estrada index of a the graph G. communicability communicability(G) Return communicability between all pairs of nodes in G. The communicability between pairs of nodes in G is the sum of closed walks of different lengths starting at node u and ending at node v. Parameters G: graph : Returns comm: dictionary of dictionaries : Dictionary of dictionaries keyed by nodes with communicability as the value. Raises NetworkXError : If the graph is not undirected and simple. See Also: communicability_centrality_exp Communicability centrality for each node of G using matrix ex- ponential. communicability_centrality Communicability centrality for each node in G using spectral decom- position. communicability Communicability between pairs of nodes in G. Notes This algorithm uses a spectral decomposition of the adjacency matrix. Let G=(V,E) be a simple undirected graph. Using the connection between the powers of the adjacency matrix and the number of walks in the graph, the communicability between nodes u and v based on the graph spectrum is [R133] C(u, v)= n Xj=1 j(u)j(v)ej , 170 Chapter 4. Algorithms NetworkX Reference, Release 1.7 where j(u) is the uth element of the jth orthonormal eigenvector of the adjacency matrix associated with the eigenvalue j. References [R133] Examples >>> G = nx.Graph([(0,1),(1,2),(1,5),(5,4),(2,4),(2,3),(4,3),(3,6)]) >>> c = nx.communicability(G) communicability_exp communicability_exp(G) Return communicability between all pairs of nodes in G. Communicability between pair of node (u,v) of node in G is the sum of closed walks of different lengths starting at node u and ending at node v. Parameters G: graph : Returns comm: dictionary of dictionaries : Dictionary of dictionaries keyed by nodes with communicability as the value. Raises NetworkXError : If the graph is not undirected and simple. See Also: communicability_centrality_exp Communicability centrality for each node of G using matrix ex- ponential. communicability_centrality Communicability centrality for each node in G using spectral decom- position. communicability_exp Communicability between all pairs of nodes in G using spectral decomposition. Notes This algorithm uses matrix exponentiation of the adjacency matrix. Let G=(V,E) be a simple undirected graph. Using the connection between the powers of the adjacency matrix and the number of walks in the graph, the communicability between nodes u and v is [R140], C(u, v)=(eA)uv, where A is the adjacency matrix of G. References [R140] 4.6. Centrality 171 NetworkX Reference, Release 1.7 Examples >>> G = nx.Graph([(0,1),(1,2),(1,5),(5,4),(2,4),(2,3),(4,3),(3,6)]) >>> c = nx.communicability_exp(G) communicability_centrality communicability_centrality(G) Return communicability centrality for each node in G. Communicability centrality, also called subgraph centrality, of a node n is the sum of closed walks of all lengths starting and ending at node n. Parameters G: graph : Returns nodes: dictionary : Dictionary of nodes with communicability centrality as the value. Raises NetworkXError : If the graph is not undirected and simple. See Also: communicability Communicability between all pairs of nodes in G. communicability_centrality Communicability centrality for each node of G. Notes This version of the algorithm computes eigenvalues and eigenvectors of the adjacency matrix. Communicability centrality of a node u in G can be found using a spectral decomposition of the adjacency matrix [R136] [R137], SC(u)= N Xj=1 (vu j )2ej , where vj is an eigenvector of the adjacency matrix A of G corresponding corresponding to the eigenvalue j. References [R136], [R137] Examples >>> G = nx.Graph([(0,1),(1,2),(1,5),(5,4),(2,4),(2,3),(4,3),(3,6)]) >>> sc = nx.communicability_centrality(G) 172 Chapter 4. Algorithms NetworkX Reference, Release 1.7 communicability_centrality_exp communicability_centrality_exp(G) Return the communicability centrality for each node of G Communicability centrality, also called subgraph centrality, of a node n is the sum of closed walks of all lengths starting and ending at node n. Parameters G: graph : Returns nodes:dictionary : Dictionary of nodes with communicability centrality as the value. Raises NetworkXError : If the graph is not undirected and simple. See Also: communicability Communicability between all pairs of nodes in G. communicability_centrality Communicability centrality for each node of G. Notes This version of the algorithm exponentiates the adjacency matrix. The communicability centrality of a node u in G can be found using the matrix exponential of the adjacency matrix of G [R138] [R139], SC(u)=(eA)uu. References [R138], [R139] Examples >>> G = nx.Graph([(0,1),(1,2),(1,5),(5,4),(2,4),(2,3),(4,3),(3,6)]) >>> sc = nx.communicability_centrality_exp(G) communicability_betweenness_centrality communicability_betweenness_centrality(G, normalized=True) Return communicability betweenness for all pairs of nodes in G. Communicability betweenness measure makes use of the number of walks connecting every pair of nodes as the basis of a betweenness centrality measure. Parameters G: graph : Returns nodes:dictionary : Dictionary of nodes with communicability betweenness as the value. Raises NetworkXError : If the graph is not undirected and simple. 4.6. Centrality 173 NetworkX Reference, Release 1.7 See Also: communicability Communicability between all pairs of nodes in G. communicability_centrality Communicability centrality for each node of G using matrix exponen- tial. communicability_centrality_exp Communicability centrality for each node in G using spectral de- composition. Notes Let G =(V,E) be a simple undirected graph with n nodes and m edges, and A denote the adjacency matrix of G. Let G(r)=(V,E(r)) be the graph resulting from removing all edges connected to node r but not the node itself. The adjacency matrix for G(r) is A + E(r), where E(r) has nonzeros only in row and column r. The communicability betweenness of a node r is [R135] !r = 1 C Xp Xq Gprq Gpq ,p6= q, q 6= r, where Gprq =(eA pq (eA+E(r))pq is the number of walks involving node r, Gpq =(eA)pq is the number of closed walks starting at node p and ending at node q, and C =(n 1)2 (n 1) is a normalization factor equal to the number of terms in the sum. The resulting !r takes values between zero and one. The lower bound cannot be attained for a connected graph, and the upper bound is attained in the star graph. References [R135] Examples >>> G = nx.Graph([(0,1),(1,2),(1,5),(5,4),(2,4),(2,3),(4,3),(3,6)]) >>> cbc = nx.communicability_betweenness_centrality(G) estrada_index estrada_index(G) Return the Estrada index of a the graph G. Parameters G: graph : Returns estrada index: ﬂoat : Raises NetworkXError : If the graph is not undirected and simple. 174 Chapter 4. Algorithms NetworkX Reference, Release 1.7 See Also: estrada_index_exp Notes Let G =(V,E) be a simple undirected graph with n nodes and let 1 2 ···n be a non-increasing ordering of the eigenvalues of its adjacency matrix A. The Estrada index is EE(G)= n Xj=1 ej . References [R149] Examples >>> G=nx.Graph([(0,1),(1,2),(1,5),(5,4),(2,4),(2,3),(4,3),(3,6)]) >>> ei=nx.estrada_index(G) 4.6.8 Load load_centrality(G[, v, cutoff, normalized, ...]) Compute load centrality for nodes. edge_load(G[, nodes, cutoff]) Compute edge load. load_centrality load_centrality(G, v=None, cutoff=None, normalized=True, weight=None) Compute load centrality for nodes. The load centrality of a node is the fraction of all shortest paths that pass through that node. Parameters G : graph A networkx graph normalized : bool, optional If True the betweenness values are normalized by b=b/(n-1)(n-2) where n is the number of nodes in G. weight : None or string, optional If None, edge weights are ignored. Otherwise holds the name of the edge attribute used as weight. cutoff : bool, optional If speciﬁed, only consider paths of length <= cutoff. Returns nodes : dictionary 4.6. Centrality 175 NetworkX Reference, Release 1.7 Dictionary of nodes with centrality as the value. See Also: betweenness_centrality Notes Load centrality is slightly different than betweenness. For this load algorithm see the reference Scientiﬁc col- laboration networks: II. Shortest paths, weighted networks, and centrality, M. E. J. Newman, Phys. Rev. E 64, 016132 (2001). edge_load edge_load(G, nodes=None, cutoff=False) Compute edge load. WARNING: This module is for demonstration and testing purposes. 4.7 Chordal Algorithms for chordal graphs. A graph is chordal if every cycle of length at least 4 has a chord (an edge joining two nodes not adjacent in the cycle). http://en.wikipedia.org/wiki/Chordal_graph is_chordal(G) Checks whether G is a chordal graph. chordal_graph_cliques(G) Returns the set of maximal cliques of a chordal graph. chordal_graph_treewidth(G) Returns the treewidth of the chordal graph G. find_induced_nodes(G, s, t[, treewidth_bound]) Returns the set of induced nodes in the path from s to t. 4.7.1 is_chordal is_chordal(G) Checks whether G is a chordal graph. A graph is chordal if every cycle of length at least 4 has a chord (an edge joining two nodes not adjacent in the cycle). Parameters G : graph A NetworkX graph. Returns chordal : bool True if G is a chordal graph and False otherwise. Raises NetworkXError : The algorithm does not support DiGraph, MultiGraph and MultiDiGraph. If the input graph is an instance of one of these classes, a NetworkXError is raised. 176 Chapter 4. Algorithms NetworkX Reference, Release 1.7 Notes The routine tries to go through every node following maximum cardinality search. It returns False when it ﬁnds that the separator for any node is not a clique. Based on the algorithms in [R152]. References [R152] Examples >>> import networkx as nx >>> e=[(1,2),(1,3),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)] >>> G=nx.Graph(e) >>> nx.is_chordal(G) True 4.7.2 chordal_graph_cliques chordal_graph_cliques(G) Returns the set of maximal cliques of a chordal graph. The algorithm breaks the graph in connected components and performs a maximum cardinality search in each component to get the cliques. Parameters G : graph A NetworkX graph Returns cliques : A set containing the maximal cliques in G. Raises NetworkXError : The algorithm does not support DiGraph, MultiGraph and MultiDiGraph. If the input graph is an instance of one of these classes, a NetworkXError is raised. The algorithm can only be applied to chordal graphs. If the input graph is found to be non-chordal, a NetworkXError is raised. Examples >>> import networkx as nx >>> e= [(1,2),(1,3),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6),(7,8)] >>> G = nx.Graph(e) >>> G.add_node(9) >>> setlist = nx.chordal_graph_cliques(G) 4.7.3 chordal_graph_treewidth chordal_graph_treewidth(G) Returns the treewidth of the chordal graph G. Parameters G : graph 4.7. Chordal 177 NetworkX Reference, Release 1.7 A NetworkX graph Returns treewidth : int The size of the largest clique in the graph minus one. Raises NetworkXError : The algorithm does not support DiGraph, MultiGraph and MultiDiGraph. If the input graph is an instance of one of these classes, a NetworkXError is raised. The algorithm can only be applied to chordal graphs. If the input graph is found to be non-chordal, a NetworkXError is raised. References [R150] Examples >>> import networkx as nx >>> e = [(1,2),(1,3),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6),(7,8)] >>> G = nx.Graph(e) >>> G.add_node(9) >>> nx.chordal_graph_treewidth(G) 3 4.7.4 ﬁnd_induced_nodes find_induced_nodes(G, s, t, treewidth_bound=9223372036854775807) Returns the set of induced nodes in the path from s to t. Parameters G : graph A chordal NetworkX graph s : node Source node to look for induced nodes t : node Destination node to look for induced nodes treewith_bound: ﬂoat : Maximum treewidth acceptable for the graph H. The search for induced nodes will end as soon as the treewidth_bound is exceeded. Returns I : Set of nodes The set of induced nodes in the path from s to t in G Raises NetworkXError : The algorithm does not support DiGraph, MultiGraph and MultiDiGraph. If the input graph is an instance of one of these classes, a NetworkXError is raised. The algorithm can only be applied to chordal graphs. If the input graph is found to be non-chordal, a NetworkXError is raised. 178 Chapter 4. Algorithms NetworkX Reference, Release 1.7 Notes G must be a chordal graph and (s,t) an edge that is not in G. If a treewidth_bound is provided, the search for induced nodes will end as soon as the treewidth_bound is exceeded. The algorithm is inspired by Algorithm 4 in [R151]. A formal deﬁnition of induced node can also be found on that reference. References [R151] Examples >>> import networkx as nx >>> G=nx.Graph() >>> G = nx.generators.classic.path_graph(10) >>> I = nx.find_induced_nodes(G,1,9,2) >>> list(I) [1, 2, 3, 4, 5, 6, 7, 8, 9] 4.8 Clique Find and manipulate cliques of graphs. Note that ﬁnding the largest clique of a graph has been shown to be an NP-complete problem; the algorithms here could take a long time to run. http://en.wikipedia.org/wiki/Clique_problem find_cliques(G) Search for all maximal cliques in a graph. make_max_clique_graph(G[, create_using, name]) Create the maximal clique graph of a graph. make_clique_bipartite(G[, fpos, ...]) Create a bipartite clique graph from a graph G. graph_clique_number(G[, cliques]) Return the clique number (size of the largest clique) for G. graph_number_of_cliques(G[, cliques]) Returns the number of maximal cliques in G. node_clique_number(G[, nodes, cliques]) Returns the size of the largest maximal clique containing each given node. number_of_cliques(G[, nodes, cliques]) Returns the number of maximal cliques for each node. cliques_containing_node(G[, nodes, cliques]) Returns a list of cliques containing the given node. 4.8.1 ﬁnd_cliques find_cliques(G) Search for all maximal cliques in a graph. Maximal cliques are the largest complete subgraph containing a given node. The largest maximal clique is sometimes called the maximum clique. Returns generator of lists: genetor of member list for each maximal clique : See Also: 4.8. Clique 179 NetworkX Reference, Release 1.7 find_cliques_recursive, A Notes To obtain a list of cliques, use list(ﬁnd_cliques(G)). Based on the algorithm published by Bron & Kerbosch (1973) [R153] as adapated by Tomita, Tanaka and Takahashi (2006) [R154] and discussed in Cazals and Karande (2008) [R155]. The method essentially unrolls the recursion used in the references to avoid issues of recursion stack depth. This algorithm is not suitable for directed graphs. This algorithm ignores self-loops and parallel edges as clique is not conventionally deﬁned with such edges. There are often many cliques in graphs. This algorithm can run out of memory for large graphs. References [R153], [R154], [R155] 4.8.2 make_max_clique_graph make_max_clique_graph(G, create_using=None, name=None) Create the maximal clique graph of a graph. Finds the maximal cliques and treats these as nodes. The nodes are connected if they have common members in the original graph. Theory has done a lot with clique graphs, but I haven’t seen much on maximal clique graphs. Notes This should be the same as make_clique_bipartite followed by project_up, but it saves all the intermediate steps. 4.8.3 make_clique_bipartite make_clique_bipartite(G, fpos=None, create_using=None, name=None) Create a bipartite clique graph from a graph G. Nodes of G are retained as the “bottom nodes” of B and cliques of G become “top nodes” of B. Edges are present if a bottom node belongs to the clique represented by the top node. Returns a Graph with additional attribute dict B.node_type which is keyed by nodes to “Bottom” or “Top” appropriately. if fpos is not None, a second additional attribute dict B.pos is created to hold the position tuple of each node for viewing the bipartite graph. 4.8.4 graph_clique_number graph_clique_number(G, cliques=None) Return the clique number (size of the largest clique) for G. An optional list of cliques can be input if already computed. 180 Chapter 4. Algorithms NetworkX Reference, Release 1.7 4.8.5 graph_number_of_cliques graph_number_of_cliques(G, cliques=None) Returns the number of maximal cliques in G. An optional list of cliques can be input if already computed. 4.8.6 node_clique_number node_clique_number(G, nodes=None, cliques=None) Returns the size of the largest maximal clique containing each given node. Returns a single or list depending on input nodes. Optional list of cliques can be input if already computed. 4.8.7 number_of_cliques number_of_cliques(G, nodes=None, cliques=None) Returns the number of maximal cliques for each node. Returns a single or list depending on input nodes. Optional list of cliques can be input if already computed. 4.8.8 cliques_containing_node cliques_containing_node(G, nodes=None, cliques=None) Returns a list of cliques containing the given node. Returns a single list or list of lists depending on input nodes. Optional list of cliques can be input if already computed. 4.9 Clustering Algorithms to characterize the number of triangles in a graph. triangles(G[, nodes]) Compute the number of triangles. transitivity(G) Compute graph transitivity, the fraction of all possible triangles clustering(G[, nodes, weight]) Compute the clustering coefﬁcient for nodes. average_clustering(G[, nodes, weight, ...]) Compute the average clustering coefﬁcient for the graph G. square_clustering(G[, nodes]) Compute the squares clustering coefﬁcient for nodes. 4.9.1 triangles triangles(G, nodes=None) Compute the number of triangles. Finds the number of triangles that include a node as one vertex. Parameters G : graph A networkx graph nodes : container of nodes, optional (default= all nodes in G) Compute triangles for nodes in this container. 4.9. Clustering 181 NetworkX Reference, Release 1.7 Returns out : dictionary Number of triangles keyed by node label. Notes When computing triangles for the entire graph each triangle is counted three times, once at each node. Self loops are ignored. Examples >>> G=nx.complete_graph(5) >>> print(nx.triangles(G,0)) 6 >>> print(nx.triangles(G)) {0: 6, 1: 6, 2: 6, 3: 6, 4: 6} >>> print(list(nx.triangles(G,(0,1)).values())) [6, 6] 4.9.2 transitivity transitivity(G) Compute graph transitivity, the fraction of all possible triangles present in G. Possible triangles are identiﬁed by the number of “triads” (two edges with a shared vertex). The transitivity is T =3#triangles #triads . Parameters G : graph Returns out : ﬂoat Transitivity Examples >>> G = nx.complete_graph(5) >>> print(nx.transitivity(G)) 1.0 4.9.3 clustering clustering(G, nodes=None, weight=None) Compute the clustering coefﬁcient for nodes. For unweighted graphs the clustering of each node u is the fraction of possible triangles that exist, For each node ﬁnd the fraction of possible triangles that exist, cu = 2T(u) deg(u)(deg(u) 1), 182 Chapter 4. Algorithms NetworkX Reference, Release 1.7 where T(u) is the number of triangles through node u and deg(u) is the degree of u. For weighted graphs the clustering is deﬁned as the geometric average of the subgraph edge weights [R158], cu = 1 deg(u)(deg(u) 1)) Xuv (ˆwuv ˆwuw ˆwvw)1/3. The edge weights ˆwuv are normalized by the maximum weight in the network ˆwuv = wuv/ max(w). The value of cu is assigned to 0 if deg(u) < 2. Parameters G : graph nodes : container of nodes, optional (default=all nodes in G) Compute clustering for nodes in this container. weight : string or None, optional (default=None) The edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1. Returns out : ﬂoat, or dictionary Clustering coefﬁcient at speciﬁed nodes Notes Self loops are ignored. References [R158] Examples >>> G=nx.complete_graph(5) >>> print(nx.clustering(G,0)) 1.0 >>> print(nx.clustering(G)) {0: 1.0, 1: 1.0, 2: 1.0, 3: 1.0, 4: 1.0} 4.9.4 average_clustering average_clustering(G, nodes=None, weight=None, count_zeros=True) Compute the average clustering coefﬁcient for the graph G. The clustering coefﬁcient for the graph is the average, C = 1 n Xv2G cv, where n is the number of nodes in G. Parameters G : graph nodes : container of nodes, optional (default=all nodes in G) 4.9. Clustering 183 NetworkX Reference, Release 1.7 Compute average clustering for nodes in this container. weight : string or None, optional (default=None) The edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1. count_zeros : bool (default=False) If False include only the nodes with nonzero clustering in the average. Returns avg : ﬂoat Average clustering Notes This is a space saving routine; it might be faster to use the clustering function to get a list and then take the average. Self loops are ignored. References [R156], [R157] Examples >>> G=nx.complete_graph(5) >>> print(nx.average_clustering(G)) 1.0 4.9.5 square_clustering square_clustering(G, nodes=None) Compute the squares clustering coefﬁcient for nodes. For each node return the fraction of possible squares that exist at the node [R159] C4(v)= P kv u=1 P kv w=u+1 qv(u, w) P kv u=1 P kv w=u+1[av(u, w)+qv(u, w)] , where qv(u, w) are the number of common neighbors of u and w other than v (ie squares), and av(u, w)= (ku (1 + qv(u, w)+✓uv))(kw (1 + qv(u, w)+✓uw)), where ✓uw =1if u and w are connected and 0 otherwise. Parameters G : graph nodes : container of nodes, optional (default=all nodes in G) Compute clustering for nodes in this container. Returns c4 : dictionary A dictionary keyed by node with the square clustering coefﬁcient value. 184 Chapter 4. Algorithms NetworkX Reference, Release 1.7 Notes While C3(v) (triangle clustering) gives the probability that two neighbors of node v are connected with each other, C4(v) is the probability that two neighbors of node v share a common neighbor different from v. This algorithm can be applied to both bipartite and unipartite networks. References [R159] Examples >>> G=nx.complete_graph(5) >>> print(nx.square_clustering(G,0)) 1.0 >>> print(nx.square_clustering(G)) {0: 1.0, 1: 1.0, 2: 1.0, 3: 1.0, 4: 1.0} 4.10 Communities 4.10.1 K-Clique k_clique_communities(G, k[, cliques]) Find k-clique communities in graph using the percolation method. k_clique_communities k_clique_communities(G, k, cliques=None) Find k-clique communities in graph using the percolation method. A k-clique community is the union of all cliques of size k that can be reached through adjacent (sharing k-1 nodes) k-cliques. Parameters G : NetworkX graph k : int Size of smallest clique cliques: list or generator : Precomputed cliques (use networkx.ﬁnd_cliques(G)) Returns Yields sets of nodes, one for each k-clique community. : References [R160] 4.10. Communities 185 NetworkX Reference, Release 1.7 Examples >>> G = nx.complete_graph(5) >>> K5 = nx.convert_node_labels_to_integers(G,first_label=2) >>> G.add_edges_from(K5.edges()) >>> c = list(nx.k_clique_communities(G, 4)) >>> list(c[0]) [0, 1, 2, 3, 4, 5, 6] >>> list(nx.k_clique_communities(G, 6)) [] 4.11 Components 4.11.1 Connectivity Connected components. is_connected(G) Test graph connectivity. number_connected_components(G) Return number of connected components in graph. connected_components(G) Return nodes in connected components of graph. connected_component_subgraphs(G) Return connected components as subgraphs. node_connected_component(G, n) Return nodes in connected components of graph containing node n. is_connected is_connected(G) Test graph connectivity. Parameters G : NetworkX Graph An undirected graph. Returns connected : bool True if the graph is connected, false otherwise. See Also: connected_components Notes For undirected graphs only. Examples >>> G=nx.path_graph(4) >>> print(nx.is_connected(G)) True 186 Chapter 4. Algorithms NetworkX Reference, Release 1.7 number_connected_components number_connected_components(G) Return number of connected components in graph. Parameters G : NetworkX Graph An undirected graph. Returns n : integer Number of connected components See Also: connected_components Notes For undirected graphs only. connected_components connected_components(G) Return nodes in connected components of graph. Parameters G : NetworkX Graph An undirected graph. Returns comp : list of lists A list of nodes for each component of G. See Also: strongly_connected_components Notes The list is ordered from largest connected component to smallest. For undirected graphs only. connected_component_subgraphs connected_component_subgraphs(G) Return connected components as subgraphs. Parameters G : NetworkX Graph An undirected graph. Returns glist : list A list of graphs, one for each connected component of G. See Also: connected_components 4.11. Components 187 NetworkX Reference, Release 1.7 Notes The list is ordered from largest connected component to smallest. For undirected graphs only. Graph, node, and edge attributes are copied to the subgraphs. Examples Get largest connected component as subgraph >>> G=nx.path_graph(4) >>> G.add_edge(5,6) >>> H=nx.connected_component_subgraphs(G)[0] node_connected_component node_connected_component(G, n) Return nodes in connected components of graph containing node n. Parameters G : NetworkX Graph An undirected graph. n : node label A node in G Returns comp : lists A list of nodes in component of G containing node n. See Also: connected_components Notes For undirected graphs only. 4.11.2 Strong connectivity Strongly connected components. is_strongly_connected(G) Test directed graph for strong connectivity. number_strongly_connected_components(G) Return number of strongly connected components in graph. strongly_connected_components(G) Return nodes in strongly connected components of graph. strongly_connected_component_subgraphs(G) Return strongly connected components as subgraphs. strongly_connected_components_recursive(G) Return nodes in strongly connected components of graph. kosaraju_strongly_connected_components(G[, ...]) Return nodes in strongly connected components of graph. condensation(G[, scc]) Returns the condensation of G. 188 Chapter 4. Algorithms NetworkX Reference, Release 1.7 is_strongly_connected is_strongly_connected(G) Test directed graph for strong connectivity. Parameters G : NetworkX Graph A directed graph. Returns connected : bool True if the graph is strongly connected, False otherwise. See Also: strongly_connected_components Notes For directed graphs only. number_strongly_connected_components number_strongly_connected_components(G) Return number of strongly connected components in graph. Parameters G : NetworkX graph A directed graph. Returns n : integer Number of strongly connected components See Also: connected_components Notes For directed graphs only. strongly_connected_components strongly_connected_components(G) Return nodes in strongly connected components of graph. Parameters G : NetworkX Graph An directed graph. Returns comp : list of lists A list of nodes for each component of G. The list is ordered from largest connected component to smallest. See Also: connected_components 4.11. Components 189 NetworkX Reference, Release 1.7 Notes Uses Tarjan’s algorithm with Nuutila’s modiﬁcations. Nonrecursive version of algorithm. References [R166], [R167] strongly_connected_component_subgraphs strongly_connected_component_subgraphs(G) Return strongly connected components as subgraphs. Parameters G : NetworkX Graph A graph. Returns glist : list A list of graphs, one for each strongly connected component of G. See Also: connected_component_subgraphs Notes The list is ordered from largest strongly connected component to smallest. Graph, node, and edge attributes are copied to the subgraphs. strongly_connected_components_recursive strongly_connected_components_recursive(G) Return nodes in strongly connected components of graph. Recursive version of algorithm. Parameters G : NetworkX Graph An directed graph. Returns comp : list of lists A list of nodes for each component of G. The list is ordered from largest connected component to smallest. See Also: connected_components Notes Uses Tarjan’s algorithm with Nuutila’s modiﬁcations. 190 Chapter 4. Algorithms NetworkX Reference, Release 1.7 References [R168], [R169] kosaraju_strongly_connected_components kosaraju_strongly_connected_components(G, source=None) Return nodes in strongly connected components of graph. Parameters G : NetworkX Graph An directed graph. Returns comp : list of lists A list of nodes for each component of G. The list is ordered from largest connected component to smallest. See Also: connected_components Notes Uses Kosaraju’s algorithm. condensation condensation(G, scc=None) Returns the condensation of G. The condensation of G is the graph with each of the strongly connected components contracted into a single node. Parameters G : NetworkX DiGraph A directed graph. scc: list (optional, default=None) : A list of strongly connected components. If provided, the elements in scc must partition the nodes in G. If not provided, it will be calculated as scc=nx.strongly_connected_components(G). Returns C : NetworkX DiGraph The condensation of G. The node labels are integers corresponding to the index of the component in the list of strongly connected components. Notes After contracting all strongly connected components to a single node, the resulting graph is a directed acyclic graph. 4.11. Components 191 NetworkX Reference, Release 1.7 4.11.3 Weak connectivity Weakly connected components. is_weakly_connected(G) Test directed graph for weak connectivity. number_weakly_connected_components(G) Return the number of connected components in G. weakly_connected_components(G) Return weakly connected components of G. weakly_connected_component_subgraphs(G) Return weakly connected components as subgraphs. is_weakly_connected is_weakly_connected(G) Test directed graph for weak connectivity. Parameters G : NetworkX Graph A directed graph. Returns connected : bool True if the graph is weakly connected, False otherwise. See Also: strongly_connected_components Notes For directed graphs only. number_weakly_connected_components number_weakly_connected_components(G) Return the number of connected components in G. For directed graphs only. weakly_connected_components weakly_connected_components(G) Return weakly connected components of G. weakly_connected_component_subgraphs weakly_connected_component_subgraphs(G) Return weakly connected components as subgraphs. Graph, node, and edge attributes are copied to the subgraphs. 4.11.4 Atrracting components Attracting components. is_attracting_component(G) Returns True if G consists of a single attracting component. Continued on next page 192 Chapter 4. Algorithms NetworkX Reference, Release 1.7 Table 4.35 – continued from previous page number_attracting_components(G) Returns the number of attracting components in G. attracting_components(G) Returns a list of attracting components in G. attracting_component_subgraphs(G) Returns a list of attracting component subgraphs from G. is_attracting_component is_attracting_component(G) Returns True if G consists of a single attracting component. Parameters G : DiGraph, MultiDiGraph The graph to be analyzed. Returns attracting : bool True if G has a single attracting component. Otherwise, False. See Also: attracting_components, number_attracting_components, attracting_component_subgraphs number_attracting_components number_attracting_components(G) Returns the number of attracting components in G. Parameters G : DiGraph, MultiDiGraph The graph to be analyzed. Returns n : int The number of attracting components in G. See Also: attracting_components, is_attracting_component, attracting_component_subgraphs attracting_components attracting_components(G) Returns a list of attracting components in G. An attracting component in a directed graph G is a strongly connected component with the property that a random walker on the graph will never leave the component, once it enters the component. The nodes in attracting components can also be thought of as recurrent nodes. If a random walker enters the attractor containing the node, then the node will be visited inﬁnitely often. Parameters G : DiGraph, MultiDiGraph The graph to be analyzed. Returns attractors : list The list of attracting components, sorted from largest attracting component to smallest attracting component. 4.11. Components 193 NetworkX Reference, Release 1.7 See Also: number_attracting_components, is_attracting_component, attracting_component_subgraphs attracting_component_subgraphs attracting_component_subgraphs(G) Returns a list of attracting component subgraphs from G. Parameters G : DiGraph, MultiDiGraph The graph to be analyzed. Returns subgraphs : list A list of node-induced subgraphs of the attracting components of G. See Also: attracting_components, number_attracting_components, is_attracting_component Notes Graph, node, and edge attributes are copied to the subgraphs. 4.11.5 Biconnected components Biconnected components and articulation points. is_biconnected(G) Return True if the graph is biconnected, False otherwise. biconnected_components(G) Return a generator of sets of nodes, one set for each biconnected biconnected_component_edges(G) Return a generator of lists of edges, one list for each biconnected component of the input graph. biconnected_component_subgraphs(G) Return a generator of graphs, one graph for each biconnected component of the input graph. articulation_points(G) Return a generator of articulation points, or cut vertices, of a graph. is_biconnected is_biconnected(G) Return True if the graph is biconnected, False otherwise. A graph is biconnected if, and only if, it cannot be disconnected by removing only one node (and all edges incident on that node). If removing a node increases the number of disconnected components in the graph, that node is called an articulation point, or cut vertex. A biconnected graph has no articulation points. Parameters G : NetworkX Graph An undirected graph. Returns biconnected : bool True if the graph is biconnected, False otherwise. Raises NetworkXError : : If the input graph is not undirected. See Also: 194 Chapter 4. Algorithms NetworkX Reference, Release 1.7 biconnected_components, articulation_points, biconnected_component_edges, biconnected_component_subgraphs Notes The algorithm to ﬁnd articulation points and biconnected components is implemented using a non-recursive depth-ﬁrst-search (DFS) that keeps track of the highest level that back edges reach in the DFS tree. A node n is an articulation point if, and only if, there exists a subtree rooted at n such that there is no back edge from any successor of n that links to a predecessor of n in the DFS tree. By keeping track of all the edges traversed by the DFS we can obtain the biconnected components because all edges of a bicomponent will be traversed consecutively between articulation points. References [R165] Examples >>> G=nx.path_graph(4) >>> print(nx.is_biconnected(G)) False >>> G.add_edge(0,3) >>> print(nx.is_biconnected(G)) True biconnected_components biconnected_components(G) Return a generator of sets of nodes, one set for each biconnected component of the graph Biconnected components are maximal subgraphs such that the removal of a node (and all edges incident on that node) will not disconnect the subgraph. Note that nodes may be part of more than one biconnected component. Those nodes are articulation points, or cut vertices. The removal of articulation points will increase the number of connected components of the graph. Notice that by convention a dyad is considered a biconnected component. Parameters G : NetworkX Graph An undirected graph. Returns nodes : generator Generator of sets of nodes, one set for each biconnected component. Raises NetworkXError : : If the input graph is not undirected. See Also: is_biconnected, articulation_points, biconnected_component_edges, biconnected_component_subgraphs 4.11. Components 195 NetworkX Reference, Release 1.7 Notes The algorithm to ﬁnd articulation points and biconnected components is implemented using a non-recursive depth-ﬁrst-search (DFS) that keeps track of the highest level that back edges reach in the DFS tree. A node n is an articulation point if, and only if, there exists a subtree rooted at n such that there is no back edge from any successor of n that links to a predecessor of n in the DFS tree. By keeping track of all the edges traversed by the DFS we can obtain the biconnected components because all edges of a bicomponent will be traversed consecutively between articulation points. References [R164] Examples >>> G = nx.barbell_graph(4,2) >>> print(nx.is_biconnected(G)) False >>> components = nx.biconnected_components(G) >>> G.add_edge(2,8) >>> print(nx.is_biconnected(G)) True >>> components = nx.biconnected_components(G) biconnected_component_edges biconnected_component_edges(G) Return a generator of lists of edges, one list for each biconnected component of the input graph. Biconnected components are maximal subgraphs such that the removal of a node (and all edges incident on that node) will not disconnect the subgraph. Note that nodes may be part of more than one biconnected compo- nent. Those nodes are articulation points, or cut vertices. However, each edge belongs to one, and only one, biconnected component. Notice that by convention a dyad is considered a biconnected component. Parameters G : NetworkX Graph An undirected graph. Returns edges : generator Generator of lists of edges, one list for each bicomponent. Raises NetworkXError : : If the input graph is not undirected. See Also: is_biconnected, biconnected_components, articulation_points, biconnected_component_subgraphs 196 Chapter 4. Algorithms NetworkX Reference, Release 1.7 Notes The algorithm to ﬁnd articulation points and biconnected components is implemented using a non-recursive depth-ﬁrst-search (DFS) that keeps track of the highest level that back edges reach in the DFS tree. A node n is an articulation point if, and only if, there exists a subtree rooted at n such that there is no back edge from any successor of n that links to a predecessor of n in the DFS tree. By keeping track of all the edges traversed by the DFS we can obtain the biconnected components because all edges of a bicomponent will be traversed consecutively between articulation points. References [R162] Examples >>> G = nx.barbell_graph(4,2) >>> print(nx.is_biconnected(G)) False >>> components = nx.biconnected_component_edges(G) >>> G.add_edge(2,8) >>> print(nx.is_biconnected(G)) True >>> components = nx.biconnected_component_edges(G) biconnected_component_subgraphs biconnected_component_subgraphs(G) Return a generator of graphs, one graph for each biconnected component of the input graph. Biconnected components are maximal subgraphs such that the removal of a node (and all edges incident on that node) will not disconnect the subgraph. Note that nodes may be part of more than one biconnected component. Those nodes are articulation points, or cut vertices. The removal of articulation points will increase the number of connected components of the graph. Notice that by convention a dyad is considered a biconnected component. Parameters G : NetworkX Graph An undirected graph. Returns graphs : generator Generator of graphs, one graph for each biconnected component. Raises NetworkXError : : If the input graph is not undirected. See Also: is_biconnected, articulation_points, biconnected_component_edges, biconnected_components 4.11. Components 197 NetworkX Reference, Release 1.7 Notes The algorithm to ﬁnd articulation points and biconnected components is implemented using a non-recursive depth-ﬁrst-search (DFS) that keeps track of the highest level that back edges reach in the DFS tree. A node n is an articulation point if, and only if, there exists a subtree rooted at n such that there is no back edge from any successor of n that links to a predecessor of n in the DFS tree. By keeping track of all the edges traversed by the DFS we can obtain the biconnected components because all edges of a bicomponent will be traversed consecutively between articulation points. Graph, node, and edge attributes are copied to the subgraphs. References [R163] Examples >>> G = nx.barbell_graph(4,2) >>> print(nx.is_biconnected(G)) False >>> subgraphs = nx.biconnected_component_subgraphs(G) articulation_points articulation_points(G) Return a generator of articulation points, or cut vertices, of a graph. An articulation point or cut vertex is any node whose removal (along with all its incident edges) increases the number of connected components of a graph. An undirected connected graph without articulation points is biconnected. Articulation points belong to more than one biconnected component of a graph. Notice that by convention a dyad is considered a biconnected component. Parameters G : NetworkX Graph An undirected graph. Returns articulation points : generator generator of nodes Raises NetworkXError : : If the input graph is not undirected. See Also: is_biconnected, biconnected_components, biconnected_component_edges, biconnected_component_subgraphs Notes The algorithm to ﬁnd articulation points and biconnected components is implemented using a non-recursive depth-ﬁrst-search (DFS) that keeps track of the highest level that back edges reach in the DFS tree. A node n is an articulation point if, and only if, there exists a subtree rooted at n such that there is no back edge from any successor of n that links to a predecessor of n in the DFS tree. By keeping track of all the edges traversed 198 Chapter 4. Algorithms NetworkX Reference, Release 1.7 by the DFS we can obtain the biconnected components because all edges of a bicomponent will be traversed consecutively between articulation points. References [R161] Examples >>> G = nx.barbell_graph(4,2) >>> print(nx.is_biconnected(G)) False >>> list(nx.articulation_points(G)) [6, 5, 4, 3] >>> G.add_edge(2,8) >>> print(nx.is_biconnected(G)) True >>> list(nx.articulation_points(G)) [] 4.12 Cores Find the k-cores of a graph. The k-core is found by recursively pruning nodes with degrees less than k. See the following reference for details: An O(m) Algorithm for Cores Decomposition of Networks Vladimir Batagelj and Matjaz Zaversnik, 2003. http://arxiv.org/abs/cs.DS/0310049 core_number(G) Return the core number for each vertex. k_core(G[, k, core_number]) Return the k-core of G. k_shell(G[, k, core_number]) Return the k-shell of G. k_crust(G[, k, core_number]) Return the k-crust of G. k_corona(G, k[, core_number]) Return the k-crust of G. 4.12.1 core_number core_number(G) Return the core number for each vertex. A k-core is a maximal subgraph that contains nodes of degree k or more. The core number of a node is the largest value k of a k-core containing that node. Parameters G : NetworkX graph A graph or directed graph Returns core_number : dictionary A dictionary keyed by node to the core number. 4.12. Cores 199 NetworkX Reference, Release 1.7 Raises NetworkXError : The k-core is not deﬁned for graphs with self loops or parallel edges. Notes Not implemented for graphs with parallel edges or self loops. For directed graphs the node degree is deﬁned to be the in-degree + out-degree. References [R170] 4.12.2 k_core k_core(G, k=None, core_number=None) Return the k-core of G. A k-core is a maximal subgraph that contains nodes of degree k or more. Parameters G : NetworkX graph A graph or directed graph k : int, optional The order of the core. If not speciﬁed return the main core. core_number : dictionary, optional Precomputed core numbers for the graph G. Returns G : NetworkX graph The k-core subgraph Raises NetworkXError : The k-core is not deﬁned for graphs with self loops or parallel edges. See Also: core_number Notes The main core is the core with the largest degree. Not implemented for graphs with parallel edges or self loops. For directed graphs the node degree is deﬁned to be the in-degree + out-degree. Graph, node, and edge attributes are copied to the subgraph. References [R171] 200 Chapter 4. Algorithms NetworkX Reference, Release 1.7 4.12.3 k_shell k_shell(G, k=None, core_number=None) Return the k-shell of G. The k-shell is the subgraph of nodes in the k-core containing nodes of exactly degree k. Parameters G : NetworkX graph A graph or directed graph. k : int, optional The order of the shell. If not speciﬁed return the main shell. core_number : dictionary, optional Precomputed core numbers for the graph G. Returns G : NetworkX graph The k-shell subgraph Raises NetworkXError : The k-shell is not deﬁned for graphs with self loops or parallel edges. See Also: core_number, k_corona, ---------- Shai Carmi, Shlomo Havlin, Scott Kirkpatrick, Yuval Shavitt, and Eran Shir, PNAS July 3, 2007 vol. 104 no. 27 11150-11154 http //www.pnas.org/content/104/27/11150.full Notes This is similar to k_corona but in that case only neighbors in the k-core are considered. Not implemented for graphs with parallel edges or self loops. For directed graphs the node degree is deﬁned to be the in-degree + out-degree. Graph, node, and edge attributes are copied to the subgraph. 4.12.4 k_crust k_crust(G, k=None, core_number=None) Return the k-crust of G. The k-crust is the graph G with the k-core removed. Parameters G : NetworkX graph A graph or directed graph. k : int, optional The order of the shell. If not speciﬁed return the main crust. core_number : dictionary, optional Precomputed core numbers for the graph G. 4.12. Cores 201 NetworkX Reference, Release 1.7 Returns G : NetworkX graph The k-crust subgraph Raises NetworkXError : The k-crust is not deﬁned for graphs with self loops or parallel edges. See Also: core_number Notes This deﬁnition of k-crust is different than the deﬁnition in [R173]. The k-crust in [R173] is equivalent to the k+1 crust of this algorithm. Not implemented for graphs with parallel edges or self loops. For directed graphs the node degree is deﬁned to be the in-degree + out-degree. Graph, node, and edge attributes are copied to the subgraph. References [R173] 4.12.5 k_corona k_corona(G, k, core_number=None) Return the k-crust of G. The k-corona is the subset of vertices in the k-core which have exactly k neighbours in the k-core. Parameters G : NetworkX graph A graph or directed graph k : int The order of the corona. core_number : dictionary, optional Precomputed core numbers for the graph G. Returns G : NetworkX graph The k-corona subgraph Raises NetworkXError : The k-cornoa is not deﬁned for graphs with self loops or parallel edges. See Also: core_number 202 Chapter 4. Algorithms NetworkX Reference, Release 1.7 Notes Not implemented for graphs with parallel edges or self loops. For directed graphs the node degree is deﬁned to be the in-degree + out-degree. Graph, node, and edge attributes are copied to the subgraph. References [R172] 4.13 Cycles cycle_basis(G[, root]) Returns a list of cycles which form a basis for cycles of G. simple_cycles(G) Find simple cycles (elementary circuits) of a directed graph. 4.13.1 cycle_basis cycle_basis(G, root=None) Returns a list of cycles which form a basis for cycles of G. A basis for cycles of a network is a minimal collection of cycles such that any cycle in the network can be written as a sum of cycles in the basis. Here summation of cycles is deﬁned as “exclusive or” of the edges. Cycle bases are useful, e.g. when deriving equations for electric circuits using Kirchhoff’s Laws. Parameters G : NetworkX Graph root : node, optional Specify starting node for basis. Returns A list of cycle lists. Each cycle list is a list of nodes : which forms a cycle (loop) in G. : See Also: simple_cycles Notes This is adapted from algorithm CACM 491 [R174]. References [R174] Examples 4.13. Cycles 203 NetworkX Reference, Release 1.7 >>> G=nx.Graph() >>> G.add_cycle([0,1,2,3]) >>> G.add_cycle([0,3,4,5]) >>> print(nx.cycle_basis(G,0)) [[3, 4, 5, 0], [1, 2, 3, 0]] 4.13.2 simple_cycles simple_cycles(G) Find simple cycles (elementary circuits) of a directed graph. An simple cycle, or elementary circuit, is a closed path where no node appears twice, except that the ﬁrst and last node are the same. Two elementary circuits are distinct if they are not cyclic permutations of each other. Parameters G : NetworkX DiGraph A directed graph Returns A list of circuits, where each circuit is a list of nodes, with the ﬁrst : and last node being the same. : Example: : >>> G = nx.DiGraph([(0, 0), (0, 1), (0, 2), (1, 2), (2, 0), (2, 1), (2, 2)]) : >>> nx.simple_cycles(G) : [[0, 0], [0, 1, 2, 0], [0, 2, 0], [1, 2, 1], [2, 2]] : See Also: cycle_basis Notes The implementation follows pp. 79-80 in [R175]. The time complexity is O((n+e)(c+1)) for n nodes, e edges and c elementary circuits. References [R175] 4.14 Directed Acyclic Graphs topological_sort(G[, nbunch]) Return a list of nodes in topological sort order. topological_sort_recursive(G[, nbunch]) Return a list of nodes in topological sort order. is_directed_acyclic_graph(G) Return True if the graph G is a directed acyclic graph (DAG) or is_aperiodic(G) Return True if G is aperiodic. 204 Chapter 4. Algorithms NetworkX Reference, Release 1.7 4.14.1 topological_sort topological_sort(G, nbunch=None) Return a list of nodes in topological sort order. A topological sort is a nonunique permutation of the nodes such that an edge from u to v implies that u appears before v in the topological sort order. Parameters G : NetworkX digraph A directed graph nbunch : container of nodes (optional) Explore graph in speciﬁed order given in nbunch Raises NetworkXError : Topological sort is deﬁned for directed graphs only. If the graph G is undirected, a NetworkXError is raised. NetworkXUnfeasible : If G is not a directed acyclic graph (DAG) no topological sort exists and a NetworkX- Unfeasible exception is raised. See Also: is_directed_acyclic_graph Notes This algorithm is based on a description and proof in The Algorithm Design Manual [R177] . References [R177] 4.14.2 topological_sort_recursive topological_sort_recursive(G, nbunch=None) Return a list of nodes in topological sort order. A topological sort is a nonunique permutation of the nodes such that an edge from u to v implies that u appears before v in the topological sort order. Parameters G : NetworkX digraph nbunch : container of nodes (optional) Explore graph in speciﬁed order given in nbunch Raises NetworkXError : Topological sort is deﬁned for directed graphs only. If the graph G is undirected, a NetworkXError is raised. NetworkXUnfeasible : If G is not a directed acyclic graph (DAG) no topological sort exists and a NetworkX- Unfeasible exception is raised. 4.14. Directed Acyclic Graphs 205 NetworkX Reference, Release 1.7 See Also: topological_sort, is_directed_acyclic_graph Notes This is a recursive version of topological sort. 4.14.3 is_directed_acyclic_graph is_directed_acyclic_graph(G) Return True if the graph G is a directed acyclic graph (DAG) or False if not. Parameters G : NetworkX graph A graph Returns is_dag : bool True if G is a DAG, false otherwise 4.14.4 is_aperiodic is_aperiodic(G) Return True if G is aperiodic. A directed graph is aperiodic if there is no integer k > 1 that divides the length of every cycle in the graph. Parameters G : NetworkX DiGraph Graph Returns aperiodic : boolean True if the graph is aperiodic False otherwise Raises NetworkXError : If G is not directed Notes This uses the method outlined in [R176], which runs in O(m) time given m edges in G. Note that a graph is not aperiodic if it is acyclic as every integer trivial divides length 0 cycles. References [R176] 4.15 Distance Measures Graph diameter, radius, eccentricity and other properties. 206 Chapter 4. Algorithms NetworkX Reference, Release 1.7 center(G[, e]) Return the periphery of the graph G. diameter(G[, e]) Return the diameter of the graph G. eccentricity(G[, v, sp]) Return the eccentricity of nodes in G. periphery(G[, e]) Return the periphery of the graph G. radius(G[, e]) Return the radius of the graph G. 4.15.1 center center(G, e=None) Return the periphery of the graph G. The center is the set of nodes with eccentricity equal to radius. Parameters G : NetworkX graph A graph e : eccentricity dictionary, optional A precomputed dictionary of eccentricities. Returns c : list List of nodes in center 4.15.2 diameter diameter(G, e=None) Return the diameter of the graph G. The diameter is the maximum eccentricity. Parameters G : NetworkX graph A graph e : eccentricity dictionary, optional A precomputed dictionary of eccentricities. Returns d : integer Diameter of graph See Also: eccentricity 4.15.3 eccentricity eccentricity(G, v=None, sp=None) Return the eccentricity of nodes in G. The eccentricity of a node v is the maximum distance from v to all other nodes in G. Parameters G : NetworkX graph A graph v : node, optional 4.15. Distance Measures 207 NetworkX Reference, Release 1.7 Return value of speciﬁed node sp : dict of dicts, optional All pairs shortest path lenghts as a dictionary of dictionaries Returns ecc : dictionary A dictionary of eccentricity values keyed by node. 4.15.4 periphery periphery(G, e=None) Return the periphery of the graph G. The periphery is the set of nodes with eccentricity equal to the diameter. Parameters G : NetworkX graph A graph e : eccentricity dictionary, optional A precomputed dictionary of eccentricities. Returns p : list List of nodes in periphery 4.15.5 radius radius(G, e=None) Return the radius of the graph G. The radius is the minimum eccentricity. Parameters G : NetworkX graph A graph e : eccentricity dictionary, optional A precomputed dictionary of eccentricities. Returns r : integer Radius of graph 4.16 Distance-Regular Graphs is_distance_regular(G) Returns True if the graph is distance regular, False otherwise. intersection_array(G) Returns the intersection array of a distance-regular graph. global_parameters(b, c) Return global parameters for a given intersection array. 4.16.1 is_distance_regular is_distance_regular(G) Returns True if the graph is distance regular, False otherwise. 208 Chapter 4. Algorithms NetworkX Reference, Release 1.7 A connected graph G is distance-regular if for any nodes x,y and any integers i,j=0,1,...,d (where d is the graph diameter), the number of vertices at distance i from x and distance j from y depends only on i,j and the graph distance between x and y, independently of the choice of x and y. Parameters G: Networkx graph (undirected) : Returns bool : True if the graph is Distance Regular, False otherwise See Also: intersection_array, global_parameters Notes For undirected and simple graphs only References [R180], [R181] Examples >>> G=nx.hypercube_graph(6) >>> nx.is_distance_regular(G) True 4.16.2 intersection_array intersection_array(G) Returns the intersection array of a distance-regular graph. Given a distance-regular graph G with integers b_i, c_i,i = 0,....,d such that for any 2 vertices x,y in G at a distance i=d(x,y), there are exactly c_i neighbors of y at a distance of i-1 from x and b_i neighbors of y at a distance of i+1 from x. A distance regular graph’sintersection array is given by, [b_0,b_1,.....b_{d-1};c_1,c_2,.....c_d] Parameters G: Networkx graph (undirected) : Returns b,c: tuple of lists : See Also: global_parameters References [R179] 4.16. Distance-Regular Graphs 209 NetworkX Reference, Release 1.7 Examples >>> G=nx.icosahedral_graph() >>> nx.intersection_array(G) ([5, 2, 1], [1, 2, 5]) 4.16.3 global_parameters global_parameters(b, c) Return global parameters for a given intersection array. Given a distance-regular graph G with integers b_i, c_i,i = 0,....,d such that for any 2 vertices x,y in G at a distance i=d(x,y), there are exactly c_i neighbors of y at a distance of i-1 from x and b_i neighbors of y at a distance of i+1 from x. Thus, a distance regular graph has the global parameters, [[c_0,a_0,b_0],[c_1,a_1,b_1],......,[c_d,a_d,b_d]] for the intersection array [b_0,b_1,.....b_{d-1};c_1,c_2,.....c_d] where a_i+b_i+c_i=k , k= degree of every vertex. Parameters b,c: tuple of lists : Returns p : list of three-tuples See Also: intersection_array References [R178] Examples >>> G=nx.dodecahedral_graph() >>> b,c=nx.intersection_array(G) >>> list(nx.global_parameters(b,c)) [(0, 0, 3), (1, 0, 2), (1, 1, 1), (1, 1, 1), (2, 0, 1), (3, 0, 0)] 4.17 Eulerian Eulerian circuits and graphs. is_eulerian(G) Return True if G is an Eulerian graph, False otherwise. eulerian_circuit(G[, source]) Return the edges of an Eulerian circuit in G. 4.17.1 is_eulerian is_eulerian(G) Return True if G is an Eulerian graph, False otherwise. An Eulerian graph is a graph with an Eulerian circuit. Parameters G : graph 210 Chapter 4. Algorithms NetworkX Reference, Release 1.7 A NetworkX Graph Notes This implementation requires the graph to be connected (or strongly connected for directed graphs). Examples >>> nx.is_eulerian(nx.DiGraph({0:[3], 1:[2], 2:[3], 3:[0, 1]})) True >>> nx.is_eulerian(nx.complete_graph(5)) True >>> nx.is_eulerian(nx.petersen_graph()) False 4.17.2 eulerian_circuit eulerian_circuit(G, source=None) Return the edges of an Eulerian circuit in G. An Eulerian circuit is a path that crosses every edge in G exactly once and ﬁnishes at the starting node. Parameters G : graph A NetworkX Graph source : node, optional Starting node for circuit. Returns edges : generator A generator that produces edges in the Eulerian circuit. Raises NetworkXError : If the graph is not Eulerian. See Also: is_eulerian Notes Uses Fleury’s algorithm [R182],[R183]_ References [R182], [R183] Examples 4.17. Eulerian 211 NetworkX Reference, Release 1.7 >>> G=nx.complete_graph(3) >>> list(nx.eulerian_circuit(G)) [(0, 1), (1, 2), (2, 0)] >>> list(nx.eulerian_circuit(G,source=1)) [(1, 0), (0, 2), (2, 1)] >>> [u for u,v in nx.eulerian_circuit(G)] # nodes in circuit [0, 1, 2] 4.18 Flows 4.18.1 Ford-Fulkerson max_flow(G, s, t[, capacity]) Find the value of a maximum single-commodity ﬂow. min_cut(G, s, t[, capacity]) Compute the value of a minimum (s, t)-cut. ford_fulkerson(G, s, t[, capacity]) Find a maximum single-commodity ﬂow using the Ford-Fulkerson ford_fulkerson_flow(G, s, t[, capacity]) Return a maximum ﬂow for a single-commodity ﬂow problem. max_ﬂow max_flow(G, s, t, capacity=’capacity’) Find the value of a maximum single-commodity ﬂow. Parameters G : NetworkX graph Edges of the graph are expected to have an attribute called ‘capacity’. If this attribute is not present, the edge is considered to have inﬁnite capacity. s : node Source node for the ﬂow. t : node Sink node for the ﬂow. capacity: string : Edges of the graph G are expected to have an attribute capacity that indicates how much ﬂow the edge can support. If this attribute is not present, the edge is considered to have inﬁnite capacity. Default value: ‘capacity’. Returns ﬂow_value : integer, ﬂoat Value of the maximum ﬂow, i.e., net outﬂow from the source. Raises NetworkXError : The algorithm does not support MultiGraph and MultiDiGraph. If the input graph is an instance of one of these two classes, a NetworkXError is raised. NetworkXUnbounded : If the graph has a path of inﬁnite capacity, the value of a feasible ﬂow on the graph is unbounded above and the function raises a NetworkXUnbounded. 212 Chapter 4. Algorithms NetworkX Reference, Release 1.7 Examples >>> import networkx as nx >>> G = nx.DiGraph() >>> G.add_edge(’x’,’a’, capacity=3.0) >>> G.add_edge(’x’,’b’, capacity=1.0) >>> G.add_edge(’a’,’c’, capacity=3.0) >>> G.add_edge(’b’,’c’, capacity=5.0) >>> G.add_edge(’b’,’d’, capacity=4.0) >>> G.add_edge(’d’,’e’, capacity=2.0) >>> G.add_edge(’c’,’y’, capacity=2.0) >>> G.add_edge(’e’,’y’, capacity=3.0) >>> flow = nx.max_flow(G, ’x’, ’y’) >>> flow 3.0 min_cut min_cut(G, s, t, capacity=’capacity’) Compute the value of a minimum (s, t)-cut. Use the max-ﬂow min-cut theorem, i.e., the capacity of a minimum capacity cut is equal to the ﬂow value of a maximum ﬂow. Parameters G : NetworkX graph Edges of the graph are expected to have an attribute called ‘capacity’. If this attribute is not present, the edge is considered to have inﬁnite capacity. s : node Source node for the ﬂow. t : node Sink node for the ﬂow. capacity: string : Edges of the graph G are expected to have an attribute capacity that indicates how much ﬂow the edge can support. If this attribute is not present, the edge is considered to have inﬁnite capacity. Default value: ‘capacity’. Returns cutValue : integer, ﬂoat Value of the minimum cut. Raises NetworkXUnbounded : If the graph has a path of inﬁnite capacity, all cuts have inﬁnite capacity and the function raises a NetworkXError. Examples >>> import networkx as nx >>> G = nx.DiGraph() >>> G.add_edge(’x’,’a’, capacity = 3.0) >>> G.add_edge(’x’,’b’, capacity = 1.0) >>> G.add_edge(’a’,’c’, capacity = 3.0) 4.18. Flows 213 NetworkX Reference, Release 1.7 >>> G.add_edge(’b’,’c’, capacity = 5.0) >>> G.add_edge(’b’,’d’, capacity = 4.0) >>> G.add_edge(’d’,’e’, capacity = 2.0) >>> G.add_edge(’c’,’y’, capacity = 2.0) >>> G.add_edge(’e’,’y’, capacity = 3.0) >>> nx.min_cut(G, ’x’, ’y’) 3.0 ford_fulkerson ford_fulkerson(G, s, t, capacity=’capacity’) Find a maximum single-commodity ﬂow using the Ford-Fulkerson algorithm. This algorithm uses Edmonds-Karp-Dinitz path selection rule which guarantees a running time of O(nm^2) for n nodes and m edges. Parameters G : NetworkX graph Edges of the graph are expected to have an attribute called ‘capacity’. If this attribute is not present, the edge is considered to have inﬁnite capacity. s : node Source node for the ﬂow. t : node Sink node for the ﬂow. capacity: string : Edges of the graph G are expected to have an attribute capacity that indicates how much ﬂow the edge can support. If this attribute is not present, the edge is considered to have inﬁnite capacity. Default value: ‘capacity’. Returns ﬂow_value : integer, ﬂoat Value of the maximum ﬂow, i.e., net outﬂow from the source. ﬂow_dict : dictionary Dictionary of dictionaries keyed by nodes such that ﬂow_dict[u][v] is the ﬂow edge (u, v). Raises NetworkXError : The algorithm does not support MultiGraph and MultiDiGraph. If the input graph is an instance of one of these two classes, a NetworkXError is raised. NetworkXUnbounded : If the graph has a path of inﬁnite capacity, the value of a feasible ﬂow on the graph is unbounded above and the function raises a NetworkXUnbounded. Examples >>> import networkx as nx >>> G = nx.DiGraph() >>> G.add_edge(’x’,’a’, capacity=3.0) >>> G.add_edge(’x’,’b’, capacity=1.0) >>> G.add_edge(’a’,’c’, capacity=3.0) 214 Chapter 4. Algorithms NetworkX Reference, Release 1.7 >>> G.add_edge(’b’,’c’, capacity=5.0) >>> G.add_edge(’b’,’d’, capacity=4.0) >>> G.add_edge(’d’,’e’, capacity=2.0) >>> G.add_edge(’c’,’y’, capacity=2.0) >>> G.add_edge(’e’,’y’, capacity=3.0) >>> flow, F = nx.ford_fulkerson(G, ’x’, ’y’) >>> flow 3.0 ford_fulkerson_ﬂow ford_fulkerson_flow(G, s, t, capacity=’capacity’) Return a maximum ﬂow for a single-commodity ﬂow problem. Parameters G : NetworkX graph Edges of the graph are expected to have an attribute called ‘capacity’. If this attribute is not present, the edge is considered to have inﬁnite capacity. s : node Source node for the ﬂow. t : node Sink node for the ﬂow. capacity: string : Edges of the graph G are expected to have an attribute capacity that indicates how much ﬂow the edge can support. If this attribute is not present, the edge is considered to have inﬁnite capacity. Default value: ‘capacity’. Returns ﬂow_dict : dictionary Dictionary of dictionaries keyed by nodes such that ﬂow_dict[u][v] is the ﬂow edge (u, v). Raises NetworkXError : The algorithm does not support MultiGraph and MultiDiGraph. If the input graph is an instance of one of these two classes, a NetworkXError is raised. NetworkXUnbounded : If the graph has a path of inﬁnite capacity, the value of a feasible ﬂow on the graph is unbounded above and the function raises a NetworkXUnbounded. Examples >>> import networkx as nx >>> G = nx.DiGraph() >>> G.add_edge(’x’,’a’, capacity=3.0) >>> G.add_edge(’x’,’b’, capacity=1.0) >>> G.add_edge(’a’,’c’, capacity=3.0) >>> G.add_edge(’b’,’c’, capacity=5.0) >>> G.add_edge(’b’,’d’, capacity=4.0) >>> G.add_edge(’d’,’e’, capacity=2.0) >>> G.add_edge(’c’,’y’, capacity=2.0) >>> G.add_edge(’e’,’y’, capacity=3.0) 4.18. Flows 215 NetworkX Reference, Release 1.7 >>> F = nx.ford_fulkerson_flow(G, ’x’, ’y’) >>> for u, v in G.edges_iter(): ... print(’(%s, %s) %.2f’ % (u, v, F[u][v])) ... (a, c) 2.00 (c, y) 2.00 (b, c) 0.00 (b, d) 1.00 (e, y) 1.00 (d, e) 1.00 (x, a) 2.00 (x, b) 1.00 4.18.2 Network Simplex network_simplex(G[, demand, capacity, weight]) Find a minimum cost ﬂow satisfying all demands in digraph G. min_cost_flow_cost(G[, demand, capacity, weight]) Find the cost of a minimum cost ﬂow satisfying all demands in digraph G. min_cost_flow(G[, demand, capacity, weight]) Return a minimum cost ﬂow satisfying all demands in digraph G. cost_of_flow(G, ﬂowDict[, weight]) Compute the cost of the ﬂow given by ﬂowDict on graph G. max_flow_min_cost(G, s, t[, capacity, weight]) Return a maximum (s, t)-ﬂow of minimum cost. network_simplex network_simplex(G, demand=’demand’, capacity=’capacity’, weight=’weight’) Find a minimum cost ﬂow satisfying all demands in digraph G. This is a primal network simplex algorithm that uses the leaving arc rule to prevent cycling. G is a digraph with edge costs and capacities and in which nodes have demand, i.e., they want to send or receive some amount of ﬂow. A negative demand means that the node wants to send ﬂow, a positive demand means that the node want to receive ﬂow. A ﬂow on the digraph G satisﬁes all demand if the net ﬂow into each node is equal to the demand of that node. Parameters G : NetworkX graph DiGraph on which a minimum cost ﬂow satisfying all demands is to be found. demand: string : Nodes of the graph G are expected to have an attribute demand that indicates how much ﬂow a node wants to send (negative demand) or receive (positive demand). Note that the sum of the demands should be 0 otherwise the problem in not feasible. If this attribute is not present, a node is considered to have 0 demand. Default value: ‘demand’. capacity: string : Edges of the graph G are expected to have an attribute capacity that indicates how much ﬂow the edge can support. If this attribute is not present, the edge is considered to have inﬁnite capacity. Default value: ‘capacity’. weight: string : Edges of the graph G are expected to have an attribute weight that indicates the cost in- curred by sending one unit of ﬂow on that edge. If not present, the weight is considered to be 0. Default value: ‘weight’. Returns ﬂowCost: integer, ﬂoat : 216 Chapter 4. Algorithms NetworkX Reference, Release 1.7 Cost of a minimum cost ﬂow satisfying all demands. ﬂowDict: dictionary : Dictionary of dictionaries keyed by nodes such that ﬂowDict[u][v] is the ﬂow edge (u, v). Raises NetworkXError : This exception is raised if the input graph is not directed, not connected or is a multi- graph. NetworkXUnfeasible : This exception is raised in the following situations: • The sum of the demands is not zero. Then, there is no ﬂow satisfying all demands. • There is no ﬂow satisfying all demand. NetworkXUnbounded : This exception is raised if the digraph G has a cycle of negative cost and inﬁnite capacity. Then, the cost of a ﬂow satisfying all demands is unbounded below. See Also: cost_of_flow, max_flow_min_cost, min_cost_flow, min_cost_flow_cost Notes This algorithm is not guaranteed to work if edge weights are ﬂoating point numbers (overﬂows and roundoff errors can cause problems). References W. J. Cook, W. H. Cunningham, W. R. Pulleyblank and A. Schrijver. Combinatorial Optimization. Wiley- Interscience, 1998. Examples A simple example of a min cost ﬂow problem. >>> import networkx as nx >>> G = nx.DiGraph() >>> G.add_node(’a’, demand = -5) >>> G.add_node(’d’, demand = 5) >>> G.add_edge(’a’, ’b’, weight = 3, capacity = 4) >>> G.add_edge(’a’, ’c’, weight = 6, capacity = 10) >>> G.add_edge(’b’, ’d’, weight = 1, capacity = 9) >>> G.add_edge(’c’, ’d’, weight = 2, capacity = 5) >>> flowCost, flowDict = nx.network_simplex(G) >>> flowCost 24 >>> flowDict {’a’: {’c’: 1, ’b’: 4}, ’c’: {’d’: 1}, ’b’: {’d’: 4}, ’d’: {}} 4.18. Flows 217 NetworkX Reference, Release 1.7 The mincost ﬂow algorithm can also be used to solve shortest path problems. To ﬁnd the shortest path between two nodes u and v, give all edges an inﬁnite capacity, give node u a demand of -1 and node v a demand a 1. Then run the network simplex. The value of a min cost ﬂow will be the distance between u and v and edges carrying positive ﬂow will indicate the path. >>> G=nx.DiGraph() >>> G.add_weighted_edges_from([(’s’,’u’,10), (’s’,’x’,5), ... (’u’,’v’,1), (’u’,’x’,2), ... (’v’,’y’,1), (’x’,’u’,3), ... (’x’,’v’,5), (’x’,’y’,2), ... (’y’,’s’,7), (’y’,’v’,6)]) >>> G.add_node(’s’, demand = -1) >>> G.add_node(’v’, demand = 1) >>> flowCost, flowDict = nx.network_simplex(G) >>> flowCost == nx.shortest_path_length(G, ’s’, ’v’, weight = ’weight’) True >>> [(u, v) for u in flowDict for v in flowDict[u] if flowDict[u][v] > 0] [(’x’, ’u’), (’s’, ’x’), (’u’, ’v’)] >>> nx.shortest_path(G, ’s’, ’v’, weight = ’weight’) [’s’, ’x’, ’u’, ’v’] It is possible to change the name of the attributes used for the algorithm. >>> G = nx.DiGraph() >>> G.add_node(’p’, spam = -4) >>> G.add_node(’q’, spam = 2) >>> G.add_node(’a’, spam = -2) >>> G.add_node(’d’, spam = -1) >>> G.add_node(’t’, spam = 2) >>> G.add_node(’w’, spam = 3) >>> G.add_edge(’p’, ’q’, cost = 7, vacancies = 5) >>> G.add_edge(’p’, ’a’, cost = 1, vacancies = 4) >>> G.add_edge(’q’, ’d’, cost = 2, vacancies = 3) >>> G.add_edge(’t’, ’q’, cost = 1, vacancies = 2) >>> G.add_edge(’a’, ’t’, cost = 2, vacancies = 4) >>> G.add_edge(’d’, ’w’, cost = 3, vacancies = 4) >>> G.add_edge(’t’, ’w’, cost = 4, vacancies = 1) >>> flowCost, flowDict = nx.network_simplex(G, demand = ’spam’, ... capacity = ’vacancies’, ... weight = ’cost’) >>> flowCost 37 >>> flowDict {’a’: {’t’: 4}, ’d’: {’w’: 2}, ’q’: {’d’: 1}, ’p’: {’q’: 2, ’a’: 2}, ’t’: {’q’: 1, ’w’: 1}, ’w’: {}} min_cost_ﬂow_cost min_cost_flow_cost(G, demand=’demand’, capacity=’capacity’, weight=’weight’) Find the cost of a minimum cost ﬂow satisfying all demands in digraph G. G is a digraph with edge costs and capacities and in which nodes have demand, i.e., they want to send or receive some amount of ﬂow. A negative demand means that the node wants to send ﬂow, a positive demand means that the node want to receive ﬂow. A ﬂow on the digraph G satisﬁes all demand if the net ﬂow into each node is equal to the demand of that node. Parameters G : NetworkX graph DiGraph on which a minimum cost ﬂow satisfying all demands is to be found. 218 Chapter 4. Algorithms NetworkX Reference, Release 1.7 demand: string : Nodes of the graph G are expected to have an attribute demand that indicates how much ﬂow a node wants to send (negative demand) or receive (positive demand). Note that the sum of the demands should be 0 otherwise the problem in not feasible. If this attribute is not present, a node is considered to have 0 demand. Default value: ‘demand’. capacity: string : Edges of the graph G are expected to have an attribute capacity that indicates how much ﬂow the edge can support. If this attribute is not present, the edge is considered to have inﬁnite capacity. Default value: ‘capacity’. weight: string : Edges of the graph G are expected to have an attribute weight that indicates the cost in- curred by sending one unit of ﬂow on that edge. If not present, the weight is considered to be 0. Default value: ‘weight’. Returns ﬂowCost: integer, ﬂoat : Cost of a minimum cost ﬂow satisfying all demands. Raises NetworkXError : This exception is raised if the input graph is not directed or not connected. NetworkXUnfeasible : This exception is raised in the following situations: • The sum of the demands is not zero. Then, there is no ﬂow satisfying all demands. • There is no ﬂow satisfying all demand. NetworkXUnbounded : This exception is raised if the digraph G has a cycle of negative cost and inﬁnite capacity. Then, the cost of a ﬂow satisfying all demands is unbounded below. See Also: cost_of_flow, max_flow_min_cost, min_cost_flow, network_simplex Examples A simple example of a min cost ﬂow problem. >>> import networkx as nx >>> G = nx.DiGraph() >>> G.add_node(’a’, demand = -5) >>> G.add_node(’d’, demand = 5) >>> G.add_edge(’a’, ’b’, weight = 3, capacity = 4) >>> G.add_edge(’a’, ’c’, weight = 6, capacity = 10) >>> G.add_edge(’b’, ’d’, weight = 1, capacity = 9) >>> G.add_edge(’c’, ’d’, weight = 2, capacity = 5) >>> flowCost = nx.min_cost_flow_cost(G) >>> flowCost 24 4.18. Flows 219 NetworkX Reference, Release 1.7 min_cost_ﬂow min_cost_flow(G, demand=’demand’, capacity=’capacity’, weight=’weight’) Return a minimum cost ﬂow satisfying all demands in digraph G. G is a digraph with edge costs and capacities and in which nodes have demand, i.e., they want to send or receive some amount of ﬂow. A negative demand means that the node wants to send ﬂow, a positive demand means that the node want to receive ﬂow. A ﬂow on the digraph G satisﬁes all demand if the net ﬂow into each node is equal to the demand of that node. Parameters G : NetworkX graph DiGraph on which a minimum cost ﬂow satisfying all demands is to be found. demand: string : Nodes of the graph G are expected to have an attribute demand that indicates how much ﬂow a node wants to send (negative demand) or receive (positive demand). Note that the sum of the demands should be 0 otherwise the problem in not feasible. If this attribute is not present, a node is considered to have 0 demand. Default value: ‘demand’. capacity: string : Edges of the graph G are expected to have an attribute capacity that indicates how much ﬂow the edge can support. If this attribute is not present, the edge is considered to have inﬁnite capacity. Default value: ‘capacity’. weight: string : Edges of the graph G are expected to have an attribute weight that indicates the cost in- curred by sending one unit of ﬂow on that edge. If not present, the weight is considered to be 0. Default value: ‘weight’. Returns ﬂowDict: dictionary : Dictionary of dictionaries keyed by nodes such that ﬂowDict[u][v] is the ﬂow edge (u, v). Raises NetworkXError : This exception is raised if the input graph is not directed or not connected. NetworkXUnfeasible : This exception is raised in the following situations: • The sum of the demands is not zero. Then, there is no ﬂow satisfying all demands. • There is no ﬂow satisfying all demand. NetworkXUnbounded : This exception is raised if the digraph G has a cycle of negative cost and inﬁnite capacity. Then, the cost of a ﬂow satisfying all demands is unbounded below. See Also: cost_of_flow, max_flow_min_cost, min_cost_flow_cost, network_simplex Examples A simple example of a min cost ﬂow problem. 220 Chapter 4. Algorithms NetworkX Reference, Release 1.7 >>> import networkx as nx >>> G = nx.DiGraph() >>> G.add_node(’a’, demand = -5) >>> G.add_node(’d’, demand = 5) >>> G.add_edge(’a’, ’b’, weight = 3, capacity = 4) >>> G.add_edge(’a’, ’c’, weight = 6, capacity = 10) >>> G.add_edge(’b’, ’d’, weight = 1, capacity = 9) >>> G.add_edge(’c’, ’d’, weight = 2, capacity = 5) >>> flowDict = nx.min_cost_flow(G) >>> flowDict {’a’: {’c’: 1, ’b’: 4}, ’c’: {’d’: 1}, ’b’: {’d’: 4}, ’d’: {}} cost_of_ﬂow cost_of_flow(G, ﬂowDict, weight=’weight’) Compute the cost of the ﬂow given by ﬂowDict on graph G. Note that this function does not check for the validity of the ﬂow ﬂowDict. This function will fail if the graph G and the ﬂow don’t have the same edge set. Parameters G : NetworkX graph DiGraph on which a minimum cost ﬂow satisfying all demands is to be found. weight: string : Edges of the graph G are expected to have an attribute weight that indicates the cost in- curred by sending one unit of ﬂow on that edge. If not present, the weight is considered to be 0. Default value: ‘weight’. ﬂowDict: dictionary : Dictionary of dictionaries keyed by nodes such that ﬂowDict[u][v] is the ﬂow edge (u, v). Returns cost: Integer, ﬂoat : The total cost of the ﬂow. This is given by the sum over all edges of the product of the edge’s ﬂow and the edge’s weight. See Also: max_flow_min_cost, min_cost_flow, min_cost_flow_cost, network_simplex max_ﬂow_min_cost max_flow_min_cost(G, s, t, capacity=’capacity’, weight=’weight’) Return a maximum (s, t)-ﬂow of minimum cost. G is a digraph with edge costs and capacities. There is a source node s and a sink node t. This function ﬁnds a maximum ﬂow from s to t whose total cost is minimized. Parameters G : NetworkX graph DiGraph on which a minimum cost ﬂow satisfying all demands is to be found. s: node label : Source of the ﬂow. t: node label : 4.18. Flows 221 NetworkX Reference, Release 1.7 Destination of the ﬂow. capacity: string : Edges of the graph G are expected to have an attribute capacity that indicates how much ﬂow the edge can support. If this attribute is not present, the edge is considered to have inﬁnite capacity. Default value: ‘capacity’. weight: string : Edges of the graph G are expected to have an attribute weight that indicates the cost in- curred by sending one unit of ﬂow on that edge. If not present, the weight is considered to be 0. Default value: ‘weight’. Returns ﬂowDict: dictionary : Dictionary of dictionaries keyed by nodes such that ﬂowDict[u][v] is the ﬂow edge (u, v). Raises NetworkXError : This exception is raised if the input graph is not directed or not connected. NetworkXUnbounded : This exception is raised if there is an inﬁnite capacity path from s to t in G. In this case there is no maximum ﬂow. This exception is also raised if the digraph G has a cycle of negative cost and inﬁnite capacity. Then, the cost of a ﬂow is unbounded below. See Also: cost_of_flow, ford_fulkerson, min_cost_flow, min_cost_flow_cost, network_simplex Examples >>> G = nx.DiGraph() >>> G.add_edges_from([(1, 2,{’capacity’: 12, ’weight’: 4}), ... (1, 3,{’capacity’: 20, ’weight’: 6}), ... (2, 3,{’capacity’: 6, ’weight’: -3}), ... (2, 6,{’capacity’: 14, ’weight’: 1}), ... (3, 4,{’weight’: 9}), ... (3, 5,{’capacity’: 10, ’weight’: 5}), ... (4, 2,{’capacity’: 19, ’weight’: 13}), ... (4, 5,{’capacity’: 4, ’weight’: 0}), ... (5, 7,{’capacity’: 28, ’weight’: 2}), ... (6, 5,{’capacity’: 11, ’weight’: 1}), ... (6, 7,{’weight’: 8}), ... (7, 4,{’capacity’: 6, ’weight’: 6})]) >>> mincostFlow = nx.max_flow_min_cost(G, 1, 7) >>> nx.cost_of_flow(G, mincostFlow) 373 >>> maxFlow = nx.ford_fulkerson_flow(G, 1, 7) >>> nx.cost_of_flow(G, maxFlow) 428 >>> mincostFlowValue = (sum((mincostFlow[u][7] for u in G.predecessors(7))) ... - sum((mincostFlow[7][v] for v in G.successors(7)))) >>> mincostFlowValue == nx.max_flow(G, 1, 7) True 222 Chapter 4. Algorithms NetworkX Reference, Release 1.7 4.19 Graphical degree sequence Generate graphs with a given degree sequence or expected degree sequence. is_valid_degree_sequence(sequence[, method]) Returns True if the sequence is a valid degree sequence. is_valid_degree_sequence_havel_hakimi(sequence) Returns True if the sequence is a valid degree sequence. is_valid_degree_sequence_erdos_gallai(sequence) Returns True if the sequence is a valid degree sequence. 4.19.1 is_valid_degree_sequence is_valid_degree_sequence(sequence, method=’hh’) Returns True if the sequence is a valid degree sequence. A degree sequence is valid if some graph can realize it. Parameters sequence : list or iterable container A sequence of integer node degrees method : “eg” | “hh” The method used to validate the degree sequence. “eg” corresponds to the Erd˝os-Gallai algorithm, and “hh” to the Havel-Hakimi algorithm. Returns valid : bool True if the sequence is a valid degree sequence and False if not. References Erd˝os-Gallai [EG1960], [choudum1986] Havel-Hakimi [havel1955], [hakimi1962], [CL1996] Examples >>> G = nx.path_graph(4) >>> sequence = G.degree().values() >>> nx.is_valid_degree_sequence(sequence) True 4.19.2 is_valid_degree_sequence_havel_hakimi is_valid_degree_sequence_havel_hakimi(sequence) Returns True if the sequence is a valid degree sequence. A degree sequence is valid if some graph can realize it. Validation proceeds via the Havel-Hakimi algorithm. Worst-case run time is: O(n(logn)) Parameters sequence : list or iterable container A sequence of integer node degrees Returns valid : bool 4.19. Graphical degree sequence 223 NetworkX Reference, Release 1.7 True if the sequence is a valid degree sequence and False if not. References [havel1955], [hakimi1962], [CL1996] 4.19.3 is_valid_degree_sequence_erdos_gallai is_valid_degree_sequence_erdos_gallai(sequence) Returns True if the sequence is a valid degree sequence. A degree sequence is valid if some graph can realize it. Validation proceeds via the Erd˝os-Gallai algorithm. Worst-case run time is: O(n2) Parameters sequence : list or iterable container A sequence of integer node degrees Returns valid : bool True if the sequence is a valid degree sequence and False if not. References [EG1960], [choudum1986] 4.20 Hierarchy Flow Hierarchy. flow_hierarchy(G[, weight]) Returns the ﬂow hierarchy of a directed network. 4.20.1 ﬂow_hierarchy flow_hierarchy(G, weight=None) Returns the ﬂow hierarchy of a directed network. Flow hierarchy is deﬁned as the fraction of edges not participating in cycles in a directed graph [R184]. Parameters G : DiGraph or MultiDiGraph A directed graph weight : key,optional (default=None) Attribute to use for node weights. If None the weight defaults to 1. Returns h : ﬂoat Flow heirarchy value 224 Chapter 4. Algorithms NetworkX Reference, Release 1.7 Notes The algorithm described in [R184] computes the ﬂow hierarchy through exponentiation of the adjacency matrix. This function implements an alternative approach that ﬁnds strongly connected components. An edge is in a cycle if and only if it is in a strongly connected component, which can be found in O(m) time using Tarjan’s algorithm. References [R184] 4.21 Isolates Functions for identifying isolate (degree zero) nodes. is_isolate(G, n) Determine of node n is an isolate (degree zero). isolates(G) Return list of isolates in the graph. 4.21.1 is_isolate is_isolate(G, n) Determine of node n is an isolate (degree zero). Parameters G : graph A networkx graph n : node A node in G Returns isolate : bool True if n has no neighbors, False otherwise. Examples >>> G=nx.Graph() >>> G.add_edge(1,2) >>> G.add_node(3) >>> nx.is_isolate(G,2) False >>> nx.is_isolate(G,3) True 4.21.2 isolates isolates(G) Return list of isolates in the graph. Isolates are nodes with no neighbors (degree zero). 4.21. Isolates 225 NetworkX Reference, Release 1.7 Parameters G : graph A networkx graph Returns isolates : list List of isolate nodes. Examples >>> G = nx.Graph() >>> G.add_edge(1,2) >>> G.add_node(3) >>> nx.isolates(G) [3] To remove all isolates in the graph use >>> G.remove_nodes_from(nx.isolates(G)) >>> G.nodes() [1, 2] For digraphs isolates have zero in-degree and zero out_degre >>> G = nx.DiGraph([(0,1),(1,2)]) >>> G.add_node(3) >>> nx.isolates(G) [3] 4.22 Isomorphism is_isomorphic(G1, G2[, node_match, edge_match]) Returns True if the graphs G1 and G2 are isomorphic and False otherwise. could_be_isomorphic(G1, G2) Returns False if graphs are deﬁnitely not isomorphic. fast_could_be_isomorphic(G1, G2) Returns False if graphs are deﬁnitely not isomorphic. faster_could_be_isomorphic(G1, G2) Returns False if graphs are deﬁnitely not isomorphic. 4.22.1 is_isomorphic is_isomorphic(G1, G2, node_match=None, edge_match=None) Returns True if the graphs G1 and G2 are isomorphic and False otherwise. Parameters G1, G2: graphs : The two graphs G1 and G2 must be the same type. node_match : callable A function that returns True if node n1 in G1 and n2 in G2 should be considered equal during the isomorphism test. If node_match is not speciﬁed then node attributes are not considered. The function will be called like node_match(G1.node[n1], G2.node[n2]). That is, the function will receive the node attribute dictionaries for n1 and n2 as inputs. edge_match : callable A function that returns True if the edge attribute dictionary for the pair of nodes (u1, v1) in G1 and (u2, v2) in G2 should be considered equal during the isomorphism test. If edge_match is not speciﬁed then edge attributes are not considered. The function will be called like 226 Chapter 4. Algorithms NetworkX Reference, Release 1.7 edge_match(G1[u1][v1], G2[u2][v2]). That is, the function will receive the edge attribute dictionaries of the edges under con- sideration. See Also: numerical_node_match, numerical_edge_match, numerical_multiedge_match, categorical_node_match, categorical_edge_match, categorical_multiedge_match Notes Uses the vf2 algorithm [R185]. References [R185] Examples >>> import networkx.algorithms.isomorphism as iso For digraphs G1 and G2, using ‘weight’ edge attribute (default: 1) >>> G1 = nx.DiGraph() >>> G2 = nx.DiGraph() >>> G1.add_path([1,2,3,4],weight=1) >>> G2.add_path([10,20,30,40],weight=2) >>> em = iso.numerical_edge_match(’weight’, 1) >>> nx.is_isomorphic(G1, G2) # no weights considered True >>> nx.is_isomorphic(G1, G2, edge_match=em) # match weights False For multidigraphs G1 and G2, using ‘ﬁll’ node attribute (default: ‘’) >>> G1 = nx.MultiDiGraph() >>> G2 = nx.MultiDiGraph() >>> G1.add_nodes_from([1,2,3],fill=’red’) >>> G2.add_nodes_from([10,20,30,40],fill=’red’) >>> G1.add_path([1,2,3,4],weight=3, linewidth=2.5) >>> G2.add_path([10,20,30,40],weight=3) >>> nm = iso.categorical_node_match(’fill’, ’red’) >>> nx.is_isomorphic(G1, G2, node_match=nm) True For multidigraphs G1 and G2, using ‘weight’ edge attribute (default: 7) >>> G1.add_edge(1,2, weight=7) >>> G2.add_edge(10,20) >>> em = iso.numerical_multiedge_match(’weight’, 7, rtol=1e-6) >>> nx.is_isomorphic(G1, G2, edge_match=em) True For multigraphs G1 and G2, using ‘weight’ and ‘linewidth’ edge attributes with default values 7 and 2.5. Also using ‘ﬁll’ node attribute with default value ‘red’. 4.22. Isomorphism 227 NetworkX Reference, Release 1.7 >>> em = iso.numerical_multiedge_match([’weight’, ’linewidth’], [7, 2.5]) >>> nm = iso.categorical_node_match(’fill’, ’red’) >>> nx.is_isomorphic(G1, G2, edge_match=em, node_match=nm) True 4.22.2 could_be_isomorphic could_be_isomorphic(G1, G2) Returns False if graphs are deﬁnitely not isomorphic. True does NOT guarantee isomorphism. Parameters G1, G2 : graphs The two graphs G1 and G2 must be the same type. Notes Checks for matching degree, triangle, and number of cliques sequences. 4.22.3 fast_could_be_isomorphic fast_could_be_isomorphic(G1, G2) Returns False if graphs are deﬁnitely not isomorphic. True does NOT guarantee isomorphism. Parameters G1, G2 : graphs The two graphs G1 and G2 must be the same type. Notes Checks for matching degree and triangle sequences. 4.22.4 faster_could_be_isomorphic faster_could_be_isomorphic(G1, G2) Returns False if graphs are deﬁnitely not isomorphic. True does NOT guarantee isomorphism. Parameters G1, G2 : graphs The two graphs G1 and G2 must be the same type. Notes Checks for matching degree sequences. 228 Chapter 4. Algorithms NetworkX Reference, Release 1.7 4.22.5 Advanced Interface to VF2 Algorithm VF2 Algorithm An implementation of VF2 algorithm for graph ismorphism testing. The simplest interface to use this module is to call networkx.is_isomorphic(). Introduction The GraphMatcher and DiGraphMatcher are responsible for matching graphs or directed graphs in a predetermined manner. This usually means a check for an isomorphism, though other checks are also possible. For example, a subgraph of one graph can be checked for isomorphism to a second graph. Matching is done via syntactic feasibility. It is also possible to check for semantic feasibility. Feasibility, then, is deﬁned as the logical AND of the two functions. To include a semantic check, the (Di)GraphMatcher class should be subclassed, and the semantic_feasibility() function should be redeﬁned. By default, the semantic feasibility function always returns True. The effect of this is that semantics are not considered in the matching of G1 and G2. Examples Suppose G1 and G2 are isomorphic graphs. Veriﬁcation is as follows: >>> from networkx.algorithms import isomorphism >>> G1 = nx.path_graph(4) >>> G2 = nx.path_graph(4) >>> GM = isomorphism.GraphMatcher(G1,G2) >>> GM.is_isomorphic() True GM.mapping stores the isomorphism mapping from G1 to G2. >>> GM.mapping {0: 0, 1: 1, 2: 2, 3: 3} Suppose G1 and G2 are isomorphic directed graphs graphs. Veriﬁcation is as follows: >>> G1 = nx.path_graph(4, create_using=nx.DiGraph()) >>> G2 = nx.path_graph(4, create_using=nx.DiGraph()) >>> DiGM = isomorphism.DiGraphMatcher(G1,G2) >>> DiGM.is_isomorphic() True DiGM.mapping stores the isomorphism mapping from G1 to G2. >>> DiGM.mapping {0: 0, 1: 1, 2: 2, 3: 3} Subgraph Isomorphism Graph theory literature can be ambiguious about the meaning of the above statement, and we seek to clarify it now. 4.22. Isomorphism 229 NetworkX Reference, Release 1.7 In the VF2 literature, a mapping M is said to be a graph-subgraph isomorphism iff M is an isomorphism between G2 and a subgraph of G1. Thus, to say that G1 and G2 are graph-subgraph isomorphic is to say that a subgraph of G1 is isomorphic to G2. Other literature uses the phrase ‘subgraph isomorphic’ as in ‘G1 does not have a subgraph isomorphic to G2’. Another use is as an in adverb for isomorphic. Thus, to say that G1 and G2 are subgraph isomorphic is to say that a subgraph of G1 is isomorphic to G2. Finally, the term ‘subgraph’ can have multiple meanings. In this context, ‘subgraph’ always means a ‘node-induced subgraph’. Edge-induced subgraph isomorphisms are not directly supported, but one should be able to perform the check by making use of nx.line_graph(). For subgraphs which are not induced, the term ‘monomorphism’ is preferred over ‘isomorphism’. Currently, it is not possible to check for monomorphisms. Let G=(N,E) be a graph with a set of nodes N and set of edges E. If G’=(N’,E’) is a subgraph, then: N’ is a subset of N E’ is a subset of E If G’=(N’,E’) is a node-induced subgraph, then: N’ is a subset of N E’ is the subset of edges in E relating nodes in N’ If G’=(N’,E’) is an edge-induced subgrpah, then: N’ is the subset of nodes in N related by edges in E’ E’ is a subset of E References [1] Luigi P. Cordella, Pasquale Foggia, Carlo Sansone, Mario Vento, “A (Sub)Graph Isomorphism Algorithm for Matching Large Graphs”, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 26, no. 10, pp. 1367-1372, Oct., 2004. http://ieeexplore.ieee.org/iel5/34/29305/01323804.pdf [2] L. P. Cordella, P. Foggia, C. Sansone, M. Vento, “An Improved Algorithm for Matching Large Graphs”, 3rd IAPR-TC15 Workshop on Graph-based Representations in Pattern Recognition, Cuen, pp. 149-159, 2001. http://amalﬁ.dis.unina.it/graph/db/papers/vf-algorithm.pdf See Also syntactic_feasibliity(), semantic_feasibility() Notes Modiﬁed to handle undirected graphs. Modiﬁed to handle multiple edges. In general, this problem is NP-Complete. Graph Matcher GraphMatcher.__init__(G1, G2[, node_match, ...]) Initialize graph matcher. GraphMatcher.initialize() Reinitializes the state of the algorithm. GraphMatcher.is_isomorphic() Returns True if G1 and G2 are isomorphic graphs. GraphMatcher.subgraph_is_isomorphic() Returns True if a subgraph of G1 is isomorphic to G2. GraphMatcher.isomorphisms_iter() Generator over isomorphisms between G1 and G2. GraphMatcher.subgraph_isomorphisms_iter() Generator over isomorphisms between a subgraph of G1 and G2. GraphMatcher.candidate_pairs_iter() Iterator over candidate pairs of nodes in G1 and G2. Continued on next page 230 Chapter 4. Algorithms NetworkX Reference, Release 1.7 Table 4.49 – continued from previous page GraphMatcher.match() Extends the isomorphism mapping. GraphMatcher.semantic_feasibility(G1_node, ...) Returns True if mapping G1_node to G2_node is semantically feasible. GraphMatcher.syntactic_feasibility(G1_node, ...) Returns True if adding (G1_node, G2_node) is syntactically feasible. __init__ GraphMatcher.__init__(G1, G2, node_match=None, edge_match=None) Initialize graph matcher. Parameters G1, G2: graph : The graphs to be tested. node_match: callable : A function that returns True iff node n1 in G1 and n2 in G2 should be considered equal during the isomorphism test. The function will be called like: node_match(G1.node[n1], G2.node[n2]) That is, the function will receive the node attribute dictionaries of the nodes under con- sideration. If None, then no attributes are considered when testing for an isomorphism. edge_match: callable : A function that returns True iff the edge attribute dictionary for the pair of nodes (u1, v1) in G1 and (u2, v2) in G2 should be considered equal during the isomorphism test. The function will be called like: edge_match(G1[u1][v1], G2[u2][v2]) That is, the function will receive the edge attribute dictionaries of the edges under con- sideration. If None, then no attributes are considered when testing for an isomorphism. initialize GraphMatcher.initialize() Reinitializes the state of the algorithm. This method should be redeﬁned if using something other than GMState. If only subclassing GraphMatcher, a redeﬁnition is not necessary. is_isomorphic GraphMatcher.is_isomorphic() Returns True if G1 and G2 are isomorphic graphs. subgraph_is_isomorphic GraphMatcher.subgraph_is_isomorphic() Returns True if a subgraph of G1 is isomorphic to G2. isomorphisms_iter GraphMatcher.isomorphisms_iter() Generator over isomorphisms between G1 and G2. 4.22. Isomorphism 231 NetworkX Reference, Release 1.7 subgraph_isomorphisms_iter GraphMatcher.subgraph_isomorphisms_iter() Generator over isomorphisms between a subgraph of G1 and G2. candidate_pairs_iter GraphMatcher.candidate_pairs_iter() Iterator over candidate pairs of nodes in G1 and G2. match GraphMatcher.match() Extends the isomorphism mapping. This function is called recursively to determine if a complete isomorphism can be found between G1 and G2. It cleans up the class variables after each recursive call. If an isomorphism is found, we yield the mapping. semantic_feasibility GraphMatcher.semantic_feasibility(G1_node, G2_node) Returns True if mapping G1_node to G2_node is semantically feasible. syntactic_feasibility GraphMatcher.syntactic_feasibility(G1_node, G2_node) Returns True if adding (G1_node, G2_node) is syntactically feasible. This function returns True if it is adding the candidate pair to the current partial isomorphism mapping is allowable. The addition is allowable if the inclusion of the candidate pair does not make it impossible for an isomorphism to be found. DiGraph Matcher DiGraphMatcher.__init__(G1, G2[, ...]) Initialize graph matcher. DiGraphMatcher.initialize() Reinitializes the state of the algorithm. DiGraphMatcher.is_isomorphic() Returns True if G1 and G2 are isomorphic graphs. DiGraphMatcher.subgraph_is_isomorphic() Returns True if a subgraph of G1 is isomorphic to G2. DiGraphMatcher.isomorphisms_iter() Generator over isomorphisms between G1 and G2. DiGraphMatcher.subgraph_isomorphisms_iter() Generator over isomorphisms between a subgraph of G1 and G2. DiGraphMatcher.candidate_pairs_iter() Iterator over candidate pairs of nodes in G1 and G2. DiGraphMatcher.match() Extends the isomorphism mapping. DiGraphMatcher.semantic_feasibility(G1_node, ...) Returns True if mapping G1_node to G2_node is semantically feasible. DiGraphMatcher.syntactic_feasibility(...) Returns True if adding (G1_node, G2_node) is syntactically feasible. __init__ DiGraphMatcher.__init__(G1, G2, node_match=None, edge_match=None) Initialize graph matcher. Parameters G1, G2 : graph The graphs to be tested. node_match : callable A function that returns True iff node n1 in G1 and n2 in G2 should be considered equal during the isomorphism test. The function will be called like: 232 Chapter 4. Algorithms NetworkX Reference, Release 1.7 node_match(G1.node[n1], G2.node[n2]) That is, the function will receive the node attribute dictionaries of the nodes under con- sideration. If None, then no attributes are considered when testing for an isomorphism. edge_match : callable A function that returns True iff the edge attribute dictionary for the pair of nodes (u1, v1) in G1 and (u2, v2) in G2 should be considered equal during the isomorphism test. The function will be called like: edge_match(G1[u1][v1], G2[u2][v2]) That is, the function will receive the edge attribute dictionaries of the edges under con- sideration. If None, then no attributes are considered when testing for an isomorphism. initialize DiGraphMatcher.initialize() Reinitializes the state of the algorithm. This method should be redeﬁned if using something other than DiGMState. If only subclassing GraphMatcher, a redeﬁnition is not necessary. is_isomorphic DiGraphMatcher.is_isomorphic() Returns True if G1 and G2 are isomorphic graphs. subgraph_is_isomorphic DiGraphMatcher.subgraph_is_isomorphic() Returns True if a subgraph of G1 is isomorphic to G2. isomorphisms_iter DiGraphMatcher.isomorphisms_iter() Generator over isomorphisms between G1 and G2. subgraph_isomorphisms_iter DiGraphMatcher.subgraph_isomorphisms_iter() Generator over isomorphisms between a subgraph of G1 and G2. candidate_pairs_iter DiGraphMatcher.candidate_pairs_iter() Iterator over candidate pairs of nodes in G1 and G2. match DiGraphMatcher.match() Extends the isomorphism mapping. This function is called recursively to determine if a complete isomorphism can be found between G1 and G2. It cleans up the class variables after each recursive call. If an isomorphism is found, we yield the mapping. 4.22. Isomorphism 233 NetworkX Reference, Release 1.7 semantic_feasibility DiGraphMatcher.semantic_feasibility(G1_node, G2_node) Returns True if mapping G1_node to G2_node is semantically feasible. syntactic_feasibility DiGraphMatcher.syntactic_feasibility(G1_node, G2_node) Returns True if adding (G1_node, G2_node) is syntactically feasible. This function returns True if it is adding the candidate pair to the current partial isomorphism mapping is allowable. The addition is allowable if the inclusion of the candidate pair does not make it impossible for an isomorphism to be found. Match helpers categorical_node_match(attr, default) Returns a comparison function for a categorical node attribute. categorical_edge_match(attr, default) Returns a comparison function for a categorical edge attribute. categorical_multiedge_match(attr, default) Returns a comparison function for a categorical edge attribute. numerical_node_match(attr, default[, rtol, atol]) Returns a comparison function for a numerical node attribute. numerical_edge_match(attr, default[, rtol, atol]) Returns a comparison function for a numerical edge attribute. numerical_multiedge_match(attr, default[, ...]) Returns a comparison function for a numerical edge attribute. generic_node_match(attr, default, op) Returns a comparison function for a generic attribute. generic_edge_match(attr, default, op) Returns a comparison function for a generic attribute. generic_multiedge_match(attr, default, op) Returns a comparison function for a generic attribute. categorical_node_match categorical_node_match(attr, default) Returns a comparison function for a categorical node attribute. The value(s) of the attr(s) must be hashable and comparable via the == operator since they are placed into a set([]) object. If the sets from G1 and G2 are the same, then the constructed function returns True. Parameters attr : string | list The categorical node attribute to compare, or a list of categorical node attributes to compare. default : value | list The default value for the categorical node attribute, or a list of default values for the categorical node attributes. Returns match : function The customized, categorical nodematch function. Examples >>> import networkx.algorithms.isomorphism as iso >>> nm = iso.categorical_node_match(’size’, 1) >>> nm = iso.categorical_node_match([’color’, ’size’], [’red’, 2]) categorical_edge_match 234 Chapter 4. Algorithms NetworkX Reference, Release 1.7 categorical_edge_match(attr, default) Returns a comparison function for a categorical edge attribute. The value(s) of the attr(s) must be hashable and comparable via the == operator since they are placed into a set([]) object. If the sets from G1 and G2 are the same, then the constructed function returns True. Parameters attr : string | list The categorical edge attribute to compare, or a list of categorical edge attributes to compare. default : value | list The default value for the categorical edge attribute, or a list of default values for the categorical edge attributes. Returns match : function The customized, categorical edgematch function. Examples >>> import networkx.algorithms.isomorphism as iso >>> nm = iso.categorical_edge_match(’size’, 1) >>> nm = iso.categorical_edge_match([’color’, ’size’], [’red’, 2]) categorical_multiedge_match categorical_multiedge_match(attr, default) Returns a comparison function for a categorical edge attribute. The value(s) of the attr(s) must be hashable and comparable via the == operator since they are placed into a set([]) object. If the sets from G1 and G2 are the same, then the constructed function returns True. Parameters attr : string | list The categorical edge attribute to compare, or a list of categorical edge attributes to compare. default : value | list The default value for the categorical edge attribute, or a list of default values for the categorical edge attributes. Returns match : function The customized, categorical edgematch function. Examples >>> import networkx.algorithms.isomorphism as iso >>> nm = iso.categorical_multiedge_match(’size’, 1) >>> nm = iso.categorical_multiedge_match([’color’, ’size’], [’red’, 2]) numerical_node_match 4.22. Isomorphism 235 NetworkX Reference, Release 1.7 numerical_node_match(attr, default, rtol=1e-05, atol=1e-08) Returns a comparison function for a numerical node attribute. The value(s) of the attr(s) must be numerical and sortable. If the sorted list of values from G1 and G2 are the same within some tolerance, then the constructed function returns True. Parameters attr : string | list The numerical node attribute to compare, or a list of numerical node attributes to com- pare. default : value | list The default value for the numerical node attribute, or a list of default values for the numerical node attributes. rtol : ﬂoat The relative error tolerance. atol : ﬂoat The absolute error tolerance. Returns match : function The customized, numerical nodematch function. Examples >>> import networkx.algorithms.isomorphism as iso >>> nm = iso.numerical_node_match(’weight’, 1.0) >>> nm = iso.numerical_node_match([’weight’, ’linewidth’], [.25, .5]) numerical_edge_match numerical_edge_match(attr, default, rtol=1e-05, atol=1e-08) Returns a comparison function for a numerical edge attribute. The value(s) of the attr(s) must be numerical and sortable. If the sorted list of values from G1 and G2 are the same within some tolerance, then the constructed function returns True. Parameters attr : string | list The numerical edge attribute to compare, or a list of numerical edge attributes to com- pare. default : value | list The default value for the numerical edge attribute, or a list of default values for the numerical edge attributes. rtol : ﬂoat The relative error tolerance. atol : ﬂoat The absolute error tolerance. Returns match : function The customized, numerical edgematch function. 236 Chapter 4. Algorithms NetworkX Reference, Release 1.7 Examples >>> import networkx.algorithms.isomorphism as iso >>> nm = iso.numerical_edge_match(’weight’, 1.0) >>> nm = iso.numerical_edge_match([’weight’, ’linewidth’], [.25, .5]) numerical_multiedge_match numerical_multiedge_match(attr, default, rtol=1e-05, atol=1e-08) Returns a comparison function for a numerical edge attribute. The value(s) of the attr(s) must be numerical and sortable. If the sorted list of values from G1 and G2 are the same within some tolerance, then the constructed function returns True. Parameters attr : string | list The numerical edge attribute to compare, or a list of numerical edge attributes to com- pare. default : value | list The default value for the numerical edge attribute, or a list of default values for the numerical edge attributes. rtol : ﬂoat The relative error tolerance. atol : ﬂoat The absolute error tolerance. Returns match : function The customized, numerical edgematch function. Examples >>> import networkx.algorithms.isomorphism as iso >>> nm = iso.numerical_multiedge_match(’weight’, 1.0) >>> nm = iso.numerical_multiedge_match([’weight’, ’linewidth’], [.25, .5]) generic_node_match generic_node_match(attr, default, op) Returns a comparison function for a generic attribute. The value(s) of the attr(s) are compared using the speciﬁed operators. If all the attributes are equal, then the constructed function returns True. Parameters attr : string | list The node attribute to compare, or a list of node attributes to compare. default : value | list The default value for the node attribute, or a list of default values for the node attributes. op : callable | list The operator to use when comparing attribute values, or a list of operators to use when comparing values for each attribute. 4.22. Isomorphism 237 NetworkX Reference, Release 1.7 Returns match : function The customized, generic nodematch function. Examples >>> from operator import eq >>> from networkx.algorithms.isomorphism.matchhelpers import close >>> from networkx.algorithms.isomorphism import generic_node_match >>> nm = generic_node_match(’weight’, 1.0, close) >>> nm = generic_node_match(’color’, ’red’, eq) >>> nm = generic_node_match([’weight’, ’color’], [1.0, ’red’], [close, eq]) generic_edge_match generic_edge_match(attr, default, op) Returns a comparison function for a generic attribute. The value(s) of the attr(s) are compared using the speciﬁed operators. If all the attributes are equal, then the constructed function returns True. Parameters attr : string | list The edge attribute to compare, or a list of edge attributes to compare. default : value | list The default value for the edge attribute, or a list of default values for the edge attributes. op : callable | list The operator to use when comparing attribute values, or a list of operators to use when comparing values for each attribute. Returns match : function The customized, generic edgematch function. Examples >>> from operator import eq >>> from networkx.algorithms.isomorphism.matchhelpers import close >>> from networkx.algorithms.isomorphism import generic_edge_match >>> nm = generic_edge_match(’weight’, 1.0, close) >>> nm = generic_edge_match(’color’, ’red’, eq) >>> nm = generic_edge_match([’weight’, ’color’], [1.0, ’red’], [close, eq]) generic_multiedge_match generic_multiedge_match(attr, default, op) Returns a comparison function for a generic attribute. The value(s) of the attr(s) are compared using the speciﬁed operators. If all the attributes are equal, then the constructed function returns True. Potentially, the constructed edge_match function can be slow since it must verify that no isomorphism exists between the multiedges before it returns False. Parameters attr : string | list The edge attribute to compare, or a list of node attributes to compare. 238 Chapter 4. Algorithms NetworkX Reference, Release 1.7 default : value | list The default value for the edge attribute, or a list of default values for the dgeattributes. op : callable | list The operator to use when comparing attribute values, or a list of operators to use when comparing values for each attribute. Returns match : function The customized, generic edgematch function. Examples >>> from operator import eq >>> from networkx.algorithms.isomorphism.matchhelpers import close >>> from networkx.algorithms.isomorphism import generic_node_match >>> nm = generic_node_match(’weight’, 1.0, close) >>> nm = generic_node_match(’color’, ’red’, eq) >>> nm = generic_node_match([’weight’, ’color’], ... [1.0, ’red’], ... [close, eq]) ... 4.23 Link Analysis 4.23.1 PageRank PageRank analysis of graph structure. pagerank(G[, alpha, personalization, ...]) Return the PageRank of the nodes in the graph. pagerank_numpy(G[, alpha, personalization, ...]) Return the PageRank of the nodes in the graph. pagerank_scipy(G[, alpha, personalization, ...]) Return the PageRank of the nodes in the graph. google_matrix(G[, alpha, personalization, ...]) Return the Google matrix of the graph. pagerank pagerank(G, alpha=0.85, personalization=None, max_iter=100, tol=1e-08, nstart=None, weight=’weight’) Return the PageRank of the nodes in the graph. PageRank computes a ranking of the nodes in the graph G based on the structure of the incoming links. It was originally designed as an algorithm to rank web pages. Parameters G : graph A NetworkX graph alpha : ﬂoat, optional Damping parameter for PageRank, default=0.85 personalization: dict, optional : 4.23. Link Analysis 239 NetworkX Reference, Release 1.7 The “personalization vector” consisting of a dictionary with a key for every graph node and nonzero personalization value for each node. max_iter : integer, optional Maximum number of iterations in power method eigenvalue solver. tol : ﬂoat, optional Error tolerance used to check convergence in power method solver. nstart : dictionary, optional Starting value of PageRank iteration for each node. weight : key, optional Edge data key to use as weight. If None weights are set to 1. Returns pagerank : dictionary Dictionary of nodes with PageRank as value See Also: pagerank_numpy, pagerank_scipy, google_matrix Notes The eigenvector calculation is done by the power iteration method and has no guarantee of convergence. The iteration will stop after max_iter iterations or an error tolerance of number_of_nodes(G)*tol has been reached. The PageRank algorithm was designed for directed graphs but this algorithm does not check if the input graph is directed and will execute on undirected graphs by converting each oriented edge in the directed graph to two edges. References [R192], [R193] Examples >>> G=nx.DiGraph(nx.path_graph(4)) >>> pr=nx.pagerank(G,alpha=0.9) pagerank_numpy pagerank_numpy(G, alpha=0.85, personalization=None, weight=’weight’) Return the PageRank of the nodes in the graph. PageRank computes a ranking of the nodes in the graph G based on the structure of the incoming links. It was originally designed as an algorithm to rank web pages. Parameters G : graph A NetworkX graph alpha : ﬂoat, optional 240 Chapter 4. Algorithms NetworkX Reference, Release 1.7 Damping parameter for PageRank, default=0.85 personalization: dict, optional : The “personalization vector” consisting of a dictionary with a key for every graph node and nonzero personalization value for each node. weight : key, optional Edge data key to use as weight. If None weights are set to 1. Returns pagerank : dictionary Dictionary of nodes with PageRank as value See Also: pagerank, pagerank_scipy, google_matrix Notes The eigenvector calculation uses NumPy’s interface to the LAPACK eigenvalue solvers. This will be the fastest and most accurate for small graphs. This implementation works with Multi(Di)Graphs. References [R194], [R195] Examples >>> G=nx.DiGraph(nx.path_graph(4)) >>> pr=nx.pagerank_numpy(G,alpha=0.9) pagerank_scipy pagerank_scipy(G, alpha=0.85, personalization=None, max_iter=100, tol=1e-06, weight=’weight’) Return the PageRank of the nodes in the graph. PageRank computes a ranking of the nodes in the graph G based on the structure of the incoming links. It was originally designed as an algorithm to rank web pages. Parameters G : graph A NetworkX graph alpha : ﬂoat, optional Damping parameter for PageRank, default=0.85 personalization: dict, optional : The “personalization vector” consisting of a dictionary with a key for every graph node and nonzero personalization value for each node. max_iter : integer, optional Maximum number of iterations in power method eigenvalue solver. 4.23. Link Analysis 241 NetworkX Reference, Release 1.7 tol : ﬂoat, optional Error tolerance used to check convergence in power method solver. weight : key, optional Edge data key to use as weight. If None weights are set to 1. Returns pagerank : dictionary Dictionary of nodes with PageRank as value See Also: pagerank, pagerank_numpy, google_matrix Notes The eigenvector calculation uses power iteration with a SciPy sparse matrix representation. References [R196], [R197] Examples >>> G=nx.DiGraph(nx.path_graph(4)) >>> pr=nx.pagerank_scipy(G,alpha=0.9) google_matrix google_matrix(G, alpha=0.85, personalization=None, nodelist=None, weight=’weight’) Return the Google matrix of the graph. Parameters G : graph A NetworkX graph alpha : ﬂoat The damping factor personalization: dict, optional : The “personalization vector” consisting of a dictionary with a key for every graph node and nonzero personalization value for each node. nodelist : list, optional The rows and columns are ordered according to the nodes in nodelist. If nodelist is None, then the ordering is produced by G.nodes(). weight : key, optional Edge data key to use as weight. If None weights are set to 1. Returns A : NumPy matrix Google matrix of the graph 242 Chapter 4. Algorithms NetworkX Reference, Release 1.7 See Also: pagerank, pagerank_numpy, pagerank_scipy 4.23.2 Hits Hubs and authorities analysis of graph structure. hits(G[, max_iter, tol, nstart]) Return HITS hubs and authorities values for nodes. hits_numpy(G) Return HITS hubs and authorities values for nodes. hits_scipy(G[, max_iter, tol]) Return HITS hubs and authorities values for nodes. hub_matrix(G[, nodelist]) Return the HITS hub matrix. authority_matrix(G[, nodelist]) Return the HITS authority matrix. hits hits(G, max_iter=100, tol=1e-08, nstart=None) Return HITS hubs and authorities values for nodes. The HITS algorithm computes two numbers for a node. Authorities estimates the node value based on the incoming links. Hubs estimates the node value based on outgoing links. Parameters G : graph A NetworkX graph max_iter : interger, optional Maximum number of iterations in power method. tol : ﬂoat, optional Error tolerance used to check convergence in power method iteration. nstart : dictionary, optional Starting value of each node for power method iteration. Returns (hubs,authorities) : two-tuple of dictionaries Two dictionaries keyed by node containing the hub and authority values. Notes The eigenvector calculation is done by the power iteration method and has no guarantee of convergence. The iteration will stop after max_iter iterations or an error tolerance of number_of_nodes(G)*tol has been reached. The HITS algorithm was designed for directed graphs but this algorithm does not check if the input graph is directed and will execute on undirected graphs. References [R186], [R187] 4.23. Link Analysis 243 NetworkX Reference, Release 1.7 Examples >>> G=nx.path_graph(4) >>> h,a=nx.hits(G) hits_numpy hits_numpy(G) Return HITS hubs and authorities values for nodes. The HITS algorithm computes two numbers for a node. Authorities estimates the node value based on the incoming links. Hubs estimates the node value based on outgoing links. Parameters G : graph A NetworkX graph Returns (hubs,authorities) : two-tuple of dictionaries Two dictionaries keyed by node containing the hub and authority values. Notes The eigenvector calculation uses NumPy’s interface to LAPACK. The HITS algorithm was designed for directed graphs but this algorithm does not check if the input graph is directed and will execute on undirected graphs. References [R188], [R189] Examples >>> G=nx.path_graph(4) >>> h,a=nx.hits(G) hits_scipy hits_scipy(G, max_iter=100, tol=1e-06) Return HITS hubs and authorities values for nodes. The HITS algorithm computes two numbers for a node. Authorities estimates the node value based on the incoming links. Hubs estimates the node value based on outgoing links. Parameters G : graph A NetworkX graph max_iter : interger, optional Maximum number of iterations in power method. tol : ﬂoat, optional 244 Chapter 4. Algorithms NetworkX Reference, Release 1.7 Error tolerance used to check convergence in power method iteration. nstart : dictionary, optional Starting value of each node for power method iteration. Returns (hubs,authorities) : two-tuple of dictionaries Two dictionaries keyed by node containing the hub and authority values. Notes This implementation uses SciPy sparse matrices. The eigenvector calculation is done by the power iteration method and has no guarantee of convergence. The iteration will stop after max_iter iterations or an error tolerance of number_of_nodes(G)*tol has been reached. The HITS algorithm was designed for directed graphs but this algorithm does not check if the input graph is directed and will execute on undirected graphs. References [R190], [R191] Examples >>> G=nx.path_graph(4) >>> h,a=nx.hits(G) hub_matrix hub_matrix(G, nodelist=None) Return the HITS hub matrix. authority_matrix authority_matrix(G, nodelist=None) Return the HITS authority matrix. 4.24 Matching maximal_matching(graph) Find a maximal matching in the graph. max_weight_matching(G[, maxcardinality]) Compute a maximum-weighted matching of G. 4.24.1 maximal_matching maximal_matching(graph) Find a maximal matching in the graph. Parameters graph : NetworkX graph 4.24. Matching 245 NetworkX Reference, Release 1.7 Undirected graph Returns matching : set of edges. A maximal mathing of the graph. 4.24.2 max_weight_matching max_weight_matching(G, maxcardinality=False) Compute a maximum-weighted matching of G. A matching is a subset of edges in which no node occurs more than once. The cardinality of a matching is the number of matched edges. The weight of a matching is the sum of the weights of its edges. Parameters G : NetworkX graph Undirected graph maxcardinality: bool, optional : If maxcardinality is True, compute the maximum-cardinality matching with maximum weight among all maximum-cardinality matchings. Returns mate : dictionary The matching is returned as a dictionary, mate, such that mate[v] == w if node v is matched to node w. Unmatched nodes do not occur as a key in mate. Notes If G has edges with ‘weight’ attribute the edge data are used as weight values else the weights are assumed to be 1. This function takes time O(number_of_nodes ** 3). If all edge weights are integers, the algorithm uses only integer computations. If ﬂoating point weights are used, the algorithm could return a slightly suboptimal matching due to numeric precision errors. This method is based on the “blossom” method for ﬁnding augmenting paths and the “primal-dual” method for ﬁnding a matching of maximum weight, both methods invented by Jack Edmonds [R198]. References [R198] 4.25 Maximal independent set Algorithm to ﬁnd a maximal (not maximum) independent set. maximal_independent_set(G[, nodes]) Return a random maximal independent set guaranteed to contain a given set of nodes. 246 Chapter 4. Algorithms NetworkX Reference, Release 1.7 4.25.1 maximal_independent_set maximal_independent_set(G, nodes=None) Return a random maximal independent set guaranteed to contain a given set of nodes. An independent set is a set of nodes such that the subgraph of G induced by these nodes contains no edges. A maximal independent set is an independent set such that it is not possible to add a new node and still get an independent set. Parameters G : NetworkX graph nodes : list or iterable Nodes that must be part of the independent set. This set of nodes must be independent. Returns indep_nodes : list List of nodes that are part of a maximal independent set. Raises NetworkXUnfeasible : If the nodes in the provided list are not part of the graph or do not form an independent set, an exception is raised. Notes This algorithm does not solve the maximum independent set problem. Examples >>> G = nx.path_graph(5) >>> nx.maximal_independent_set(G) [4, 0, 2] >>> nx.maximal_independent_set(G, [1]) [1, 3] 4.26 Minimum Spanning Tree Computes minimum spanning tree of a weighted graph. minimum_spanning_tree(G[, weight]) Return a minimum spanning tree or forest of an undirected weighted graph. minimum_spanning_edges(G[, weight, data]) Generate edges in a minimum spanning forest of an undirected weighted graph. 4.26.1 minimum_spanning_tree minimum_spanning_tree(G, weight=’weight’) Return a minimum spanning tree or forest of an undirected weighted graph. A minimum spanning tree is a subgraph of the graph (a tree) with the minimum sum of edge weights. If the graph is not connected a spanning forest is constructed. A spanning forest is a union of the spanning trees for each connected component of the graph. Parameters G : NetworkX Graph 4.26. Minimum Spanning Tree 247 NetworkX Reference, Release 1.7 weight : string Edge data key to use for weight (default ‘weight’). Returns G : NetworkX Graph A minimum spanning tree or forest. Notes Uses Kruskal’s algorithm. If the graph edges do not have a weight attribute a default weight of 1 will be used. Examples >>> G=nx.cycle_graph(4) >>> G.add_edge(0,3,weight=2) # assign weight 2 to edge 0-3 >>> T=nx.minimum_spanning_tree(G) >>> print(sorted(T.edges(data=True))) [(0, 1, {}), (1, 2, {}), (2, 3, {})] 4.26.2 minimum_spanning_edges minimum_spanning_edges(G, weight=’weight’, data=True) Generate edges in a minimum spanning forest of an undirected weighted graph. A minimum spanning tree is a subgraph of the graph (a tree) with the minimum sum of edge weights. A spanning forest is a union of the spanning trees for each connected component of the graph. Parameters G : NetworkX Graph weight : string Edge data key to use for weight (default ‘weight’). data : bool, optional If True yield the edge data along with the edge. Returns edges : iterator A generator that produces edges in the minimum spanning tree. The edges are three- tuples (u,v,w) where w is the weight. Notes Uses Kruskal’s algorithm. If the graph edges do not have a weight attribute a default weight of 1 will be used. Modiﬁed code from David Eppstein, April 2006 http://www.ics.uci.edu/~eppstein/PADS/ 248 Chapter 4. Algorithms NetworkX Reference, Release 1.7 Examples >>> G=nx.cycle_graph(4) >>> G.add_edge(0,3,weight=2) # assign weight 2 to edge 0-3 >>> mst=nx.minimum_spanning_edges(G,data=False) # a generator of MST edges >>> edgelist=list(mst) # make a list of the edges >>> print(sorted(edgelist)) [(0, 1), (1, 2), (2, 3)] 4.27 Operators Unary operations on graphs complement(G[, name]) Return the graph complement of G. 4.27.1 complement complement(G, name=None) Return the graph complement of G. Parameters G : graph A NetworkX graph name : string Specify name for new graph Returns GC : A new graph. Notes Note that complement() does not create self-loops and also does not produce parallel edges for MultiGraphs. Graph, node, and edge data are not propagated to the new graph. Operations on graphs including union, intersection, difference. compose(G, H[, name]) Return a new graph of G composed with H. union(G, H[, rename, name]) Return the union of graphs G and H. disjoint_union(G, H) Return the disjoint union of graphs G and H. intersection(G, H) Return a new graph that contains only the edges that exist in difference(G, H) Return a new graph that contains the edges that exist in G but not in H. symmetric_difference(G, H) Return new graph with edges that exist in either G or H but not both. 4.27.2 compose compose(G, H, name=None) Return a new graph of G composed with H. Composition is the simple union of the node sets and edge sets. The node sets of G and H need not be disjoint. Parameters G,H : graph 4.27. Operators 249 NetworkX Reference, Release 1.7 A NetworkX graph name : string Specify name for new graph Returns C: A new graph with the same type as G : Notes It is recommended that G and H be either both directed or both undirected. Attributes from H take precedent over attributes from G. 4.27.3 union union(G, H, rename=(None, None), name=None) Return the union of graphs G and H. Graphs G and H must be disjoint, otherwise an exception is raised. Parameters G,H : graph A NetworkX graph create_using : NetworkX graph Use speciﬁed graph for result. Otherwise rename : bool , default=(None, None) Node names of G and H can be changed by specifying the tuple rename=(‘G-‘,’H-‘) (for example). Node “u” in G is then renamed “G-u” and “v” in H is renamed “H-v”. name : string Specify the name for the union graph Returns U : A union graph with the same type as G. See Also: disjoint_union Notes To force a disjoint union with node relabeling, use disjoint_union(G,H) or convert_node_labels_to integers(). Graph, edge, and node attributes are propagated from G and H to the union graph. If a graph attribute is present in both G and H the value from H is used. 4.27.4 disjoint_union disjoint_union(G, H) Return the disjoint union of graphs G and H. This algorithm forces distinct integer node labels. Parameters G,H : graph A NetworkX graph 250 Chapter 4. Algorithms NetworkX Reference, Release 1.7 Returns U : A union graph with the same type as G. Notes A new graph is created, of the same class as G. It is recommended that G and H be either both directed or both undirected. The nodes of G are relabeled 0 to len(G)-1, and the nodes of H are relabeled len(G) to len(G)+len(H)-1. Graph, edge, and node attributes are propagated from G and H to the union graph. If a graph attribute is present in both G and H the value from H is used. 4.27.5 intersection intersection(G, H) Return a new graph that contains only the edges that exist in both G and H. The node sets of H and G must be the same. Parameters G,H : graph A NetworkX graph. G and H must have the same node sets. Returns GH : A new graph with the same type as G. Notes Attributes from the graph, nodes, and edges are not copied to the new graph. If you want a new graph of the intersection of G and H with the attributes (including edge data) from G use remove_nodes_from() as follows >>> G=nx.path_graph(3) >>> H=nx.path_graph(5) >>> R=G.copy() >>> R.remove_nodes_from(n for n in G if n not in H) 4.27.6 difference difference(G, H) Return a new graph that contains the edges that exist in G but not in H. The node sets of H and G must be the same. Parameters G,H : graph A NetworkX graph. G and H must have the same node sets. Returns D : A new graph with the same type as G. Notes Attributes from the graph, nodes, and edges are not copied to the new graph. If you want a new graph of the difference of G and H with with the attributes (including edge data) from G use remove_nodes_from() as follows: 4.27. Operators 251 NetworkX Reference, Release 1.7 >>> G=nx.path_graph(3) >>> H=nx.path_graph(5) >>> R=G.copy() >>> R.remove_nodes_from(n for n in G if n in H) 4.27.7 symmetric_difference symmetric_difference(G, H) Return new graph with edges that exist in either G or H but not both. The node sets of H and G must be the same. Parameters G,H : graph A NetworkX graph. G and H must have the same node sets. Returns D : A new graph with the same type as G. Notes Attributes from the graph, nodes, and edges are not copied to the new graph. Operations on many graphs. compose_all(graphs[, name]) Return the composition of all graphs. union_all(graphs[, rename, name]) Return the union of all graphs. disjoint_union_all(graphs) Return the disjoint union of all graphs. intersection_all(graphs) Return a new graph that contains only the edges that exist in all graphs. 4.27.8 compose_all compose_all(graphs, name=None) Return the composition of all graphs. Composition is the simple union of the node sets and edge sets. The node sets of the supplied graphs need not be disjoint. Parameters graphs : list List of NetworkX graphs name : string Specify name for new graph Returns C : A graph with the same type as the ﬁrst graph in list Notes It is recommended that the supplied graphs be either all directed or all undirected. Graph, edge, and node attributes are propagated to the union graph. If a graph attribute is present in multiple graphs, then the value from the last graph in the list with that attribute is used. 252 Chapter 4. Algorithms NetworkX Reference, Release 1.7 4.27.9 union_all union_all(graphs, rename=(None, ), name=None) Return the union of all graphs. The graphs must be disjoint, otherwise an exception is raised. Parameters graphs : list of graphs List of NetworkX graphs rename : bool , default=(None, None) Node names of G and H can be changed by specifying the tuple rename=(‘G-‘,’H-‘) (for example). Node “u” in G is then renamed “G-u” and “v” in H is renamed “H-v”. name : string Specify the name for the union graph@not_implemnted_for(‘direct Returns U : a graph with the same type as the ﬁrst graph in list See Also: union, disjoint_union_all Notes To force a disjoint union with node relabeling, use disjoint_union_all(G,H) or convert_node_labels_to integers(). Graph, edge, and node attributes are propagated to the union graph. If a graph attribute is present in multiple graphs, then the value from the last graph in the list with that attribute is used. 4.27.10 disjoint_union_all disjoint_union_all(graphs) Return the disjoint union of all graphs. This operation forces distinct integer node labels starting with 0 for the ﬁrst graph in the list and numbering consecutively. Parameters graphs : list List of NetworkX graphs Returns U : A graph with the same type as the ﬁrst graph in list Notes It is recommended that the graphs be either all directed or all undirected. Graph, edge, and node attributes are propagated to the union graph. If a graph attribute is present in multiple graphs, then the value from the last graph in the list with that attribute is used. 4.27. Operators 253 NetworkX Reference, Release 1.7 4.27.11 intersection_all intersection_all(graphs) Return a new graph that contains only the edges that exist in all graphs. All supplied graphs must have the same node set. Parameters graphs_list : list List of NetworkX graphs Returns R : A new graph with the same type as the ﬁrst graph in list Notes Attributes from the graph, nodes, and edges are not copied to the new graph. Graph products. cartesian_product(G, H) Return the Cartesian product of G and H. lexicographic_product(G, H) Return the lexicographic product of G and H. strong_product(G, H) Return the strong product of G and H. tensor_product(G, H) Return the tensor product of G and H. 4.27.12 cartesian_product cartesian_product(G, H) Return the Cartesian product of G and H. The tensor product P of the graphs G and H has a node set that is the Cartesian product of the node sets, $V(P)=V(G) imes V(H)$. P has an edge ((u,v),(x,y)) if and only if (u,v) is an edge in G and x==y or and (x,y) is an edge in H and u==v. and (x,y) is an edge in H. Parameters G, H: graphs : Networkx graphs. Returns P: NetworkX graph : The Cartesian product of G and H. P will be a multi-graph if either G or H is a multi- graph. Will be a directed if G and H are directed, and undirected if G and H are undi- rected. Raises NetworkXError : If G and H are not both directed or both undirected. Notes Node attributes in P are two-tuple of the G and H node attributes. Missing attributes are assigned None. For example >>> G = nx.Graph() >>> H = nx.Graph() >>> G.add_node(0,a1=True) >>> H.add_node(‘a’,a2=’Spam’) >>> P = nx.tensor_product(G,H) >>> P.nodes(data=True) [((0, ‘a’), {‘a1’: (True, None), ‘a2’: (None, ‘Spam’)})] Edge attributes and edge keys (for multigraphs) are also copied to the new product graph 254 Chapter 4. Algorithms NetworkX Reference, Release 1.7 4.27.13 lexicographic_product lexicographic_product(G, H) Return the lexicographic product of G and H. The lexicographical product P of the graphs G and H has a node set that is the Cartesian product of the node sets, $V(P)=V(G) imes V(H)$. P has an edge ((u,v),(x,y)) if and only if (u,v) is an edge in G or u==v and (x,y) is an edge in H. Parameters G, H: graphs : Networkx graphs. Returns P: NetworkX graph : The Cartesian product of G and H. P will be a multi-graph if either G or H is a multi- graph. Will be a directed if G and H are directed, and undirected if G and H are undi- rected. Raises NetworkXError : If G and H are not both directed or both undirected. Notes Node attributes in P are two-tuple of the G and H node attributes. Missing attributes are assigned None. For example >>> G = nx.Graph() >>> H = nx.Graph() >>> G.add_node(0,a1=True) >>> H.add_node(‘a’,a2=’Spam’) >>> P = nx.tensor_product(G,H) >>> P.nodes(data=True) [((0, ‘a’), {‘a1’: (True, None), ‘a2’: (None, ‘Spam’)})] Edge attributes and edge keys (for multigraphs) are also copied to the new product graph 4.27.14 strong_product strong_product(G, H) Return the strong product of G and H. The strong product P of the graphs G and H has a node set that is the Cartesian product of the node sets, $V(P)=V(G) imes V(H)$. P has an edge ((u,v),(x,y)) if and only if u==v and (x,y) is an edge in H, or x==y and (u,v) is an edge in G, or (u,v) is an edge in G and (x,y) is an edge in H. Parameters G, H: graphs : Networkx graphs. Returns P: NetworkX graph : The Cartesian product of G and H. P will be a multi-graph if either G or H is a multi- graph. Will be a directed if G and H are directed, and undirected if G and H are undi- rected. Raises NetworkXError : If G and H are not both directed or both undirected. 4.27. Operators 255 NetworkX Reference, Release 1.7 Notes Node attributes in P are two-tuple of the G and H node attributes. Missing attributes are assigned None. For example >>> G = nx.Graph() >>> H = nx.Graph() >>> G.add_node(0,a1=True) >>> H.add_node(‘a’,a2=’Spam’) >>> P = nx.tensor_product(G,H) >>> P.nodes(data=True) [((0, ‘a’), {‘a1’: (True, None), ‘a2’: (None, ‘Spam’)})] Edge attributes and edge keys (for multigraphs) are also copied to the new product graph 4.27.15 tensor_product tensor_product(G, H) Return the tensor product of G and H. The tensor product P of the graphs G and H has a node set that is the Cartesian product of the node sets, $V(P)=V(G) times V(H)$. P has an edge ((u,v),(x,y)) if and only if (u,v) is an edge in G and (x,y) is an edge in H. Sometimes referred to as the categorical product. Parameters G, H: graphs : Networkx graphs. Returns P: NetworkX graph : The tensor product of G and H. P will be a multi-graph if either G or H is a multi-graph. Will be a directed if G and H are directed, and undirected if G and H are undirected. Raises NetworkXError : If G and H are not both directed or both undirected. Notes Node attributes in P are two-tuple of the G and H node attributes. Missing attributes are assigned None. For example >>> G = nx.Graph() >>> H = nx.Graph() >>> G.add_node(0,a1=True) >>> H.add_node(‘a’,a2=’Spam’) >>> P = nx.tensor_product(G,H) >>> P.nodes(data=True) [((0, ‘a’), {‘a1’: (True, None), ‘a2’: (None, ‘Spam’)})] Edge attributes and edge keys (for multigraphs) are also copied to the new product graph 4.28 Rich Club rich_club_coefficient(G[, normalized, Q]) Return the rich-club coefﬁcient of the graph G. 4.28.1 rich_club_coefﬁcient rich_club_coefficient(G, normalized=True, Q=100) Return the rich-club coefﬁcient of the graph G. The rich-club coefﬁcient is the ratio, for every degree k, of the number of actual to the number of potential edges 256 Chapter 4. Algorithms NetworkX Reference, Release 1.7 for nodes with degree greater than k: (k)= 2Ek Nk(Nk 1) where Nk is the number of nodes with degree larger than k, and Ek be the number of edges among those nodes. Parameters G : NetworkX graph normalized : bool (optional) Normalize using randomized network (see [R199]) Q : ﬂoat (optional, default=100) If normalized=True build a random network by performing Q*M double-edge swaps, where M is the number of edges in G, to use as a null-model for normalization. Returns rc : dictionary A dictionary, keyed by degree, with rich club coefﬁcient values. Notes The rich club deﬁnition and algorithm are found in [R199]. This algorithm ignores any edge weights and is not deﬁned for directed graphs or graphs with parallel edges or self loops. Estimates for appropriate values of Q are found in [R200]. References [R199], [R200] Examples >>> G = nx.Graph([(0,1),(0,2),(1,2),(1,3),(1,4),(4,5)]) >>> rc = nx.rich_club_coefficient(G,normalized=False) >>> rc[0] 0.4 4.29 Shortest Paths Compute the shortest paths and path lengths between nodes in the graph. These algorithms work with undirected and directed graphs. For directed graphs the paths can be computed in the reverse order by ﬁrst ﬂipping the edge orientation using R=G.reverse(copy=False). shortest_path(G[, source, target, weight]) Compute shortest paths in the graph. all_shortest_paths(G, source, target[, weight]) Compute all shortest paths in the graph. shortest_path_length(G[, source, target, weight]) Compute shortest path lengths in the graph. average_shortest_path_length(G[, weight]) Return the average shortest path length. has_path(G, source, target) Return True if G has a path from source to target, False otherwise. 4.29. Shortest Paths 257 NetworkX Reference, Release 1.7 4.29.1 shortest_path shortest_path(G, source=None, target=None, weight=None) Compute shortest paths in the graph. Parameters G : NetworkX graph source : node, optional Starting node for path. If not speciﬁed compute shortest paths for all connected node pairs. target : node, optional Ending node for path. If not speciﬁed compute shortest paths for every node reachable from the source. weight : None or string, optional (default = None) If None, every edge has weight/distance/cost 1. If a string, use this edge attribute as the edge weight. Any edge attribute not present defaults to 1. Returns path: list or dictionary : If the source and target are both speciﬁed return a single list of nodes in a shortest path. If only the source is speciﬁed return a dictionary keyed by targets with a list of nodes in a shortest path. If neither the source or target is speciﬁed return a dictionary of dictionaries with path[source][target]=[list of nodes in path]. See Also: all_pairs_shortest_path, all_pairs_dijkstra_path, single_source_shortest_path, single_source_dijkstra_path Notes There may be more than one shortest path between a source and target. This returns only one of them. For digraphs this returns a shortest directed path. To ﬁnd paths in the reverse direction ﬁrst use G.reverse(copy=False) to ﬂip the edge orientation. Examples >>> G=nx.path_graph(5) >>> print(nx.shortest_path(G,source=0,target=4)) [0, 1, 2, 3, 4] >>> p=nx.shortest_path(G,source=0) # target not specified >>> p[4] [0, 1, 2, 3, 4] >>> p=nx.shortest_path(G) # source,target not specified >>> p[0][4] [0, 1, 2, 3, 4] 4.29.2 all_shortest_paths all_shortest_paths(G, source, target, weight=None) Compute all shortest paths in the graph. 258 Chapter 4. Algorithms NetworkX Reference, Release 1.7 Parameters G : NetworkX graph source : node Starting node for path. target : node Ending node for path. weight : None or string, optional (default = None) If None, every edge has weight/distance/cost 1. If a string, use this edge attribute as the edge weight. Any edge attribute not present defaults to 1. Returns paths: generator of lists : A generator of all paths between source and target. See Also: shortest_path, single_source_shortest_path, all_pairs_shortest_path Notes There may be many shortest paths between the source and target. Examples >>> G=nx.Graph() >>> G.add_path([0,1,2]) >>> G.add_path([0,10,2]) >>> print([p for p in nx.all_shortest_paths(G,source=0,target=2)]) [[0, 1, 2], [0, 10, 2]] 4.29.3 shortest_path_length shortest_path_length(G, source=None, target=None, weight=None) Compute shortest path lengths in the graph. This function can compute the single source shortest path lengths by specifying only the source or all pairs shortest path lengths by specifying neither the source or target. Parameters G : NetworkX graph source : node, optional Starting node for path. If not speciﬁed compute shortest path lengths for all connected node pairs. target : node, optional Ending node for path. If not speciﬁed compute shortest path lengths for every node reachable from the source. weight : None or string, optional (default = None) If None, every edge has weight/distance/cost 1. If a string, use this edge attribute as the edge weight. Any edge attribute not present defaults to 1. Returns length : number, or container of numbers 4.29. Shortest Paths 259 NetworkX Reference, Release 1.7 If the source and target are both speciﬁed return a single number for the shortest path. If only the source is speciﬁed return a dictionary keyed by targets with a the shortest path as keys. If neither the source or target is speciﬁed return a dictionary of dictionaries with length[source][target]=value. Raises NetworkXNoPath : If no path exists between source and target. See Also: all_pairs_shortest_path_length, all_pairs_dijkstra_path_length, single_source_shortest_path_length, single_source_dijkstra_path_length Notes For digraphs this returns the shortest directed path. To ﬁnd path lengths in the reverse direction use G.reverse(copy=False) ﬁrst to ﬂip the edge orientation. Examples >>> G=nx.path_graph(5) >>> print(nx.shortest_path_length(G,source=0,target=4)) 4 >>> p=nx.shortest_path_length(G,source=0) # target not specified >>> p[4] 4 >>> p=nx.shortest_path_length(G) # source,target not specified >>> p[0][4] 4 4.29.4 average_shortest_path_length average_shortest_path_length(G, weight=None) Return the average shortest path length. The average shortest path length is a = Xs,t2V d(s, t) n(n 1) where V is the set of nodes in G, d(s, t) is the shortest path from s to t, and n is the number of nodes in G. Parameters G : NetworkX graph weight : None or string, optional (default = None) If None, every edge has weight/distance/cost 1. If a string, use this edge attribute as the edge weight. Any edge attribute not present defaults to 1. Raises NetworkXError: : if the graph is not connected. 260 Chapter 4. Algorithms NetworkX Reference, Release 1.7 Examples >>> G=nx.path_graph(5) >>> print(nx.average_shortest_path_length(G)) 2.0 For disconnected graphs you can compute the average shortest path length for each compo- nent: >>> G=nx.Graph([(1,2),(3,4)]) >>> for g in nx.connected_component_subgraphs(G): ... print(nx.average_shortest_path_length(g)) 1.0 1.0 4.29.5 has_path has_path(G, source, target) Return True if G has a path from source to target, False otherwise. Parameters G : NetworkX graph source : node Starting node for path target : node Ending node for path 4.29.6 Advanced Interface Shortest path algorithms for unweighted graphs. single_source_shortest_path(G, source[, cutoff]) Compute shortest path between source and all other nodes reachable from source. single_source_shortest_path_length(G, source) Compute the shortest path lengths from source to all reachable nodes. all_pairs_shortest_path(G[, cutoff]) Compute shortest paths between all nodes. all_pairs_shortest_path_length(G[, cutoff]) Compute the shortest path lengths between all nodes in G. predecessor(G, source[, target, cutoff, ...]) Returns dictionary of predecessors for the path from source to all nodes in G. single_source_shortest_path single_source_shortest_path(G, source, cutoff=None) Compute shortest path between source and all other nodes reachable from source. Parameters G : NetworkX graph source : node label Starting node for path cutoff : integer, optional Depth to stop the search. Only paths of length <= cutoff are returned. Returns lengths : dictionary Dictionary, keyed by target, of shortest paths. See Also: shortest_path 4.29. Shortest Paths 261 NetworkX Reference, Release 1.7 Notes The shortest path is not necessarily unique. So there can be multiple paths between the source and each target node, all of which have the same ‘shortest’ length. For each target node, this function returns only one of those paths. Examples >>> G=nx.path_graph(5) >>> path=nx.single_source_shortest_path(G,0) >>> path[4] [0, 1, 2, 3, 4] single_source_shortest_path_length single_source_shortest_path_length(G, source, cutoff=None) Compute the shortest path lengths from source to all reachable nodes. Parameters G : NetworkX graph source : node Starting node for path cutoff : integer, optional Depth to stop the search. Only paths of length <= cutoff are returned. Returns lengths : dictionary Dictionary of shortest path lengths keyed by target. See Also: shortest_path_length Examples >>> G=nx.path_graph(5) >>> length=nx.single_source_shortest_path_length(G,0) >>> length[4] 4 >>> print(length) {0: 0, 1: 1, 2: 2, 3: 3, 4: 4} all_pairs_shortest_path all_pairs_shortest_path(G, cutoff=None) Compute shortest paths between all nodes. Parameters G : NetworkX graph cutoff : integer, optional Depth to stop the search. Only paths of length <= cutoff are returned. Returns lengths : dictionary 262 Chapter 4. Algorithms NetworkX Reference, Release 1.7 Dictionary, keyed by source and target, of shortest paths. See Also: floyd_warshall Examples >>> G=nx.path_graph(5) >>> path=nx.all_pairs_shortest_path(G) >>> print(path[0][4]) [0, 1, 2, 3, 4] all_pairs_shortest_path_length all_pairs_shortest_path_length(G, cutoff=None) Compute the shortest path lengths between all nodes in G. Parameters G : NetworkX graph cutoff : integer, optional depth to stop the search. Only paths of length <= cutoff are returned. Returns lengths : dictionary Dictionary of shortest path lengths keyed by source and target. Notes The dictionary returned only has keys for reachable node pairs. Examples >>> G=nx.path_graph(5) >>> length=nx.all_pairs_shortest_path_length(G) >>> print(length[1][4]) 3 >>> length[1] {0: 1, 1: 0, 2: 1, 3: 2, 4: 3} predecessor predecessor(G, source, target=None, cutoff=None, return_seen=None) Returns dictionary of predecessors for the path from source to all nodes in G. Parameters G : NetworkX graph source : node label Starting node for path target : node label, optional Ending node for path. If provided only predecessors between source and target are returned 4.29. Shortest Paths 263 NetworkX Reference, Release 1.7 cutoff : integer, optional Depth to stop the search. Only paths of length <= cutoff are returned. Returns pred : dictionary Dictionary, keyed by node, of predecessors in the shortest path. Examples >>> G=nx.path_graph(4) >>> print(G.nodes()) [0, 1, 2, 3] >>> nx.predecessor(G,0) {0: [], 1: [0], 2: [1], 3: [2]} Shortest path algorithms for weighed graphs. dijkstra_path(G, source, target[, weight]) Returns the shortest path from source to target in a weighted graph G. dijkstra_path_length(G, source, target[, weight]) Returns the shortest path length from source to target in a weighted graph. single_source_dijkstra_path(G, source[, ...]) Compute shortest path between source and all other reachable nodes for a weighted graph. single_source_dijkstra_path_length(G, source) Compute the shortest path length between source and all other reachable nodes for a weighted graph. all_pairs_dijkstra_path(G[, cutoff, weight]) Compute shortest paths between all nodes in a weighted graph. all_pairs_dijkstra_path_length(G[, cutoff, ...]) Compute shortest path lengths between all nodes in a weighted graph. single_source_dijkstra(G, source[, target, ...]) Compute shortest paths and lengths in a weighted graph G. bidirectional_dijkstra(G, source, target[, ...]) Dijkstra’s algorithm for shortest paths using bidirectional search. dijkstra_predecessor_and_distance(G, source) Compute shortest path length and predecessors on shortest paths in weighted graphs. bellman_ford(G, source[, weight]) Compute shortest path lengths and predecessors on shortest paths in weighted graphs. negative_edge_cycle(G[, weight]) Return True if there exists a negative edge cycle anywhere in G. dijkstra_path dijkstra_path(G, source, target, weight=’weight’) Returns the shortest path from source to target in a weighted graph G. Parameters G : NetworkX graph source : node Starting node target : node Ending node weight: string, optional (default=’weight’) : Edge data key corresponding to the edge weight Returns path : list List of nodes in a shortest path. Raises NetworkXNoPath : If no path exists between source and target. See Also: bidirectional_dijkstra 264 Chapter 4. Algorithms NetworkX Reference, Release 1.7 Notes Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed. Examples >>> G=nx.path_graph(5) >>> print(nx.dijkstra_path(G,0,4)) [0, 1, 2, 3, 4] dijkstra_path_length dijkstra_path_length(G, source, target, weight=’weight’) Returns the shortest path length from source to target in a weighted graph. Parameters G : NetworkX graph source : node label starting node for path target : node label ending node for path weight: string, optional (default=’weight’) : Edge data key corresponding to the edge weight Returns length : number Shortest path length. Raises NetworkXNoPath : If no path exists between source and target. See Also: bidirectional_dijkstra Notes Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed. Examples >>> G=nx.path_graph(5) >>> print(nx.dijkstra_path_length(G,0,4)) 4 4.29. Shortest Paths 265 NetworkX Reference, Release 1.7 single_source_dijkstra_path single_source_dijkstra_path(G, source, cutoff=None, weight=’weight’) Compute shortest path between source and all other reachable nodes for a weighted graph. Parameters G : NetworkX graph source : node Starting node for path. weight: string, optional (default=’weight’) : Edge data key corresponding to the edge weight cutoff : integer or ﬂoat, optional Depth to stop the search. Only paths of length <= cutoff are returned. Returns paths : dictionary Dictionary of shortest path lengths keyed by target. See Also: single_source_dijkstra Notes Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed. Examples >>> G=nx.path_graph(5) >>> path=nx.single_source_dijkstra_path(G,0) >>> path[4] [0, 1, 2, 3, 4] single_source_dijkstra_path_length single_source_dijkstra_path_length(G, source, cutoff=None, weight=’weight’) Compute the shortest path length between source and all other reachable nodes for a weighted graph. Parameters G : NetworkX graph source : node label Starting node for path weight: string, optional (default=’weight’) : Edge data key corresponding to the edge weight. cutoff : integer or ﬂoat, optional Depth to stop the search. Only paths of length <= cutoff are returned. Returns length : dictionary Dictionary of shortest lengths keyed by target. 266 Chapter 4. Algorithms NetworkX Reference, Release 1.7 See Also: single_source_dijkstra Notes Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed. Examples >>> G=nx.path_graph(5) >>> length=nx.single_source_dijkstra_path_length(G,0) >>> length[4] 4 >>> print(length) {0: 0, 1: 1, 2: 2, 3: 3, 4: 4} all_pairs_dijkstra_path all_pairs_dijkstra_path(G, cutoff=None, weight=’weight’) Compute shortest paths between all nodes in a weighted graph. Parameters G : NetworkX graph weight: string, optional (default=’weight’) : Edge data key corresponding to the edge weight cutoff : integer or ﬂoat, optional Depth to stop the search. Only paths of length <= cutoff are returned. Returns distance : dictionary Dictionary, keyed by source and target, of shortest paths. See Also: floyd_warshall Notes Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed. Examples >>> G=nx.path_graph(5) >>> path=nx.all_pairs_dijkstra_path(G) >>> print(path[0][4]) [0, 1, 2, 3, 4] 4.29. Shortest Paths 267 NetworkX Reference, Release 1.7 all_pairs_dijkstra_path_length all_pairs_dijkstra_path_length(G, cutoff=None, weight=’weight’) Compute shortest path lengths between all nodes in a weighted graph. Parameters G : NetworkX graph weight: string, optional (default=’weight’) : Edge data key corresponding to the edge weight cutoff : integer or ﬂoat, optional Depth to stop the search. Only paths of length <= cutoff are returned. Returns distance : dictionary Dictionary, keyed by source and target, of shortest path lengths. Notes Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed. The dictionary returned only has keys for reachable node pairs. Examples >>> G=nx.path_graph(5) >>> length=nx.all_pairs_dijkstra_path_length(G) >>> print(length[1][4]) 3 >>> length[1] {0: 1, 1: 0, 2: 1, 3: 2, 4: 3} single_source_dijkstra single_source_dijkstra(G, source, target=None, cutoff=None, weight=’weight’) Compute shortest paths and lengths in a weighted graph G. Uses Dijkstra’s algorithm for shortest paths. Parameters G : NetworkX graph source : node label Starting node for path target : node label, optional Ending node for path cutoff : integer or ﬂoat, optional Depth to stop the search. Only paths of length <= cutoff are returned. Returns distance,path : dictionaries Returns a tuple of two dictionaries keyed by node. The ﬁrst dictionary stores distance from the source. The second stores the path from the source to that node. 268 Chapter 4. Algorithms NetworkX Reference, Release 1.7 See Also: single_source_dijkstra_path, single_source_dijkstra_path_length Notes Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed. Based on the Python cookbook recipe (119466) at http://aspn.activestate.com/ASPN/Cookbook/Python/Recipe/119466 This algorithm is not guaranteed to work if edge weights are negative or are ﬂoating point numbers (overﬂows and roundoff errors can cause problems). Examples >>> G=nx.path_graph(5) >>> length,path=nx.single_source_dijkstra(G,0) >>> print(length[4]) 4 >>> print(length) {0: 0, 1: 1, 2: 2, 3: 3, 4: 4} >>> path[4] [0, 1, 2, 3, 4] bidirectional_dijkstra bidirectional_dijkstra(G, source, target, weight=’weight’) Dijkstra’s algorithm for shortest paths using bidirectional search. Parameters G : NetworkX graph source : node Starting node. target : node Ending node. weight: string, optional (default=’weight’) : Edge data key corresponding to the edge weight Returns length : number Shortest path length. Returns a tuple of two dictionaries keyed by node. : The ﬁrst dictionary stores distance from the source. : The second stores the path from the source to that node. : Raises NetworkXNoPath : If no path exists between source and target. See Also: shortest_path, shortest_path_length 4.29. Shortest Paths 269 NetworkX Reference, Release 1.7 Notes Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed. In practice bidirectional Dijkstra is much more than twice as fast as ordinary Dijkstra. Ordinary Dijkstra expands nodes in a sphere-like manner from the source. The radius of this sphere will even- tually be the length of the shortest path. Bidirectional Dijkstra will expand nodes from both the source and the target, making two spheres of half this radius. Volume of the ﬁrst sphere is pi*r*r while the others are 2*pi*r/2*r/2, making up half the volume. This algorithm is not guaranteed to work if edge weights are negative or are ﬂoating point numbers (overﬂows and roundoff errors can cause problems). Examples >>> G=nx.path_graph(5) >>> length,path=nx.bidirectional_dijkstra(G,0,4) >>> print(length) 4 >>> print(path) [0, 1, 2, 3, 4] dijkstra_predecessor_and_distance dijkstra_predecessor_and_distance(G, source, cutoff=None, weight=’weight’) Compute shortest path length and predecessors on shortest paths in weighted graphs. Parameters G : NetworkX graph source : node label Starting node for path weight: string, optional (default=’weight’) : Edge data key corresponding to the edge weight cutoff : integer or ﬂoat, optional Depth to stop the search. Only paths of length <= cutoff are returned. Returns pred,distance : dictionaries Returns two dictionaries representing a list of predecessors of a node and the distance to each node. Notes Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed. The list of predecessors contains more than one element only when there are more than one shortest paths to the key node. 270 Chapter 4. Algorithms NetworkX Reference, Release 1.7 bellman_ford bellman_ford(G, source, weight=’weight’) Compute shortest path lengths and predecessors on shortest paths in weighted graphs. The algorithm has a running time of O(mn) where n is the number of nodes and m is the number of edges. It is slower than Dijkstra but can handle negative edge weights. Parameters G : NetworkX graph The algorithm works for all types of graphs, including directed graphs and multigraphs. source: node label : Starting node for path weight: string, optional (default=’weight’) : Edge data key corresponding to the edge weight Returns pred, dist : dictionaries Returns two dictionaries keyed by node to predecessor in the path and to the distance from the source respectively. Raises NetworkXUnbounded : If the (di)graph contains a negative cost (di)cycle, the algorithm raises an exception to indicate the presence of the negative cost (di)cycle. Note: any negative weight edge in an undirected graph is a negative cost cycle. Notes Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed. The dictionaries returned only have keys for nodes reachable from the source. In the case where the (di)graph is not connected, if a component not containing the source contains a negative cost (di)cycle, it will not be detected. Examples >>> import networkx as nx >>> G = nx.path_graph(5, create_using = nx.DiGraph()) >>> pred, dist = nx.bellman_ford(G, 0) >>> pred {0: None, 1: 0, 2: 1, 3: 2, 4: 3} >>> dist {0: 0, 1: 1, 2: 2, 3: 3, 4: 4} >>> from nose.tools import assert_raises >>> G = nx.cycle_graph(5, create_using = nx.DiGraph()) >>> G[1][2][’weight’] = -7 >>> assert_raises(nx.NetworkXUnbounded, nx.bellman_ford, G, 0) negative_edge_cycle negative_edge_cycle(G, weight=’weight’) Return True if there exists a negative edge cycle anywhere in G. 4.29. Shortest Paths 271 NetworkX Reference, Release 1.7 Parameters G : NetworkX graph weight: string, optional (default=’weight’) : Edge data key corresponding to the edge weight Returns negative_cycle : bool True if a negative edge cycle exists, otherwise False. Notes Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed. This algorithm uses bellman_ford() but ﬁnds negative cycles on any component by ﬁrst adding a new node connected to every node, and starting bellman_ford on that node. It then removes that extra node. Examples >>> import networkx as nx >>> G = nx.cycle_graph(5, create_using = nx.DiGraph()) >>> print(nx.negative_edge_cycle(G)) False >>> G[1][2][’weight’] = -7 >>> print(nx.negative_edge_cycle(G)) True 4.29.7 Dense Graphs Floyd-Warshall algorithm for shortest paths. floyd_warshall(G[, weight]) Find all-pairs shortest path lengths using Floyd’s algorithm. floyd_warshall_predecessor_and_distance(G[, ...]) Find all-pairs shortest path lengths using Floyd’s algorithm. floyd_warshall_numpy(G[, nodelist, weight]) Find all-pairs shortest path lengths using Floyd’s algorithm. ﬂoyd_warshall floyd_warshall(G, weight=’weight’) Find all-pairs shortest path lengths using Floyd’s algorithm. Parameters G : NetworkX graph weight: string, optional (default= ‘weight’) : Edge data key corresponding to the edge weight. Returns distance : dict A dictionary, keyed by source and target, of shortest paths distances between nodes. See Also: floyd_warshall_predecessor_and_distance, floyd_warshall_numpy, all_pairs_shortest_path, all_pairs_shortest_path_length 272 Chapter 4. Algorithms NetworkX Reference, Release 1.7 Notes Floyd’s algorithm is appropriate for ﬁnding shortest paths in dense graphs or graphs with negative weights when Dijkstra’s algorithm fails. This algorithm can still fail if there are negative cycles. It has running time O(n^3) with running space of O(n^2). ﬂoyd_warshall_predecessor_and_distance floyd_warshall_predecessor_and_distance(G, weight=’weight’) Find all-pairs shortest path lengths using Floyd’s algorithm. Parameters G : NetworkX graph weight: string, optional (default= ‘weight’) : Edge data key corresponding to the edge weight. Returns predecessor,distance : dictionaries Dictionaries, keyed by source and target, of predecessors and distances in the shortest path. See Also: floyd_warshall, floyd_warshall_numpy, all_pairs_shortest_path, all_pairs_shortest_path_length Notes Floyd’s algorithm is appropriate for ﬁnding shortest paths in dense graphs or graphs with negative weights when Dijkstra’s algorithm fails. This algorithm can still fail if there are negative cycles. It has running time O(n^3) with running space of O(n^2). ﬂoyd_warshall_numpy floyd_warshall_numpy(G, nodelist=None, weight=’weight’) Find all-pairs shortest path lengths using Floyd’s algorithm. Parameters G : NetworkX graph nodelist : list, optional The rows and columns are ordered by the nodes in nodelist. If nodelist is None then the ordering is produced by G.nodes(). weight: string, optional (default= ‘weight’) : Edge data key corresponding to the edge weight. Returns distance : NumPy matrix A matrix of shortest path distances between nodes. If there is no path between to nodes the corresponding matrix entry will be Inf. 4.29. Shortest Paths 273 NetworkX Reference, Release 1.7 Notes Floyd’s algorithm is appropriate for ﬁnding shortest paths in dense graphs or graphs with negative weights when Dijkstra’s algorithm fails. This algorithm can still fail if there are negative cycles. It has running time O(n^3) with running space of O(n^2). 4.29.8 A* Algorithm Shortest paths and path lengths using A* (“A star”) algorithm. astar_path(G, source, target[, heuristic, ...]) Return a list of nodes in a shortest path between source and target astar_path_length(G, source, target[, ...]) Return the length of the shortest path between source and target using astar_path astar_path(G, source, target, heuristic=None, weight=’weight’) Return a list of nodes in a shortest path between source and target using the A* (“A-star”) algorithm. There may be more than one shortest path. This returns only one. Parameters G : NetworkX graph source : node Starting node for path target : node Ending node for path heuristic : function A function to evaluate the estimate of the distance from the a node to the target. The function takes two nodes arguments and must return a number. weight: string, optional (default=’weight’) : Edge data key corresponding to the edge weight. Raises NetworkXNoPath : If no path exists between source and target. See Also: shortest_path, dijkstra_path Examples >>> G=nx.path_graph(5) >>> print(nx.astar_path(G,0,4)) [0, 1, 2, 3, 4] >>> G=nx.grid_graph(dim=[3,3]) # nodes are two-tuples (x,y) >>> def dist(a, b): ... (x1, y1) = a ... (x2, y2) = b ... return ((x1 - x2) ** 2 + (y1 - y2) ** 2) ** 0.5 274 Chapter 4. Algorithms NetworkX Reference, Release 1.7 >>> print(nx.astar_path(G,(0,0),(2,2),dist)) [(0, 0), (0, 1), (1, 1), (1, 2), (2, 2)] astar_path_length astar_path_length(G, source, target, heuristic=None, weight=’weight’) Return the length of the shortest path between source and target using the A* (“A-star”) algorithm. Parameters G : NetworkX graph source : node Starting node for path target : node Ending node for path heuristic : function A function to evaluate the estimate of the distance from the a node to the target. The function takes two nodes arguments and must return a number. Raises NetworkXNoPath : If no path exists between source and target. See Also: astar_path 4.30 Simple Paths all_simple_paths(G, source, target[, cutoff]) Generate all simple paths in the graph G from source to target. 4.30.1 all_simple_paths all_simple_paths(G, source, target, cutoff=None) Generate all simple paths in the graph G from source to target. A simple path is a path with no repeated nodes. Parameters G : NetworkX graph source : node Starting node for path target : node Ending node for path cutoff : integer, optional Depth to stop the search. Only paths of length <= cutoff are returned. Returns path_generator: generator : A generator that produces lists of simple paths. If there are no paths between the source and target within the given cutoff the generator produces no output. 4.30. Simple Paths 275 NetworkX Reference, Release 1.7 See Also: all_shortest_paths, shortest_path Notes This algorithm uses a modiﬁed depth-ﬁrst search to generate the paths [R201]. A single path can be found in O(V + E) time but the number of simple paths in a graph can be very large, e.g. O(n!) in the complete graph of order n. References [R201] Examples >>> G = nx.complete_graph(4) >>> for path in nx.all_simple_paths(G, source=0, target=3): ... print(path) ... [0, 1, 2, 3] [0, 1, 3] [0, 2, 1, 3] [0, 2, 3] [0, 3] >>> paths = nx.all_simple_paths(G, source=0, target=3, cutoff=2) >>> print(list(paths)) [[0, 1, 3], [0, 2, 3], [0, 3]] 4.31 Swap Swap edges in a graph. double_edge_swap(G[, nswap, max_tries]) Swap two edges in the graph while keeping the node degrees ﬁxed. connected_double_edge_swap(G[, nswap]) Attempt nswap double-edge swaps in the graph G. 4.31.1 double_edge_swap double_edge_swap(G, nswap=1, max_tries=100) Swap two edges in the graph while keeping the node degrees ﬁxed. A double-edge swap removes two randomly chosen edges u-v and x-y and creates the new edges u-x and v-y: u--v u v becomes | | x--y x y If either the edge u-x or v-y already exist no swap is performed and another attempt is made to ﬁnd a suitable edge pair. Parameters G : graph 276 Chapter 4. Algorithms NetworkX Reference, Release 1.7 An undirected graph nswap : integer (optional, default=1) Number of double-edge swaps to perform max_tries : integer (optional) Maximum number of attempts to swap edges Returns G : graph The graph after double edge swaps. Notes Does not enforce any connectivity constraints. The graph G is modiﬁed in place. 4.31.2 connected_double_edge_swap connected_double_edge_swap(G, nswap=1) Attempt nswap double-edge swaps in the graph G. A double-edge swap removes two randomly chosen edges u-v and x-y and creates the new edges u-x and v-y: u--v u v becomes | | x--y x y If either the edge u-x or v-y already exist no swap is performed so the actual count of swapped edges is always <= nswap Parameters G : graph An undirected graph nswap : integer (optional, default=1) Number of double-edge swaps to perform Returns G : int The number of successful swaps Notes The initial graph G must be connected, and the resulting graph is connected. The graph G is modiﬁed in place. References [R202] 4.31. Swap 277 NetworkX Reference, Release 1.7 4.32 Traversal 4.32.1 Depth First Search Basic algorithms for depth-ﬁrst searching. Based on http://www.ics.uci.edu/~eppstein/PADS/DFS.py by D. Eppstein, July 2004. dfs_edges(G[, source]) Produce edges in a depth-ﬁrst-search starting at source. dfs_tree(G[, source]) Return directed tree of depth-ﬁrst-search from source. dfs_predecessors(G[, source]) Return dictionary of predecessors in depth-ﬁrst-search from source. dfs_successors(G[, source]) Return dictionary of successors in depth-ﬁrst-search from source. dfs_preorder_nodes(G[, source]) Produce nodes in a depth-ﬁrst-search pre-ordering starting at source. dfs_postorder_nodes(G[, source]) Produce nodes in a depth-ﬁrst-search post-ordering starting dfs_labeled_edges(G[, source]) Produce edges in a depth-ﬁrst-search starting at source and dfs_edges dfs_edges(G, source=None) Produce edges in a depth-ﬁrst-search starting at source. dfs_tree dfs_tree(G, source=None) Return directed tree of depth-ﬁrst-search from source. dfs_predecessors dfs_predecessors(G, source=None) Return dictionary of predecessors in depth-ﬁrst-search from source. dfs_successors dfs_successors(G, source=None) Return dictionary of successors in depth-ﬁrst-search from source. dfs_preorder_nodes dfs_preorder_nodes(G, source=None) Produce nodes in a depth-ﬁrst-search pre-ordering starting at source. dfs_postorder_nodes dfs_postorder_nodes(G, source=None) Produce nodes in a depth-ﬁrst-search post-ordering starting from source. 278 Chapter 4. Algorithms NetworkX Reference, Release 1.7 dfs_labeled_edges dfs_labeled_edges(G, source=None) Produce edges in a depth-ﬁrst-search starting at source and labeled by direction type (forward, reverse, nontree). 4.32.2 Breadth First Search Basic algorithms for breadth-ﬁrst searching. bfs_edges(G, source) Produce edges in a breadth-ﬁrst-search starting at source. bfs_tree(G, source) Return directed tree of breadth-ﬁrst-search from source. bfs_predecessors(G, source) Return dictionary of predecessors in breadth-ﬁrst-search from source. bfs_successors(G, source) Return dictionary of successors in breadth-ﬁrst-search from source. bfs_edges bfs_edges(G, source) Produce edges in a breadth-ﬁrst-search starting at source. bfs_tree bfs_tree(G, source) Return directed tree of breadth-ﬁrst-search from source. bfs_predecessors bfs_predecessors(G, source) Return dictionary of predecessors in breadth-ﬁrst-search from source. bfs_successors bfs_successors(G, source) Return dictionary of successors in breadth-ﬁrst-search from source. 4.33 Vitality Vitality measures. closeness_vitality(G[, weight]) Compute closeness vitality for nodes. 4.33.1 closeness_vitality closeness_vitality(G, weight=None) Compute closeness vitality for nodes. Closeness vitality of a node is the change in the sum of distances between all node pairs when excluding that node. 4.33. Vitality 279 NetworkX Reference, Release 1.7 Parameters G : graph weight : None or string (optional) The name of the edge attribute used as weight. If None the edge weights are ignored. Returns nodes : dictionary Dictionary with nodes as keys and closeness vitality as the value. See Also: closeness_centrality References [R203] Examples >>> G=nx.cycle_graph(3) >>> nx.closeness_vitality(G) {0: 4.0, 1: 4.0, 2: 4.0} 280 Chapter 4. Algorithms CHAPTER FIVE FUNCTIONS Functional interface to graph methods and assorted utilities. 5.1 Graph degree(G[, nbunch, weight]) Return degree of single node or of nbunch of nodes. degree_histogram(G) Return a list of the frequency of each degree value. density(G) Return the density of a graph. info(G[, n]) Print short summary of information for the graph G or the node n. create_empty_copy(G[, with_nodes]) Return a copy of the graph G with all of the edges removed. is_directed(G) Return True if graph is directed. 5.1.1 degree degree(G, nbunch=None, weight=None) Return degree of single node or of nbunch of nodes. If nbunch is ommitted, then return degrees of all nodes. 5.1.2 degree_histogram degree_histogram(G) Return a list of the frequency of each degree value. Parameters G : Networkx graph A graph Returns hist : list A list of frequencies of degrees. The degree values are the index in the list. Notes Note: the bins are width one, hence len(list) can be large (Order(number_of_edges)) 281 NetworkX Reference, Release 1.7 5.1.3 density density(G) Return the density of a graph. The density for undirected graphs is d = 2m n(n 1), and for directed graphs is d = m n(n 1), where n is the number of nodes and m is the number of edges in G. Notes The density is 0 for an graph without edges and 1.0 for a complete graph. The density of multigraphs can be higher than 1. 5.1.4 info info(G, n=None) Print short summary of information for the graph G or the node n. Parameters G : Networkx graph A graph n : node (any hashable) A node in the graph G 5.1.5 create_empty_copy create_empty_copy(G, with_nodes=True) Return a copy of the graph G with all of the edges removed. Parameters G : graph A NetworkX graph with_nodes : bool (default=True) Include nodes. Notes Graph, node, and edge data is not propagated to the new graph. 5.1.6 is_directed is_directed(G) Return True if graph is directed. 282 Chapter 5. Functions NetworkX Reference, Release 1.7 5.2 Nodes nodes(G) Return a copy of the graph nodes in a list. number_of_nodes(G) Return the number of nodes in the graph. nodes_iter(G) Return an iterator over the graph nodes. all_neighbors(graph, node) Returns all of the neighbors of a node in the graph. non_neighbors(graph, node) Returns the non-neighbors of the node in the graph. 5.2.1 nodes nodes(G) Return a copy of the graph nodes in a list. 5.2.2 number_of_nodes number_of_nodes(G) Return the number of nodes in the graph. 5.2.3 nodes_iter nodes_iter(G) Return an iterator over the graph nodes. 5.2.4 all_neighbors all_neighbors(graph, node) Returns all of the neighbors of a node in the graph. If the graph is directed returns predecessors as well as successors. Parameters graph : NetworkX graph Graph to ﬁnd neighbors. node : node The node whose neighbors will be returned. Returns neighbors : iterator Iterator of neighbors 5.2.5 non_neighbors non_neighbors(graph, node) Returns the non-neighbors of the node in the graph. Parameters graph : NetworkX graph Graph to ﬁnd neighbors. node : node 5.2. Nodes 283 NetworkX Reference, Release 1.7 The node whose neighbors will be returned. Returns non_neighbors : iterator Iterator of nodes in the graph that are not neighbors of the node. 5.3 Edges edges(G[, nbunch]) Return list of edges adjacent to nodes in nbunch. number_of_edges(G) Return the number of edges in the graph. edges_iter(G[, nbunch]) Return iterator over edges adjacent to nodes in nbunch. 5.3.1 edges edges(G, nbunch=None) Return list of edges adjacent to nodes in nbunch. Return all edges if nbunch is unspeciﬁed or nbunch=None. For digraphs, edges=out_edges 5.3.2 number_of_edges number_of_edges(G) Return the number of edges in the graph. 5.3.3 edges_iter edges_iter(G, nbunch=None) Return iterator over edges adjacent to nodes in nbunch. Return all edges if nbunch is unspeciﬁed or nbunch=None. For digraphs, edges=out_edges 5.4 Attributes set_node_attributes(G, name, attributes) Set node attributes from dictionary of nodes and values get_node_attributes(G, name) Get node attributes from graph set_edge_attributes(G, name, attributes) Set edge attributes from dictionary of edge tuples and values get_edge_attributes(G, name) Get edge attributes from graph 5.4.1 set_node_attributes set_node_attributes(G, name, attributes) Set node attributes from dictionary of nodes and values Parameters G : NetworkX Graph 284 Chapter 5. Functions NetworkX Reference, Release 1.7 name : string Attribute name attributes: dict : Dictionary of attributes keyed by node. Examples >>> G=nx.path_graph(3) >>> bb=nx.betweenness_centrality(G) >>> nx.set_node_attributes(G,’betweenness’,bb) >>> G.node[1][’betweenness’] 1.0 5.4.2 get_node_attributes get_node_attributes(G, name) Get node attributes from graph Parameters G : NetworkX Graph name : string Attribute name Returns Dictionary of attributes keyed by node. : Examples >>> G=nx.Graph() >>> G.add_nodes_from([1,2,3],color=’red’) >>> color=nx.get_node_attributes(G,’color’) >>> color[1] ’red’ 5.4.3 set_edge_attributes set_edge_attributes(G, name, attributes) Set edge attributes from dictionary of edge tuples and values Parameters G : NetworkX Graph name : string Attribute name attributes: dict : Dictionary of attributes keyed by edge (tuple). 5.4. Attributes 285 NetworkX Reference, Release 1.7 Examples >>> G=nx.path_graph(3) >>> bb=nx.edge_betweenness_centrality(G, normalized=False) >>> nx.set_edge_attributes(G,’betweenness’,bb) >>> G[1][2][’betweenness’] 2.0 5.4.4 get_edge_attributes get_edge_attributes(G, name) Get edge attributes from graph Parameters G : NetworkX Graph name : string Attribute name Returns Dictionary of attributes keyed by node. : Examples >>> G=nx.Graph() >>> G.add_path([1,2,3],color=’red’) >>> color=nx.get_edge_attributes(G,’color’) >>> color[(1,2)] ’red’ 5.5 Freezing graph structure freeze(G) Modify graph to prevent further change by adding or removing nodes or edges. is_frozen(G) Return True if graph is frozen. 5.5.1 freeze freeze(G) Modify graph to prevent further change by adding or removing nodes or edges. Node and edge data can still be modiﬁed. Parameters G : graph A NetworkX graph See Also: is_frozen Notes To “unfreeze” a graph you must make a copy by creating a new graph object: 286 Chapter 5. Functions NetworkX Reference, Release 1.7 >>> graph = nx.path_graph(4) >>> frozen_graph = nx.freeze(graph) >>> unfrozen_graph = nx.Graph(frozen_graph) >>> nx.is_frozen(unfrozen_graph) False Examples >>> G=nx.Graph() >>> G.add_path([0,1,2,3]) >>> G=nx.freeze(G) >>> try: ... G.add_edge(4,5) ... except nx.NetworkXError as e: ... print(str(e)) Frozen graph can’t be modified 5.5.2 is_frozen is_frozen(G) Return True if graph is frozen. Parameters G : graph A NetworkX graph See Also: freeze 5.5. Freezing graph structure 287 NetworkX Reference, Release 1.7 288 Chapter 5. Functions CHAPTER SIX GRAPH GENERATORS 6.1 Atlas Generators for the small graph atlas. See “An Atlas of Graphs” by Ronald C. Read and Robin J. Wilson, Oxford University Press, 1998. Because of its size, this module is not imported by default. graph_atlas_g() Return the list [G0,G1,...,G1252] of graphs as named in the Graph Atlas. 6.1.1 graph_atlas_g graph_atlas_g() Return the list [G0,G1,...,G1252] of graphs as named in the Graph Atlas. G0,G1,...,G1252 are all graphs with up to 7 nodes. The graphs are listed: 1. in increasing order of number of nodes; 2. for a ﬁxed number of nodes, in increasing order of the number of edges; 3. for ﬁxed numbers of nodes and edges, in increasing order of the degree sequence, for example 111223 < 112222; 4. for ﬁxed degree sequence, in increasing number of automorphisms. Note that indexing is set up so that for GAG=graph_atlas_g(), then G123=GAG[123] and G[0]=empty_graph(0) 6.2 Classic Generators for some classic graphs. The typical graph generator is called as follows: >>> G=nx.complete_graph(100) returning the complete graph on n nodes labeled 0,..,99 as a simple graph. Except for empty_graph, all the generators in this module return a Graph class (i.e. a simple, undirected graph). balanced_tree(r, h[, create_using]) Return the perfectly balanced r-tree of height h. Continued on next page 289 NetworkX Reference, Release 1.7 Table 6.2 – continued from previous page barbell_graph(m1, m2[, create_using]) Return the Barbell Graph: two complete graphs connected by a path. complete_graph(n[, create_using]) Return the complete graph K_n with n nodes. complete_bipartite_graph(n1, n2[, create_using]) Return the complete bipartite graph K_{n1_n2}. circular_ladder_graph(n[, create_using]) Return the circular ladder graph CL_n of length n. cycle_graph(n[, create_using]) Return the cycle graph C_n over n nodes. dorogovtsev_goltsev_mendes_graph(n[, ...]) Return the hierarchically constructed Dorogovtsev-Goltsev-Mendes graph. empty_graph([n, create_using]) Return the empty graph with n nodes and zero edges. grid_2d_graph(m, n[, periodic, create_using]) Return the 2d grid graph of mxn nodes, each connected to its nearest neighbors. grid_graph(dim[, periodic]) Return the n-dimensional grid graph. hypercube_graph(n) Return the n-dimensional hypercube. ladder_graph(n[, create_using]) Return the Ladder graph of length n. lollipop_graph(m, n[, create_using]) Return the Lollipop Graph; K_m connected to P_n. null_graph([create_using]) Return the Null graph with no nodes or edges. path_graph(n[, create_using]) Return the Path graph P_n of n nodes linearly connected by n-1 edges. star_graph(n[, create_using]) Return the Star graph with n+1 nodes: one center node, connected to n outer nodes. trivial_graph([create_using]) Return the Trivial graph with one node (with integer label 0) and no edges. wheel_graph(n[, create_using]) Return the wheel graph: a single hub node connected to each node of the (n-1)-node cycle graph. 6.2.1 balanced_tree balanced_tree(r, h, create_using=None) Return the perfectly balanced r-tree of height h. Parameters r : int Branching factor of the tree h : int Height of the tree create_using : NetworkX graph type, optional Use speciﬁed type to construct graph (default = networkx.Graph) Returns G : networkx Graph A tree with n nodes Notes This is the rooted tree where all leaves are at distance h from the root. The root has degree r and all other internal nodes have degree r+1. Node labels are the integers 0 (the root) up to number_of_nodes - 1. Also refered to as a complete r-ary tree. 6.2.2 barbell_graph barbell_graph(m1, m2, create_using=None) Return the Barbell Graph: two complete graphs connected by a path. For m1 > 1 and m2 >= 0. Two identical complete graphs K_{m1} form the left and right bells, and are connected by a path P_{m2}. 290 Chapter 6. Graph generators NetworkX Reference, Release 1.7 The 2*m1+m2 nodes are numbered 0,...,m1-1 for the left barbell, m1,...,m1+m2-1 for the path, and m1+m2,...,2*m1+m2-1 for the right barbell. The 3 subgraphs are joined via the edges (m1-1,m1) and (m1+m2-1,m1+m2). If m2=0, this is merely two complete graphs joined together. This graph is an extremal example in David Aldous and Jim Fill’s etext on Random Walks on Graphs. 6.2.3 complete_graph complete_graph(n, create_using=None) Return the complete graph K_n with n nodes. Node labels are the integers 0 to n-1. 6.2.4 complete_bipartite_graph complete_bipartite_graph(n1, n2, create_using=None) Return the complete bipartite graph K_{n1_n2}. Composed of two partitions with n1 nodes in the ﬁrst and n2 nodes in the second. Each node in the ﬁrst is connected to each node in the second. Node labels are the integers 0 to n1+n2-1 6.2.5 circular_ladder_graph circular_ladder_graph(n, create_using=None) Return the circular ladder graph CL_n of length n. CL_n consists of two concentric n-cycles in which each of the n pairs of concentric nodes are joined by an edge. Node labels are the integers 0 to n-1 6.2.6 cycle_graph cycle_graph(n, create_using=None) Return the cycle graph C_n over n nodes. C_n is the n-path with two end-nodes connected. Node labels are the integers 0 to n-1 If create_using is a DiGraph, the direction is in increasing order. 6.2.7 dorogovtsev_goltsev_mendes_graph dorogovtsev_goltsev_mendes_graph(n, create_using=None) Return the hierarchically constructed Dorogovtsev-Goltsev-Mendes graph. n is the generation. See: arXiv:/cond-mat/0112143 by Dorogovtsev, Goltsev and Mendes. 6.2. Classic 291 NetworkX Reference, Release 1.7 6.2.8 empty_graph empty_graph(n=0, create_using=None) Return the empty graph with n nodes and zero edges. Node labels are the integers 0 to n-1 For example: >>> G=nx.empty_graph(10) >>> G.number_of_nodes() 10 >>> G.number_of_edges() 0 The variable create_using should point to a “graph”-like object that will be cleaned (nodes and edges will be removed) and reﬁtted as an empty “graph” with n nodes with integer labels. This capability is useful for specifying the class-nature of the resulting empty “graph” (i.e. Graph, DiGraph, MyWeirdGraphClass, etc.). The variable create_using has two main uses: Firstly, the variable create_using can be used to create an empty digraph, network,etc. For example, >>> n=10 >>> G=nx.empty_graph(n,create_using=nx.DiGraph()) will create an empty digraph on n nodes. Secondly, one can pass an existing graph (digraph, pseudograph, etc.) via create_using. For example, if G is an existing graph (resp. digraph, pseudograph, etc.), then empty_graph(n,create_using=G) will empty G (i.e. delete all nodes and edges using G.clear() in base) and then add n nodes and zero edges, and return the modiﬁed graph (resp. digraph, pseudograph, etc.). See also create_empty_copy(G). 6.2.9 grid_2d_graph grid_2d_graph(m, n, periodic=False, create_using=None) Return the 2d grid graph of mxn nodes, each connected to its nearest neighbors. Optional argument peri- odic=True will connect boundary nodes via periodic boundary conditions. 6.2.10 grid_graph grid_graph(dim, periodic=False) Return the n-dimensional grid graph. The dimension is the length of the list ‘dim’ and the size in each dimension is the value of the list element. E.g. G=grid_graph(dim=[2,3]) produces a 2x3 grid graph. If periodic=True then join grid edges with periodic boundary conditions. 6.2.11 hypercube_graph hypercube_graph(n) Return the n-dimensional hypercube. Node labels are the integers 0 to 2**n - 1. 292 Chapter 6. Graph generators NetworkX Reference, Release 1.7 6.2.12 ladder_graph ladder_graph(n, create_using=None) Return the Ladder graph of length n. This is two rows of n nodes, with each pair connected by a single edge. Node labels are the integers 0 to 2*n - 1. 6.2.13 lollipop_graph lollipop_graph(m, n, create_using=None) Return the Lollipop Graph; K_m connected to P_n. This is the Barbell Graph without the right barbell. For m>1 and n>=0, the complete graph K_m is connected to the path P_n. The resulting m+n nodes are labelled 0,...,m-1 for the complete graph and m,...,m+n-1 for the path. The 2 subgraphs are joined via the edge (m-1,m). If n=0, this is merely a complete graph. Node labels are the integers 0 to number_of_nodes - 1. (This graph is an extremal example in David Aldous and Jim Fill’s etext on Random Walks on Graphs.) 6.2.14 null_graph null_graph(create_using=None) Return the Null graph with no nodes or edges. See empty_graph for the use of create_using. 6.2.15 path_graph path_graph(n, create_using=None) Return the Path graph P_n of n nodes linearly connected by n-1 edges. Node labels are the integers 0 to n - 1. If create_using is a DiGraph then the edges are directed in increasing order. 6.2.16 star_graph star_graph(n, create_using=None) Return the Star graph with n+1 nodes: one center node, connected to n outer nodes. Node labels are the integers 0 to n. 6.2.17 trivial_graph trivial_graph(create_using=None) Return the Trivial graph with one node (with integer label 0) and no edges. 6.2. Classic 293 NetworkX Reference, Release 1.7 6.2.18 wheel_graph wheel_graph(n, create_using=None) Return the wheel graph: a single hub node connected to each node of the (n-1)-node cycle graph. Node labels are the integers 0 to n - 1. 6.3 Small Various small and named graphs, together with some compact generators. make_small_graph(graph_description[, ...]) Return the small graph described by graph_description. LCF_graph(n, shift_list, repeats[, create_using]) Return the cubic graph speciﬁed in LCF notation. bull_graph([create_using]) Return the Bull graph. chvatal_graph([create_using]) Return the Chvátal graph. cubical_graph([create_using]) Return the 3-regular Platonic Cubical graph. desargues_graph([create_using]) Return the Desargues graph. diamond_graph([create_using]) Return the Diamond graph. dodecahedral_graph([create_using]) Return the Platonic Dodecahedral graph. frucht_graph([create_using]) Return the Frucht Graph. heawood_graph([create_using]) Return the Heawood graph, a (3,6) cage. house_graph([create_using]) Return the House graph (square with triangle on top). house_x_graph([create_using]) Return the House graph with a cross inside the house square. icosahedral_graph([create_using]) Return the Platonic Icosahedral graph. krackhardt_kite_graph([create_using]) Return the Krackhardt Kite Social Network. moebius_kantor_graph([create_using]) Return the Moebius-Kantor graph. octahedral_graph([create_using]) Return the Platonic Octahedral graph. pappus_graph() Return the Pappus graph. petersen_graph([create_using]) Return the Petersen graph. sedgewick_maze_graph([create_using]) Return a small maze with a cycle. tetrahedral_graph([create_using]) Return the 3-regular Platonic Tetrahedral graph. truncated_cube_graph([create_using]) Return the skeleton of the truncated cube. truncated_tetrahedron_graph([create_using]) Return the skeleton of the truncated Platonic tetrahedron. tutte_graph([create_using]) Return the Tutte graph. 6.3.1 make_small_graph make_small_graph(graph_description, create_using=None) Return the small graph described by graph_description. graph_description is a list of the form [ltype,name,n,xlist] Here ltype is one of “adjacencylist” or “edgelist”, name is the name of the graph and n the number of nodes. This constructs a graph of n nodes with integer labels 0,..,n-1. If ltype=”adjacencylist” then xlist is an adjacency list with exactly n entries, in with the j’th entry (which can be empty) speciﬁes the nodes connected to vertex j. e.g. the “square” graph C_4 can be obtained by >>> G=nx.make_small_graph(["adjacencylist","C_4",4,[[2,4],[1,3],[2,4],[1,3]]]) or, since we do not need to add edges twice, 294 Chapter 6. Graph generators NetworkX Reference, Release 1.7 >>> G=nx.make_small_graph(["adjacencylist","C_4",4,[[2,4],[3],[4],[]]]) If ltype=”edgelist” then xlist is an edge list written as [[v1,w2],[v2,w2],...,[vk,wk]], where vj and wj integers in the range 1,..,n e.g. the “square” graph C_4 can be obtained by >>> G=nx.make_small_graph(["edgelist","C_4",4,[[1,2],[3,4],[2,3],[4,1]]]) Use the create_using argument to choose the graph class/type. 6.3.2 LCF_graph LCF_graph(n, shift_list, repeats, create_using=None) Return the cubic graph speciﬁed in LCF notation. LCF notation (LCF=Lederberg-Coxeter-Fruchte) is a compressed notation used in the generation of various cubic Hamiltonian graphs of high symmetry. See, for example, dodecahedral_graph, desargues_graph, hea- wood_graph and pappus_graph below. n (number of nodes) The starting graph is the n-cycle with nodes 0,...,n-1. (The null graph is returned if n < 0.) shift_list = [s1,s2,..,sk], a list of integer shifts mod n, repeats integer specifying the number of times that shifts in shift_list are successively applied to each v_current in the n-cycle to generate an edge between v_current and v_current+shift mod n. For v1 cycling through the n-cycle a total of k*repeats with shift cycling through shiftlist repeats times connect v1 with v1+shift mod n The utility graph K_{3,3} >>> G=nx.LCF_graph(6,[3,-3],3) The Heawood graph >>> G=nx.LCF_graph(14,[5,-5],7) See http://mathworld.wolfram.com/LCFNotation.html for a description and references. 6.3.3 bull_graph bull_graph(create_using=None) Return the Bull graph. 6.3.4 chvatal_graph chvatal_graph(create_using=None) Return the Chvátal graph. 6.3.5 cubical_graph cubical_graph(create_using=None) Return the 3-regular Platonic Cubical graph. 6.3. Small 295 NetworkX Reference, Release 1.7 6.3.6 desargues_graph desargues_graph(create_using=None) Return the Desargues graph. 6.3.7 diamond_graph diamond_graph(create_using=None) Return the Diamond graph. 6.3.8 dodecahedral_graph dodecahedral_graph(create_using=None) Return the Platonic Dodecahedral graph. 6.3.9 frucht_graph frucht_graph(create_using=None) Return the Frucht Graph. The Frucht Graph is the smallest cubical graph whose automorphism group consists only of the identity element. 6.3.10 heawood_graph heawood_graph(create_using=None) Return the Heawood graph, a (3,6) cage. 6.3.11 house_graph house_graph(create_using=None) Return the House graph (square with triangle on top). 6.3.12 house_x_graph house_x_graph(create_using=None) Return the House graph with a cross inside the house square. 6.3.13 icosahedral_graph icosahedral_graph(create_using=None) Return the Platonic Icosahedral graph. 296 Chapter 6. Graph generators NetworkX Reference, Release 1.7 6.3.14 krackhardt_kite_graph krackhardt_kite_graph(create_using=None) Return the Krackhardt Kite Social Network. A 10 actor social network introduced by David Krackhardt to illustrate: degree, betweenness, centrality, close- ness, etc. The traditional labeling is: Andre=1, Beverley=2, Carol=3, Diane=4, Ed=5, Fernando=6, Garth=7, Heather=8, Ike=9, Jane=10. 6.3.15 moebius_kantor_graph moebius_kantor_graph(create_using=None) Return the Moebius-Kantor graph. 6.3.16 octahedral_graph octahedral_graph(create_using=None) Return the Platonic Octahedral graph. 6.3.17 pappus_graph pappus_graph() Return the Pappus graph. 6.3.18 petersen_graph petersen_graph(create_using=None) Return the Petersen graph. 6.3.19 sedgewick_maze_graph sedgewick_maze_graph(create_using=None) Return a small maze with a cycle. This is the maze used in Sedgewick,3rd Edition, Part 5, Graph Algorithms, Chapter 18, e.g. Figure 18.2 and following. Nodes are numbered 0,..,7 6.3.20 tetrahedral_graph tetrahedral_graph(create_using=None) Return the 3-regular Platonic Tetrahedral graph. 6.3.21 truncated_cube_graph truncated_cube_graph(create_using=None) Return the skeleton of the truncated cube. 6.3. Small 297 NetworkX Reference, Release 1.7 6.3.22 truncated_tetrahedron_graph truncated_tetrahedron_graph(create_using=None) Return the skeleton of the truncated Platonic tetrahedron. 6.3.23 tutte_graph tutte_graph(create_using=None) Return the Tutte graph. 6.4 Random Graphs Generators for random graphs. fast_gnp_random_graph(n, p[, seed, directed]) Return a random graph G_{n,p} (Erd˝os-Rényi graph, binomial graph). gnp_random_graph(n, p[, seed, directed]) Return a random graph G_{n,p} (Erd˝os-Rényi graph, binomial graph). dense_gnm_random_graph(n, m[, seed]) Return the random graph G_{n,m}. gnm_random_graph(n, m[, seed, directed]) Return the random graph G_{n,m}. erdos_renyi_graph(n, p[, seed, directed]) Return a random graph G_{n,p} (Erd˝os-Rényi graph, binomial graph). binomial_graph(n, p[, seed, directed]) Return a random graph G_{n,p} (Erd˝os-Rényi graph, binomial graph). newman_watts_strogatz_graph(n, k, p[, seed]) Return a Newman-Watts-Strogatz small world graph. watts_strogatz_graph(n, k, p[, seed]) Return a Watts-Strogatz small-world graph. connected_watts_strogatz_graph(n, k, p[, ...]) Return a connected Watts-Strogatz small-world graph. random_regular_graph(d, n[, seed]) Return a random regular graph of n nodes each with degree d. barabasi_albert_graph(n, m[, seed]) Return random graph using Barabási-Albert preferential attachment model. powerlaw_cluster_graph(n, m, p[, seed]) Holme and Kim algorithm for growing graphs with powerlaw random_lobster(n, p1, p2[, seed]) Return a random lobster. random_shell_graph(constructor[, seed]) Return a random shell graph for the constructor given. random_powerlaw_tree(n[, gamma, seed, tries]) Return a tree with a powerlaw degree distribution. random_powerlaw_tree_sequence(n[, gamma, ...]) Return a degree sequence for a tree with a powerlaw distribution. 6.4.1 fast_gnp_random_graph fast_gnp_random_graph(n, p, seed=None, directed=False) Return a random graph G_{n,p} (Erd˝os-Rényi graph, binomial graph). Parameters n : int The number of nodes. p : ﬂoat Probability for edge creation. seed : int, optional Seed for random number generator (default=None). directed : bool, optional (default=False) If True return a directed graph See Also: gnp_random_graph 298 Chapter 6. Graph generators NetworkX Reference, Release 1.7 Notes The G_{n,p} graph algorithm chooses each of the [n(n-1)]/2 (undirected) or n(n-1) (directed) possible edges with probability p. This algorithm is O(n+m) where m is the expected number of edges m=p*n*(n-1)/2. It should be faster than gnp_random_graph when p is small and the expected number of edges is small (sparse graph). References [R234] 6.4.2 gnp_random_graph gnp_random_graph(n, p, seed=None, directed=False) Return a random graph G_{n,p} (Erd˝os-Rényi graph, binomial graph). Chooses each of the possible edges with probability p. This is also called binomial_graph and erdos_renyi_graph. Parameters n : int The number of nodes. p : ﬂoat Probability for edge creation. seed : int, optional Seed for random number generator (default=None). directed : bool, optional (default=False) If True return a directed graph See Also: fast_gnp_random_graph Notes This is an O(n^2) algorithm. For sparse graphs (small p) see fast_gnp_random_graph for a faster algorithm. References [R235], [R236] 6.4. Random Graphs 299 NetworkX Reference, Release 1.7 6.4.3 dense_gnm_random_graph dense_gnm_random_graph(n, m, seed=None) Return the random graph G_{n,m}. Gives a graph picked randomly out of the set of all graphs with n nodes and m edges. This algorithm should be faster than gnm_random_graph for dense graphs. Parameters n : int The number of nodes. m : int The number of edges. seed : int, optional Seed for random number generator (default=None). See Also: gnm_random_graph Notes Algorithm by Keith M. Briggs Mar 31, 2006. Inspired by Knuth’s Algorithm S (Selection sampling technique), in section 3.4.2 of [R231]. References [R231] 6.4.4 gnm_random_graph gnm_random_graph(n, m, seed=None, directed=False) Return the random graph G_{n,m}. Produces a graph picked randomly out of the set of all graphs with n nodes and m edges. Parameters n : int The number of nodes. m : int The number of edges. seed : int, optional Seed for random number generator (default=None). directed : bool, optional (default=False) If True return a directed graph 300 Chapter 6. Graph generators NetworkX Reference, Release 1.7 6.4.5 erdos_renyi_graph erdos_renyi_graph(n, p, seed=None, directed=False) Return a random graph G_{n,p} (Erd˝os-Rényi graph, binomial graph). Chooses each of the possible edges with probability p. This is also called binomial_graph and erdos_renyi_graph. Parameters n : int The number of nodes. p : ﬂoat Probability for edge creation. seed : int, optional Seed for random number generator (default=None). directed : bool, optional (default=False) If True return a directed graph See Also: fast_gnp_random_graph Notes This is an O(n^2) algorithm. For sparse graphs (small p) see fast_gnp_random_graph for a faster algorithm. References [R232], [R233] 6.4.6 binomial_graph binomial_graph(n, p, seed=None, directed=False) Return a random graph G_{n,p} (Erd˝os-Rényi graph, binomial graph). Chooses each of the possible edges with probability p. This is also called binomial_graph and erdos_renyi_graph. Parameters n : int The number of nodes. p : ﬂoat Probability for edge creation. seed : int, optional Seed for random number generator (default=None). directed : bool, optional (default=False) If True return a directed graph 6.4. Random Graphs 301 NetworkX Reference, Release 1.7 See Also: fast_gnp_random_graph Notes This is an O(n^2) algorithm. For sparse graphs (small p) see fast_gnp_random_graph for a faster algorithm. References [R229], [R230] 6.4.7 newman_watts_strogatz_graph newman_watts_strogatz_graph(n, k, p, seed=None) Return a Newman-Watts-Strogatz small world graph. Parameters n : int The number of nodes k : int Each node is connected to k nearest neighbors in ring topology p : ﬂoat The probability of adding a new edge for each edge seed : int, optional seed for random number generator (default=None) See Also: watts_strogatz_graph Notes First create a ring over n nodes. Then each node in the ring is connected with its k nearest neighbors (k-1 neighbors if k is odd). Then shortcuts are created by adding new edges as follows: for each edge u-v in the underlying “n-ring with k nearest neighbors” with probability p add a new edge u-w with randomly-chosen existing node w. In contrast with watts_strogatz_graph(), no edges are removed. References [R237] 6.4.8 watts_strogatz_graph watts_strogatz_graph(n, k, p, seed=None) Return a Watts-Strogatz small-world graph. Parameters n : int 302 Chapter 6. Graph generators NetworkX Reference, Release 1.7 The number of nodes k : int Each node is connected to k nearest neighbors in ring topology p : ﬂoat The probability of rewiring each edge seed : int, optional Seed for random number generator (default=None) See Also: newman_watts_strogatz_graph, connected_watts_strogatz_graph Notes First create a ring over n nodes. Then each node in the ring is connected with its k nearest neighbors (k-1 neighbors if k is odd). Then shortcuts are created by replacing some edges as follows: for each edge u-v in the underlying “n-ring with k nearest neighbors” with probability p replace it with a new edge u-w with uniformly random choice of existing node w. In contrast with newman_watts_strogatz_graph(), the random rewiring does not increase the number of edges. The rewired graph is not guaranteed to be connected as in connected_watts_strogatz_graph(). References [R241] 6.4.9 connected_watts_strogatz_graph connected_watts_strogatz_graph(n, k, p, tries=100, seed=None) Return a connected Watts-Strogatz small-world graph. Attempt to generate a connected realization by repeated generation of Watts-Strogatz small-world graphs. An exception is raised if the maximum number of tries is exceeded. Parameters n : int The number of nodes k : int Each node is connected to k nearest neighbors in ring topology p : ﬂoat The probability of rewiring each edge tries : int Number of attempts to generate a connected graph. seed : int, optional The seed for random number generator. 6.4. Random Graphs 303 NetworkX Reference, Release 1.7 See Also: newman_watts_strogatz_graph, watts_strogatz_graph 6.4.10 random_regular_graph random_regular_graph(d, n, seed=None) Return a random regular graph of n nodes each with degree d. The resulting graph G has no self-loops or parallel edges. Parameters d : int Degree n : integer Number of nodes. The value of n*d must be even. seed : hashable object The seed for random number generator. Notes The nodes are numbered form 0 to n-1. Kim and Vu’s paper [R240] shows that this algorithm samples in an asymptotically uniform way from the space of random graphs when d = O(n**(1/3-epsilon)). References [R239], [R240] 6.4.11 barabasi_albert_graph barabasi_albert_graph(n, m, seed=None) Return random graph using Barabási-Albert preferential attachment model. A graph of n nodes is grown by attaching new nodes each with m edges that are preferentially attached to existing nodes with high degree. Parameters n : int Number of nodes m : int Number of edges to attach from a new node to existing nodes seed : int, optional Seed for random number generator (default=None). Returns G : Graph 304 Chapter 6. Graph generators NetworkX Reference, Release 1.7 Notes The initialization is a graph with with m nodes and no edges. References [R228] 6.4.12 powerlaw_cluster_graph powerlaw_cluster_graph(n, m, p, seed=None) Holme and Kim algorithm for growing graphs with powerlaw degree distribution and approximate average clustering. Parameters n : int the number of nodes m : int the number of random edges to add for each new node p : ﬂoat, Probability of adding a triangle after adding a random edge seed : int, optional Seed for random number generator (default=None). Notes The average clustering has a hard time getting above a certain cutoff that depends on m. This cutoff is often quite low. Note that the transitivity (fraction of triangles to possible triangles) seems to go down with network size. It is essentially the Barabási-Albert (B-A) growth model with an extra step that each random edge is followed by a chance of making an edge to one of its neighbors too (and thus a triangle). This algorithm improves on B-A in the sense that it enables a higher average clustering to be attained if desired. It seems possible to have a disconnected graph with this algorithm since the initial m nodes may not be all linked to a new node on the ﬁrst iteration like the B-A model. References [R238] 6.4.13 random_lobster random_lobster(n, p1, p2, seed=None) Return a random lobster. A lobster is a tree that reduces to a caterpillar when pruning all leaf nodes. A caterpillar is a tree that reduces to a path graph when pruning all leaf nodes (p2=0). 6.4. Random Graphs 305 NetworkX Reference, Release 1.7 Parameters n : int The expected number of nodes in the backbone p1 : ﬂoat Probability of adding an edge to the backbone p2 : ﬂoat Probability of adding an edge one level beyond backbone seed : int, optional Seed for random number generator (default=None). 6.4.14 random_shell_graph random_shell_graph(constructor, seed=None) Return a random shell graph for the constructor given. Parameters constructor: a list of three-tuples : (n,m,d) for each shell starting at the center shell. n : int The number of nodes in the shell m : int The number or edges in the shell d : ﬂoat The ratio of inter-shell (next) edges to intra-shell edges. d=0 means no intra shell edges, d=1 for the last shell seed : int, optional Seed for random number generator (default=None). Examples >>> constructor=[(10,20,0.8),(20,40,0.8)] >>> G=nx.random_shell_graph(constructor) 6.4.15 random_powerlaw_tree random_powerlaw_tree(n, gamma=3, seed=None, tries=100) Return a tree with a powerlaw degree distribution. Parameters n : int, The number of nodes gamma : ﬂoat Exponent of the power-law seed : int, optional 306 Chapter 6. Graph generators NetworkX Reference, Release 1.7 Seed for random number generator (default=None). tries : int Number of attempts to adjust sequence to make a tree Notes A trial powerlaw degree sequence is chosen and then elements are swapped with new elements from a powerlaw distribution until the sequence makes a tree (#edges=#nodes-1). 6.4.16 random_powerlaw_tree_sequence random_powerlaw_tree_sequence(n, gamma=3, seed=None, tries=100) Return a degree sequence for a tree with a powerlaw distribution. Parameters n : int, The number of nodes gamma : ﬂoat Exponent of the power-law seed : int, optional Seed for random number generator (default=None). tries : int Number of attempts to adjust sequence to make a tree Notes A trial powerlaw degree sequence is chosen and then elements are swapped with new elements from a powerlaw distribution until the sequence makes a tree (#edges=#nodes-1). 6.5 Degree Sequence Generate graphs with a given degree sequence or expected degree sequence. configuration_model(deg_sequence[, ...]) Return a random graph with the given degree sequence. directed_configuration_model(...[, ...]) Return a directed_random graph with the given degree sequences. expected_degree_graph(w[, seed, selﬂoops]) Return a random graph with given expected degrees. havel_hakimi_graph(deg_sequence[, create_using]) Return a simple graph with given degree sequence constructed degree_sequence_tree(deg_sequence[, ...]) Make a tree for the given degree sequence. random_degree_sequence_graph(sequence[, ...]) Return a simple random graph with the given degree sequence. 6.5.1 conﬁguration_model configuration_model(deg_sequence, create_using=None, seed=None) Return a random graph with the given degree sequence. The conﬁguration model generates a random pseudograph (graph with parallel edges and self loops) by ran- 6.5. Degree Sequence 307 NetworkX Reference, Release 1.7 domly assigning edges to match the given degree sequence. Parameters deg_sequence : list of integers Each list entry corresponds to the degree of a node. create_using : graph, optional (default MultiGraph) Return graph of this type. The instance will be cleared. seed : hashable object, optional Seed for random number generator. Returns G : MultiGraph A graph with the speciﬁed degree sequence. Nodes are labeled starting at 0 with an index corresponding to the position in deg_sequence. Raises NetworkXError : If the degree sequence does not have an even sum. See Also: is_valid_degree_sequence Notes As described by Newman [R207]. A non-graphical degree sequence (not realizable by some simple graph) is allowed since this function returns graphs with self loops and parallel edges. An exception is raised if the degree sequence does not have an even sum. This conﬁguration model construction process can lead to duplicate edges and loops. You can remove the self-loops and parallel edges (see below) which will likely result in a graph that doesn’t have the exact degree sequence speciﬁed. This “ﬁnite-size effect” decreases as the size of the graph increases. References [R207] Examples >>> from networkx.utils import powerlaw_sequence >>> z=nx.utils.create_degree_sequence(100,powerlaw_sequence) >>> G=nx.configuration_model(z) To remove parallel edges: >>> G=nx.Graph(G) To remove self loops: >>> G.remove_edges_from(G.selfloop_edges()) 308 Chapter 6. Graph generators NetworkX Reference, Release 1.7 6.5.2 directed_conﬁguration_model directed_configuration_model(in_degree_sequence, out_degree_sequence, create_using=None, seed=None) Return a directed_random graph with the given degree sequences. The conﬁguration model generates a random directed pseudograph (graph with parallel edges and self loops) by randomly assigning edges to match the given degree sequences. Parameters in_degree_sequence : list of integers Each list entry corresponds to the in-degree of a node. out_degree_sequence : list of integers Each list entry corresponds to the out-degree of a node. create_using : graph, optional (default MultiDiGraph) Return graph of this type. The instance will be cleared. seed : hashable object, optional Seed for random number generator. Returns G : MultiDiGraph A graph with the speciﬁed degree sequences. Nodes are labeled starting at 0 with an index corresponding to the position in deg_sequence. Raises NetworkXError : If the degree sequences do not have the same sum. See Also: configuration_model Notes Algorithm as described by Newman [R208]. A non-graphical degree sequence (not realizable by some simple graph) is allowed since this function returns graphs with self loops and parallel edges. An exception is raised if the degree sequences does not have the same sum. This conﬁguration model construction process can lead to duplicate edges and loops. You can remove the self-loops and parallel edges (see below) which will likely result in a graph that doesn’t have the exact degree sequence speciﬁed. This “ﬁnite-size effect” decreases as the size of the graph increases. References [R208] Examples >>> D=nx.DiGraph([(0,1),(1,2),(2,3)]) # directed path graph >>> din=list(D.in_degree().values()) >>> dout=list(D.out_degree().values()) >>> din.append(1) 6.5. Degree Sequence 309 NetworkX Reference, Release 1.7 >>> dout[0]=2 >>> D=nx.directed_configuration_model(din,dout) To remove parallel edges: >>> D=nx.DiGraph(D) To remove self loops: >>> D.remove_edges_from(D.selfloop_edges()) 6.5.3 expected_degree_graph expected_degree_graph(w, seed=None, selﬂoops=True) Return a random graph with given expected degrees. Given a sequence of expected degrees W =(w0,w1,...,wn1) of length n this algorithm assigns an edge between node u and node v with probability puv = wuwv P k wk . Parameters w : list The list of expected degrees. selﬂoops: bool (default=True) : Set to False to remove the possibility of self-loop edges. seed : hashable object, optional The seed for the random number generator. Returns Graph : Notes The nodes have integer labels corresponding to index of expected degrees input sequence. The complexity of this algorithm is O(n + m) where n is the number of nodes and m is the expected number of edges. The model in [R209] includes the possibility of self-loop edges. Set selﬂoops=False to produce a graph without self loops. For ﬁnite graphs this model doesn’t produce exactly the given expected degree sequence. Instead the expected degrees are as follows. For the case without self loops (selﬂoops=False), E[deg(u)] = Xv6=u puv = wu ✓ 1 wu P k wk ◆ . NetworkX uses the standard convention that a self-loop edge counts 2 in the degree of a node, so with self loops (selﬂoops=True), E[deg(u)] = Xv6=u puv +2puu = wu ✓ 1+ wu P k wk ◆ . 310 Chapter 6. Graph generators NetworkX Reference, Release 1.7 References [R209], [R210] Examples >>> z=[10 for i in range(100)] >>> G=nx.expected_degree_graph(z) 6.5.4 havel_hakimi_graph havel_hakimi_graph(deg_sequence, create_using=None) Return a simple graph with given degree sequence constructed using the Havel-Hakimi algorithm. Parameters deg_sequence: list of integers : Each integer corresponds to the degree of a node (need not be sorted). create_using : graph, optional (default Graph) Return graph of this type. The instance will be cleared. Multigraphs and directed graphs are not allowed. Raises NetworkXException : For a non-graphical degree sequence (i.e. one not realizable by some simple graph). Notes The Havel-Hakimi algorithm constructs a simple graph by successively connecting the node of highest degree to other nodes of highest degree, resorting remaining nodes by degree, and repeating the process. The resulting graph has a high degree-associativity. Nodes are labeled 1,.., len(deg_sequence), corresponding to their position in deg_sequence. See Theorem 1.4 in [R211]. This algorithm is also used in the function is_valid_degree_sequence. References [R211] 6.5.5 degree_sequence_tree degree_sequence_tree(deg_sequence, create_using=None) Make a tree for the given degree sequence. A tree has #nodes-#edges=1 so the degree sequence must have len(deg_sequence)-sum(deg_sequence)/2=1 6.5. Degree Sequence 311 NetworkX Reference, Release 1.7 6.5.6 random_degree_sequence_graph random_degree_sequence_graph(sequence, seed=None, tries=10) Return a simple random graph with the given degree sequence. If the maximum degree dm in the sequence is O(m1/4) then the algorithm produces almost uniform random graphs in O(mdm) time where m is the number of edges. Parameters sequence : list of integers Sequence of degrees seed : hashable object, optional Seed for random number generator tries : int, optional Maximum number of tries to create a graph Returns G : Graph A graph with the speciﬁed degree sequence. Nodes are labeled starting at 0 with an index corresponding to the position in the sequence. Raises NetworkXUnfeasible : If the degree sequence is not graphical. NetworkXError : If a graph is not produced in speciﬁed number of tries See Also: is_valid_degree_sequence, configuration_model Notes The generator algorithm [R212] is not guaranteed to produce a graph. References [R212] Examples >>> sequence = [1, 2, 2, 3] >>> G = nx.random_degree_sequence_graph(sequence) >>> sorted(G.degree().values()) [1, 2, 2, 3] 6.6 Random Clustered Generate graphs with given degree and triangle sequence. 312 Chapter 6. Graph generators NetworkX Reference, Release 1.7 random_clustered_graph(joint_degree_sequence) Generate a random graph with the given joint degree and triangle degree sequence. 6.6.1 random_clustered_graph random_clustered_graph(joint_degree_sequence, create_using=None, seed=None) Generate a random graph with the given joint degree and triangle degree sequence. This uses a conﬁguration model-like approach to generate a random pseudograph (graph with parallel edges and self loops) by randomly assigning edges to match the given indepdenent edge and triangle degree sequence. Parameters joint_degree_sequence : list of integer pairs Each list entry corresponds to the independent edge degree and triangle degree of a node. create_using : graph, optional (default MultiGraph) Return graph of this type. The instance will be cleared. seed : hashable object, optional The seed for the random number generator. Returns G : MultiGraph A graph with the speciﬁed degree sequence. Nodes are labeled starting at 0 with an index corresponding to the position in deg_sequence. Raises NetworkXError : If the independent edge degree sequence sum is not even or the triangle degree sequence sum is not divisible by 3. Notes As described by Miller [R226] (see also Newman [R227] for an equivalent description). A non-graphical degree sequence (not realizable by some simple graph) is allowed since this function returns graphs with self loops and parallel edges. An exception is raised if the independent degree sequence does not have an even sum or the triangle degree sequence sum is not divisible by 3. This conﬁguration model-like construction process can lead to duplicate edges and loops. You can remove the self-loops and parallel edges (see below) which will likely result in a graph that doesn’t have the exact degree sequence speciﬁed. This “ﬁnite-size effect” decreases as the size of the graph increases. References [R226], [R227] Examples >>> deg_tri=[[1,0],[1,0],[1,0],[2,0],[1,0],[2,1],[0,1],[0,1]] >>> G = nx.random_clustered_graph(deg_tri) To remove parallel edges: 6.6. Random Clustered 313 NetworkX Reference, Release 1.7 >>> G=nx.Graph(G) To remove self loops: >>> G.remove_edges_from(G.selfloop_edges()) 6.7 Directed Generators for some directed graphs. gn_graph: growing network gnc_graph: growing network with copying gnr_graph: growing network with redirection scale_free_graph: scale free directed graph gn_graph(n[, kernel, create_using, seed]) Return the GN digraph with n nodes. gnr_graph(n, p[, create_using, seed]) Return the GNR digraph with n nodes and redirection probability p. gnc_graph(n[, create_using, seed]) Return the GNC digraph with n nodes. scale_free_graph(n[, alpha, beta, gamma, ...]) Return a scale free directed graph. 6.7.1 gn_graph gn_graph(n, kernel=None, create_using=None, seed=None) Return the GN digraph with n nodes. The GN (growing network) graph is built by adding nodes one at a time with a link to one previously added node. The target node for the link is chosen with probability based on degree. The default attachment kernel is a linear function of degree. The graph is always a (directed) tree. Parameters n : int The number of nodes for the generated graph. kernel : function The attachment kernel. create_using : graph, optional (default DiGraph) Return graph of this type. The instance will be cleared. seed : hashable object, optional The seed for the random number generator. References [R213] Examples >>> D=nx.gn_graph(10) # the GN graph >>> G=D.to_undirected() # the undirected version 314 Chapter 6. Graph generators NetworkX Reference, Release 1.7 To specify an attachment kernel use the kernel keyword >>> D=nx.gn_graph(10,kernel=lambda x:x**1.5) # A_k=k^1.5 6.7.2 gnr_graph gnr_graph(n, p, create_using=None, seed=None) Return the GNR digraph with n nodes and redirection probability p. The GNR (growing network with redirection) graph is built by adding nodes one at a time with a link to one previously added node. The previous target node is chosen uniformly at random. With probabiliy p the link is instead “redirected” to the successor node of the target. The graph is always a (directed) tree. Parameters n : int The number of nodes for the generated graph. p : ﬂoat The redirection probability. create_using : graph, optional (default DiGraph) Return graph of this type. The instance will be cleared. seed : hashable object, optional The seed for the random number generator. References [R215] Examples >>> D=nx.gnr_graph(10,0.5) # the GNR graph >>> G=D.to_undirected() # the undirected version 6.7.3 gnc_graph gnc_graph(n, create_using=None, seed=None) Return the GNC digraph with n nodes. The GNC (growing network with copying) graph is built by adding nodes one at a time with a links to one previously added node (chosen uniformly at random) and to all of that node’s successors. Parameters n : int The number of nodes for the generated graph. create_using : graph, optional (default DiGraph) Return graph of this type. The instance will be cleared. seed : hashable object, optional The seed for the random number generator. 6.7. Directed 315 NetworkX Reference, Release 1.7 References [R214] 6.7.4 scale_free_graph scale_free_graph(n, alpha=0.41, beta=0.54, gamma=0.05, delta_in=0.2, delta_out=0, cre- ate_using=None, seed=None) Return a scale free directed graph. Parameters n : integer Number of nodes in graph alpha : ﬂoat Probability for adding a new node connected to an existing node chosen randomly ac- cording to the in-degree distribution. beta : ﬂoat Probability for adding an edge between two existing nodes. One existing node is chosen randomly according the in-degree distribution and the other chosen randomly according to the out-degree distribution. gamma : ﬂoat Probability for adding a new node conecgted to an existing node chosen randomly ac- cording to the out-degree distribution. delta_in : ﬂoat Bias for choosing ndoes from in-degree distribution. delta_out : ﬂoat Bias for choosing ndoes from out-degree distribution. create_using : graph, optional (default MultiDiGraph) Use this graph instance to start the process (default=3-cycle). seed : integer, optional Seed for random number generator Notes The sum of alpha, beta, and gamma must be 1. References [R216] Examples >>> G=nx.scale_free_graph(100) 316 Chapter 6. Graph generators NetworkX Reference, Release 1.7 6.8 Geometric Generators for geometric graphs. random_geometric_graph(n, radius[, dim, pos]) Return the random geometric graph in the unit cube. geographical_threshold_graph(n, theta[, ...]) Return a geographical threshold graph. waxman_graph(n[, alpha, beta, L, domain]) Return a Waxman random graph. navigable_small_world_graph(n[, p, q, r, ...]) Return a navigable small-world graph. 6.8.1 random_geometric_graph random_geometric_graph(n, radius, dim=2, pos=None) Return the random geometric graph in the unit cube. The random geometric graph model places n nodes uniformly at random in the unit cube Two nodes u, v are connected with an edge if d(u, v) <= r where d is the Euclidean distance and r is a radius threshold. Parameters n : int Number of nodes radius: ﬂoat : Distance threshold value dim : int, optional Dimension of graph pos : dict, optional A dictionary keyed by node with node positions as values. Returns Graph : Notes This uses an n2 algorithm to build the graph. A faster algorithm is possible using k-d trees. The pos keyword can be used to specify node positions so you can create an arbitrary distribution and domain for positions. If you need a distance function other than Euclidean you’ll have to hack the algorithm. E.g to use a 2d Gaussian distribution of node positions with mean (0,0) and std. dev. 2 >>> import random >>> n=20 >>> p=dict((i,(random.gauss(0,2),random.gauss(0,2))) for i in range(n)) >>> G = nx.random_geometric_graph(n,0.2,pos=p) References [R220] 6.8. Geometric 317 NetworkX Reference, Release 1.7 Examples >>> G = nx.random_geometric_graph(20,0.1) 6.8.2 geographical_threshold_graph geographical_threshold_graph(n, theta, alpha=2, dim=2, pos=None, weight=None) Return a geographical threshold graph. The geographical threshold graph model places n nodes uniformly at random in a rectangular domain. Each node u is assigned a weight wu. Two nodes u, v are connected with an edge if wu + wv ✓r↵ where r is the Euclidean distance between u and v, and ✓, ↵ are parameters. Parameters n : int Number of nodes theta: ﬂoat : Threshold value alpha: ﬂoat, optional : Exponent of distance function dim : int, optional Dimension of graph pos : dict Node positions as a dictionary of tuples keyed by node. weight : dict Node weights as a dictionary of numbers keyed by node. Returns Graph : Notes If weights are not speciﬁed they are assigned to nodes by drawing randomly from an the exponential distribution with rate parameter =1. To specify a weights from a different distribution assign them to a dictionary and pass it as the weight= keyword >>> import random >>> n = 20 >>> w=dict((i,random.expovariate(5.0)) for i in range(n)) >>> G = nx.geographical_threshold_graph(20,50,weight=w) If node positions are not speciﬁed they are randomly assigned from the uniform distribution. References [R217], [R218] 318 Chapter 6. Graph generators NetworkX Reference, Release 1.7 Examples >>> G = nx.geographical_threshold_graph(20,50) 6.8.3 waxman_graph waxman_graph(n, alpha=0.4, beta=0.1, L=None, domain=(0, 0, 1, 1)) Return a Waxman random graph. The Waxman random graph models place n nodes uniformly at random in a rectangular domain. Two nodes u,v are connected with an edge with probability p = ↵ ⇤ exp(d/( ⇤ L)). This function implements both Waxman models. Waxman-1: L not speciﬁed The distance d is the Euclidean distance between the nodes u and v. L is the maximum distance between all nodes in the graph. Waxman-2: L speciﬁed The distance d is chosen randomly in [0,L]. Parameters n : int Number of nodes alpha: ﬂoat : Model parameter beta: ﬂoat : Model parameter L : ﬂoat, optional Maximum distance between nodes. If not speciﬁed the actual distance is calculated. domain : tuple of numbers, optional Domain size (xmin, ymin, xmax, ymax) Returns G: Graph : References [R221] 6.8.4 navigable_small_world_graph navigable_small_world_graph(n, p=1, q=1, r=2, dim=2, seed=None) Return a navigable small-world graph. A navigable small-world graph is a directed grid with additional long-range connections that are chosen ran- domly. From [R219]: Begin with a set of nodes that are identiﬁed with the set of lattice points in an n ⇥ n square, (i, j):i 2 1, 2,...,n,j 2 1, 2,...,nand deﬁne the lattice distance between two nodes (i, j) and (k, l) to be the number of “lattice steps” separating them: d((i, j), (k, l)) = |k i| + |l j|. 6.8. Geometric 319 NetworkX Reference, Release 1.7 For a universal constant p, the node u has a directed edge to every other node within lattice distance p (local contacts) . For universal constants q 0 and r 0 construct directed edges from u to q other nodes (long-range contacts) using independent random trials; the i’th directed edge from u has endpoint v with probability proportional to d(u, v)r. Parameters n : int The number of nodes. p : int The diameter of short range connections. Each node is connected to every other node within lattice distance p. q : int The number of long-range connections for each node. r : ﬂoat Exponent for decaying probability of connections. The probability of connecting to a node at lattice distance d is 1/d^r. dim : int Dimension of grid seed : int, optional Seed for random number generator (default=None). References [R219] 6.9 Hybrid Hybrid kl_connected_subgraph(G, k, l[, low_memory, ...]) Returns the maximum locally (k,l) connected subgraph of G. is_kl_connected(G, k, l[, low_memory]) Returns True if G is kl connected. 6.9.1 kl_connected_subgraph kl_connected_subgraph(G, k, l, low_memory=False, same_as_graph=False) Returns the maximum locally (k,l) connected subgraph of G. (k,l)-connected subgraphs are presented by Fan Chung and Li in “The Small World Phenomenon in hybrid power law graphs” to appear in “Complex Networks” (Ed. E. Ben-Naim) Lecture Notes in Physics, Springer (2004) low_memory=True then use a slightly slower, but lower memory version same_as_graph=True then return a tuple with subgraph and pﬂag for if G is kl-connected 320 Chapter 6. Graph generators NetworkX Reference, Release 1.7 6.9.2 is_kl_connected is_kl_connected(G, k, l, low_memory=False) Returns True if G is kl connected. 6.10 Bipartite Generators and functions for bipartite graphs. bipartite_configuration_model(aseq, bseq[, ...]) Return a random bipartite graph from two given degree sequences. bipartite_havel_hakimi_graph(aseq, bseq[, ...]) Return a bipartite graph from two given degree sequences using a bipartite_reverse_havel_hakimi_graph(aseq, bseq) Return a bipartite graph from two given degree sequences using a bipartite_alternating_havel_hakimi_graph(...) Return a bipartite graph from two given degree sequences using bipartite_preferential_attachment_graph(aseq, p) Create a bipartite graph with a preferential attachment model from a given single degree sequence. bipartite_random_graph(n, m, p[, seed, directed]) Return a bipartite random graph. bipartite_gnmk_random_graph(n, m, k[, seed, ...]) Return a random bipartite graph G_{n,m,k}. 6.10.1 bipartite_conﬁguration_model bipartite_configuration_model(aseq, bseq, create_using=None, seed=None) Return a random bipartite graph from two given degree sequences. Parameters aseq : list or iterator Degree sequence for node set A. bseq : list or iterator Degree sequence for node set B. create_using : NetworkX graph instance, optional Return graph of this type. seed : integer, optional Seed for random number generator. Nodes from the set A are connected to nodes in the set B by : choosing randomly from the possible free stubs, one in A and : one in B. : Notes The sum of the two sequences must be equal: sum(aseq)=sum(bseq) If no graph type is speciﬁed use MultiGraph with parallel edges. If you want a graph with no parallel edges use create_using=Graph() but then the resulting degree sequences might not be exact. The nodes are assigned the attribute ‘bipartite’ with the value 0 or 1 to indicate which bipartite set the node belongs to. 6.10. Bipartite 321 NetworkX Reference, Release 1.7 6.10.2 bipartite_havel_hakimi_graph bipartite_havel_hakimi_graph(aseq, bseq, create_using=None) Return a bipartite graph from two given degree sequences using a Havel-Hakimi style construction. Nodes from the set A are connected to nodes in the set B by connecting the highest degree nodes in set A to the highest degree nodes in set B until all stubs are connected. Parameters aseq : list or iterator Degree sequence for node set A. bseq : list or iterator Degree sequence for node set B. create_using : NetworkX graph instance, optional Return graph of this type. Notes The sum of the two sequences must be equal: sum(aseq)=sum(bseq) If no graph type is speciﬁed use MultiGraph with parallel edges. If you want a graph with no parallel edges use create_using=Graph() but then the resulting degree sequences might not be exact. The nodes are assigned the attribute ‘bipartite’ with the value 0 or 1 to indicate which bipartite set the node belongs to. 6.10.3 bipartite_reverse_havel_hakimi_graph bipartite_reverse_havel_hakimi_graph(aseq, bseq, create_using=None) Return a bipartite graph from two given degree sequences using a Havel-Hakimi style construction. Nodes from set A are connected to nodes in the set B by connecting the highest degree nodes in set A to the lowest degree nodes in set B until all stubs are connected. Parameters aseq : list or iterator Degree sequence for node set A. bseq : list or iterator Degree sequence for node set B. create_using : NetworkX graph instance, optional Return graph of this type. Notes The sum of the two sequences must be equal: sum(aseq)=sum(bseq) If no graph type is speciﬁed use MultiGraph with parallel edges. If you want a graph with no parallel edges use create_using=Graph() but then the resulting degree sequences might not be exact. The nodes are assigned the attribute ‘bipartite’ with the value 0 or 1 to indicate which bipartite set the node belongs to. 322 Chapter 6. Graph generators NetworkX Reference, Release 1.7 6.10.4 bipartite_alternating_havel_hakimi_graph bipartite_alternating_havel_hakimi_graph(aseq, bseq, create_using=None) Return a bipartite graph from two given degree sequences using an alternating Havel-Hakimi style construction. Nodes from the set A are connected to nodes in the set B by connecting the highest degree nodes in set A to alternatively the highest and the lowest degree nodes in set B until all stubs are connected. Parameters aseq : list or iterator Degree sequence for node set A. bseq : list or iterator Degree sequence for node set B. create_using : NetworkX graph instance, optional Return graph of this type. Notes The sum of the two sequences must be equal: sum(aseq)=sum(bseq) If no graph type is speciﬁed use MultiGraph with parallel edges. If you want a graph with no parallel edges use create_using=Graph() but then the resulting degree sequences might not be exact. The nodes are assigned the attribute ‘bipartite’ with the value 0 or 1 to indicate which bipartite set the node belongs to. 6.10.5 bipartite_preferential_attachment_graph bipartite_preferential_attachment_graph(aseq, p, create_using=None, seed=None) Create a bipartite graph with a preferential attachment model from a given single degree sequence. Parameters aseq : list or iterator Degree sequence for node set A. p : ﬂoat Probability that a new bottom node is added. create_using : NetworkX graph instance, optional Return graph of this type. seed : integer, optional Seed for random number generator. References [R205] 6.10. Bipartite 323 NetworkX Reference, Release 1.7 6.10.6 bipartite_random_graph bipartite_random_graph(n, m, p, seed=None, directed=False) Return a bipartite random graph. This is a bipartite version of the binomial (Erd˝os-Rényi) graph. Parameters n : int The number of nodes in the ﬁrst bipartite set. m : int The number of nodes in the second bipartite set. p : ﬂoat Probability for edge creation. seed : int, optional Seed for random number generator (default=None). directed : bool, optional (default=False) If True return a directed graph See Also: gnp_random_graph, bipartite_configuration_model Notes The bipartite random graph algorithm chooses each of the n*m (undirected) or 2*nm (directed) possible edges with probability p. This algorithm is O(n+m) where m is the expected number of edges. The nodes are assigned the attribute ‘bipartite’ with the value 0 or 1 to indicate which bipartite set the node belongs to. References [R206] 6.10.7 bipartite_gnmk_random_graph bipartite_gnmk_random_graph(n, m, k, seed=None, directed=False) Return a random bipartite graph G_{n,m,k}. Produces a bipartite graph chosen randomly out of the set of all graphs with n top nodes, m bottom nodes, and k edges. Parameters n : int The number of nodes in the ﬁrst bipartite set. m : int The number of nodes in the second bipartite set. k : int 324 Chapter 6. Graph generators NetworkX Reference, Release 1.7 The number of edges seed : int, optional Seed for random number generator (default=None). directed : bool, optional (default=False) If True return a directed graph See Also: gnm_random_graph Notes If k > m * n then a complete bipartite graph is returned. This graph is a bipartite version of the Gnm random graph model. Examples G = nx.bipartite_gnmk_random_graph(10,20,50) 6.11 Line Graph Line graphs. line_graph(G) Return the line graph of the graph or digraph G. 6.11.1 line_graph line_graph(G) Return the line graph of the graph or digraph G. The line graph of a graph G has a node for each edge in G and an edge between those nodes if the two edges in G share a common node. For DiGraphs an edge an edge represents a directed path of length 2. The original node labels are kept as two-tuple node labels in the line graph. Parameters G : graph A NetworkX Graph or DiGraph Notes Not implemented for MultiGraph or MultiDiGraph classes. Graph, node, and edge data are not propagated to the new graph. 6.11. Line Graph 325 NetworkX Reference, Release 1.7 Examples >>> G=nx.star_graph(3) >>> L=nx.line_graph(G) >>> print(sorted(L.edges())) # makes a clique, K3 [((0, 1), (0, 2)), ((0, 1), (0, 3)), ((0, 3), (0, 2))] 6.12 Ego Graph Ego graph. ego_graph(G, n[, radius, center, ...]) Returns induced subgraph of neighbors centered at node n within a given radius. 6.12.1 ego_graph ego_graph(G, n, radius=1, center=True, undirected=False, distance=None) Returns induced subgraph of neighbors centered at node n within a given radius. Parameters G : graph A NetworkX Graph or DiGraph n : node A single node radius : number, optional Include all neighbors of distance<=radius from n. center : bool, optional If False, do not include center node in graph undirected : bool, optional If True use both in- and out-neighbors of directed graphs. distance : key, optional Use speciﬁed edge data key as distance. For example, setting distance=’weight’ will use the edge weight to measure the distance from the node n. Notes For directed graphs D this produces the “out” neighborhood or successors. If you want the neighborhood of predecessors ﬁrst reverse the graph with D.reverse(). If you want both directions use the keyword argument undirected=True. Node, edge, and graph attributes are copied to the returned subgraph. 6.13 Stochastic Stocastic graph. 326 Chapter 6. Graph generators NetworkX Reference, Release 1.7 stochastic_graph(G[, copy, weight]) Return a right-stochastic representation of G. 6.13.1 stochastic_graph stochastic_graph(G, copy=True, weight=’weight’) Return a right-stochastic representation of G. A right-stochastic graph is a weighted graph in which all of the node (out) neighbors edge weights sum to 1. Parameters G : graph A NetworkX graph, must have valid edge weights copy : boolean, optional If True make a copy of the graph, otherwise modify original graph weight : key (optional) Edge data key used for weight. If None all weights are set to 1. 6.14 Intersection Generators for random intersection graphs. uniform_random_intersection_graph(n, m, p[, ...]) Return a uniform random intersection graph. k_random_intersection_graph(n, m, k) Return a intersection graph with randomly chosen attribute sets for each node that are of equal size (k). general_random_intersection_graph(n, m, p) Return a random intersection graph with independent probabilities for connections between node and attribute sets. 6.14.1 uniform_random_intersection_graph uniform_random_intersection_graph(n, m, p, seed=None) Return a uniform random intersection graph. Parameters n : int The number of nodes in the ﬁrst bipartite set (nodes) m : int The number of nodes in the second bipartite set (attributes) p : ﬂoat Probability of connecting nodes between bipartite sets seed : int, optional Seed for random number generator (default=None). See Also: gnp_random_graph References [R224], [R225] 6.14. Intersection 327 NetworkX Reference, Release 1.7 6.14.2 k_random_intersection_graph k_random_intersection_graph(n, m, k) Return a intersection graph with randomly chosen attribute sets for each node that are of equal size (k). Parameters n : int The number of nodes in the ﬁrst bipartite set (nodes) m : int The number of nodes in the second bipartite set (attributes) k : ﬂoat Size of attribute set to assign to each node. seed : int, optional Seed for random number generator (default=None). See Also: gnp_random_graph, uniform_random_intersection_graph References [R223] 6.14.3 general_random_intersection_graph general_random_intersection_graph(n, m, p) Return a random intersection graph with independent probabilities for connections between node and attribute sets. Parameters n : int The number of nodes in the ﬁrst bipartite set (nodes) m : int The number of nodes in the second bipartite set (attributes) p : list of ﬂoats of length m Probabilities for connecting nodes to each attribute seed : int, optional Seed for random number generator (default=None). See Also: gnp_random_graph, uniform_random_intersection_graph References [R222] 328 Chapter 6. Graph generators NetworkX Reference, Release 1.7 6.15 Social Networks Famous social networks. karate_club_graph() Return Zachary’s Karate club graph. davis_southern_women_graph() Return Davis Southern women social network. florentine_families_graph() Return Florentine families graph. 6.15.1 karate_club_graph karate_club_graph() Return Zachary’s Karate club graph. References [R244], [R245] 6.15.2 davis_southern_women_graph davis_southern_women_graph() Return Davis Southern women social network. This is a bipartite graph. References [R242] 6.15.3 ﬂorentine_families_graph florentine_families_graph() Return Florentine families graph. References [R243] 6.15. Social Networks 329 NetworkX Reference, Release 1.7 330 Chapter 6. Graph generators CHAPTER SEVEN LINEAR ALGEBRA 7.1 Graph Matrix Adjacency matrix and incidence matrix of graphs. adjacency_matrix(G[, nodelist, weight]) Return adjacency matrix of G. incidence_matrix(G[, nodelist, edgelist, ...]) Return incidence matrix of G. 7.1.1 adjacency_matrix adjacency_matrix(G, nodelist=None, weight=’weight’) Return adjacency matrix of G. Parameters G : graph A NetworkX graph nodelist : list, optional The rows and columns are ordered according to the nodes in nodelist. If nodelist is None, then the ordering is produced by G.nodes(). weight : string or None, optional (default=’weight’) The edge data key used to provide each value in the matrix. If None, then each edge has weight 1. Returns A : numpy matrix Adjacency matrix representation of G. See Also: to_numpy_matrix, to_dict_of_dicts Notes If you want a pure Python adjacency matrix representation try networkx.convert.to_dict_of_dicts which will return a dictionary-of-dictionaries format that can be addressed as a sparse matrix. For MultiGraph/MultiDiGraph, the edges weights are summed. See to_numpy_matrix for other options. 331 NetworkX Reference, Release 1.7 7.1.2 incidence_matrix incidence_matrix(G, nodelist=None, edgelist=None, oriented=False, weight=None) Return incidence matrix of G. The incidence matrix assigns each row to a node and each column to an edge. For a standard incidence matrix a 1 appears wherever a row’s node is incident on the column’s edge. For an oriented incidence matrix each edge is assigned an orientation (arbitrarily for undirected and aligning to direction for directed). A -1 appears for the tail of an edge and 1 for the head of the edge. The elements are zero otherwise. Parameters G : graph A NetworkX graph nodelist : list, optional (default= all nodes in G) The rows are ordered according to the nodes in nodelist. If nodelist is None, then the ordering is produced by G.nodes(). edgelist : list, optional (default= all edges in G) The columns are ordered according to the edges in edgelist. If edgelist is None, then the ordering is produced by G.edges(). oriented: bool, optional (default=False) : If True, matrix elements are +1 or -1 for the head or tail node respectively of each edge. If False, +1 occurs at both nodes. weight : string or None, optional (default=None) The edge data key used to provide each value in the matrix. If None, then each edge has weight 1. Edge weights, if used, should be positive so that the orientation can provide the sign. Returns A : NumPy matrix The incidence matrix of G. Notes For MultiGraph/MultiDiGraph, the edges in edgelist should be (u,v,key) 3-tuples. “Networks are the best discrete model for so many problems in applied mathematics” [R246]. References [R246] 7.2 Laplacian Matrix Laplacian matrix of graphs. laplacian_matrix(G[, nodelist, weight]) Return the Laplacian matrix of G. normalized_laplacian_matrix(G[, nodelist, ...]) Return the normalized Laplacian matrix of G. 332 Chapter 7. Linear algebra NetworkX Reference, Release 1.7 7.2.1 laplacian_matrix laplacian_matrix(G, nodelist=None, weight=’weight’) Return the Laplacian matrix of G. The graph Laplacian is the matrix L = D - A, where A is the adjacency matrix and D is the diagonal matrix of node degrees. Parameters G : graph A NetworkX graph nodelist : list, optional The rows and columns are ordered according to the nodes in nodelist. If nodelist is None, then the ordering is produced by G.nodes(). weight : string or None, optional (default=’weight’) The edge data key used to compute each value in the matrix. If None, then each edge has weight 1. Returns L : NumPy array Laplacian of G. See Also: to_numpy_matrix, normalized_laplacian Notes For MultiGraph/MultiDiGraph, the edges weights are summed. See to_numpy_matrix for other options. 7.2.2 normalized_laplacian_matrix normalized_laplacian_matrix(G, nodelist=None, weight=’weight’) Return the normalized Laplacian matrix of G. The normalized graph Laplacian is the matrix NL = D1/2LD1/2 where L is the graph Laplacian and D is the diagonal matrix of node degrees. Parameters G : graph A NetworkX graph nodelist : list, optional The rows and columns are ordered according to the nodes in nodelist. If nodelist is None, then the ordering is produced by G.nodes(). weight : string or None, optional (default=’weight’) The edge data key used to compute each value in the matrix. If None, then each edge has weight 1. Returns L : NumPy array Normalized Laplacian of G. 7.2. Laplacian Matrix 333 NetworkX Reference, Release 1.7 See Also: laplacian Notes For MultiGraph/MultiDiGraph, the edges weights are summed. See to_numpy_matrix for other options. References [R247] 7.3 Spectrum Eigenvalue spectrum of graphs. laplacian_spectrum(G[, weight]) Return eigenvalues of the Laplacian of G adjacency_spectrum(G[, weight]) Return eigenvalues of the adjacency matrix of G. 7.3.1 laplacian_spectrum laplacian_spectrum(G, weight=’weight’) Return eigenvalues of the Laplacian of G Parameters G : graph A NetworkX graph weight : string or None, optional (default=’weight’) The edge data key used to compute each value in the matrix. If None, then each edge has weight 1. Returns evals : NumPy array Eigenvalues See Also: laplacian_matrix Notes For MultiGraph/MultiDiGraph, the edges weights are summed. See to_numpy_matrix for other options. 7.3.2 adjacency_spectrum adjacency_spectrum(G, weight=’weight’) Return eigenvalues of the adjacency matrix of G. Parameters G : graph A NetworkX graph 334 Chapter 7. Linear algebra NetworkX Reference, Release 1.7 weight : string or None, optional (default=’weight’) The edge data key used to compute each value in the matrix. If None, then each edge has weight 1. Returns evals : NumPy array Eigenvalues See Also: adjacency_matrix Notes For MultiGraph/MultiDiGraph, the edges weights are summed. See to_numpy_matrix for other options. 7.4 Attribute Matrices Functions for constructing matrix-like objects from graph attributes. attr_matrix(G[, edge_attr, node_attr, ...]) Returns a NumPy matrix using attributes from G. attr_sparse_matrix(G[, edge_attr, ...]) Returns a SciPy sparse matrix using attributes from G. 7.4.1 attr_matrix attr_matrix(G, edge_attr=None, node_attr=None, normalized=False, rc_order=None, dtype=None, or- der=None) Returns a NumPy matrix using attributes from G. If only G is passed in, then the adjacency matrix is constructed. Let A be a discrete set of values for the node attribute nodeattr. Then the elements of A represent the rows and columns of the constructed matrix. Now, iterate through every edge e=(u,v) in G and consider the value of the edge attribute edgeattr. If ua and va are the values of the node attribute nodeattr for u and v, respectively, then the value of the edge attribute is added to the matrix element at (ua, va). Parameters G : graph The NetworkX graph used to construct the NumPy matrix. edge_attr : str, optional Each element of the matrix represents a running total of the speciﬁed edge attribute for edges whose node attributes correspond to the rows/cols of the matirx. The attribute must be present for all edges in the graph. If no attribute is speciﬁed, then we just count the number of edges whose node attributes correspond to the matrix element. node_attr : str, optional Each row and column in the matrix represents a particular value of the node attribute. The attribute must be present for all nodes in the graph. Note, the values of this attribute should be reliably hashable. So, ﬂoat values are not recommended. If no attribute is speciﬁed, then the rows and columns will be the nodes of the graph. normalized : bool, optional If True, then each row is normalized by the summation of its values. 7.4. Attribute Matrices 335 NetworkX Reference, Release 1.7 rc_order : list, optional A list of the node attribute values. This list speciﬁes the ordering of rows and columns of the array. If no ordering is provided, then the ordering will be random (and also, a return value). Returns M : NumPy matrix The attribute matrix. ordering : list If rcorder was speciﬁed, then only the matrix is returned. However, if rcorder was None, then the ordering used to construct the matrix is returned as well. Other Parameters dtype : NumPy data-type, optional A valid NumPy dtype used to initialize the array. Keep in mind certain dtypes can yield unexpected results if the array is to be normalized. The parameter is passed to numpy.zeros(). If unspeciﬁed, the NumPy default is used. order : {‘C’, ‘F’}, optional Whether to store multidimensional data in C- or Fortran-contiguous (row- or column- wise) order in memory. This parameter is passed to numpy.zeros(). If unspeciﬁed, the NumPy default is used. Examples Construct an adjacency matrix: >>> G = nx.Graph() >>> G.add_edge(0,1,thickness=1,weight=3) >>> G.add_edge(0,2,thickness=2) >>> G.add_edge(1,2,thickness=3) >>> nx.attr_matrix(G, rc_order=[0,1,2]) matrix([[ 0., 1., 1.], [ 1., 0., 1.], [ 1., 1., 0.]]) Alternatively, we can obtain the matrix describing edge thickness. >>> nx.attr_matrix(G, edge_attr=’thickness’, rc_order=[0,1,2]) matrix([[ 0., 1., 2.], [ 1., 0., 3.], [ 2., 3., 0.]]) We can also color the nodes and ask for the probability distribution over all edges (u,v) describing: Pr(v has color Y | u has color X) >>> G.node[0][’color’] = ’red’ >>> G.node[1][’color’] = ’red’ >>> G.node[2][’color’] = ’blue’ >>> rc = [’red’, ’blue’] >>> nx.attr_matrix(G, node_attr=’color’, normalized=True, rc_order=rc) matrix([[ 0.33333333, 0.66666667], [ 1. , 0. ]]) For example, the above tells us that for all edges (u,v): 336 Chapter 7. Linear algebra NetworkX Reference, Release 1.7 Pr( v is red | u is red) = 1/3 Pr( v is blue | u is red) = 2/3 Pr( v is red | u is blue) = 1 Pr( v is blue | u is blue) = 0 Finally, we can obtain the total weights listed by the node colors. >>> nx.attr_matrix(G, edge_attr=’weight’, node_attr=’color’, rc_order=rc) matrix([[ 3., 2.], [ 2., 0.]]) Thus, the total weight over all edges (u,v) with u and v having colors: (red, red) is 3 # the sole contribution is from edge (0,1) (red, blue) is 2 # contributions from edges (0,2) and (1,2) (blue, red) is 2 # same as (red, blue) since graph is undirected (blue, blue) is 0 # there are no edges with blue endpoints 7.4.2 attr_sparse_matrix attr_sparse_matrix(G, edge_attr=None, node_attr=None, normalized=False, rc_order=None, dtype=None) Returns a SciPy sparse matrix using attributes from G. If only G is passed in, then the adjacency matrix is constructed. Let A be a discrete set of values for the node attribute nodeattr. Then the elements of A represent the rows and columns of the constructed matrix. Now, iterate through every edge e=(u,v) in G and consider the value of the edge attribute edgeattr. If ua and va are the values of the node attribute nodeattr for u and v, respectively, then the value of the edge attribute is added to the matrix element at (ua, va). Parameters G : graph The NetworkX graph used to construct the NumPy matrix. edge_attr : str, optional Each element of the matrix represents a running total of the speciﬁed edge attribute for edges whose node attributes correspond to the rows/cols of the matirx. The attribute must be present for all edges in the graph. If no attribute is speciﬁed, then we just count the number of edges whose node attributes correspond to the matrix element. node_attr : str, optional Each row and column in the matrix represents a particular value of the node attribute. The attribute must be present for all nodes in the graph. Note, the values of this attribute should be reliably hashable. So, ﬂoat values are not recommended. If no attribute is speciﬁed, then the rows and columns will be the nodes of the graph. normalized : bool, optional If True, then each row is normalized by the summation of its values. rc_order : list, optional A list of the node attribute values. This list speciﬁes the ordering of rows and columns of the array. If no ordering is provided, then the ordering will be random (and also, a return value). Returns M : SciPy sparse matrix The attribute matrix. ordering : list 7.4. Attribute Matrices 337 NetworkX Reference, Release 1.7 If rcorder was speciﬁed, then only the matrix is returned. However, if rcorder was None, then the ordering used to construct the matrix is returned as well. Other Parameters dtype : NumPy data-type, optional A valid NumPy dtype used to initialize the array. Keep in mind certain dtypes can yield unexpected results if the array is to be normalized. The parameter is passed to numpy.zeros(). If unspeciﬁed, the NumPy default is used. Examples Construct an adjacency matrix: >>> G = nx.Graph() >>> G.add_edge(0,1,thickness=1,weight=3) >>> G.add_edge(0,2,thickness=2) >>> G.add_edge(1,2,thickness=3) >>> M = nx.attr_sparse_matrix(G, rc_order=[0,1,2]) >>> M.todense() matrix([[ 0., 1., 1.], [ 1., 0., 1.], [ 1., 1., 0.]]) Alternatively, we can obtain the matrix describing edge thickness. >>> M = nx.attr_sparse_matrix(G, edge_attr=’thickness’, rc_order=[0,1,2]) >>> M.todense() matrix([[ 0., 1., 2.], [ 1., 0., 3.], [ 2., 3., 0.]]) We can also color the nodes and ask for the probability distribution over all edges (u,v) describing: Pr(v has color Y | u has color X) >>> G.node[0][’color’] = ’red’ >>> G.node[1][’color’] = ’red’ >>> G.node[2][’color’] = ’blue’ >>> rc = [’red’, ’blue’] >>> M = nx.attr_sparse_matrix(G, node_attr=’color’, normalized=True, rc_order=rc) >>> M.todense() matrix([[ 0.33333333, 0.66666667], [ 1. , 0. ]]) For example, the above tells us that for all edges (u,v): Pr( v is red | u is red) = 1/3 Pr( v is blue | u is red) = 2/3 Pr( v is red | u is blue) = 1 Pr( v is blue | u is blue) = 0 Finally, we can obtain the total weights listed by the node colors. >>> M = nx.attr_sparse_matrix(G, edge_attr=’weight’, node_attr=’color’, rc_order=rc) >>> M.todense() matrix([[ 3., 2.], [ 2., 0.]]) Thus, the total weight over all edges (u,v) with u and v having colors: 338 Chapter 7. Linear algebra NetworkX Reference, Release 1.7 (red, red) is 3 # the sole contribution is from edge (0,1) (red, blue) is 2 # contributions from edges (0,2) and (1,2) (blue, red) is 2 # same as (red, blue) since graph is undirected (blue, blue) is 0 # there are no edges with blue endpoints 7.4. Attribute Matrices 339 NetworkX Reference, Release 1.7 340 Chapter 7. Linear algebra CHAPTER EIGHT CONVERTING TO AND FROM OTHER DATA FORMATS 8.1 To NetworkX Graph This module provides functions to convert NetworkX graphs to and from other formats. The preferred way of converting data to a NetworkX graph is through the graph constuctor. The constructor calls the to_networkx_graph() function which attempts to guess the input type and convert it automatically. 8.1.1 Examples Create a 10 node random graph from a numpy matrix >>> import numpy >>> a=numpy.reshape(numpy.random.random_integers(0,1,size=100),(10,10)) >>> D=nx.DiGraph(a) or equivalently >>> D=nx.to_networkx_graph(a,create_using=nx.DiGraph()) Create a graph with a single edge from a dictionary of dictionaries >>> d={0:{1: 1}} # dict-of-dicts single edge (0,1) >>> G=nx.Graph(d) 8.1.2 See Also nx_pygraphviz, nx_pydot to_networkx_graph(data[, create_using, ...]) Make a NetworkX graph from a known data structure. 8.1.3 to_networkx_graph to_networkx_graph(data, create_using=None, multigraph_input=False) Make a NetworkX graph from a known data structure. The preferred way to call this is automatically from the class constructor 341 NetworkX Reference, Release 1.7 >>> d={0:{1:{’weight’:1}}} # dict-of-dicts single edge (0,1) >>> G=nx.Graph(d) instead of the equivalent >>> G=nx.from_dict_of_dicts(d) Parameters data : a object to be converted Current known types are: any NetworkX graph dict-of-dicts dist-of-lists list of edges numpy matrix numpy ndarray scipy sparse matrix pygraphviz agraph create_using : NetworkX graph Use speciﬁed graph for result. Otherwise a new graph is created. multigraph_input : bool (default False) If True and data is a dict_of_dicts, try to create a multigraph assuming dict_of_dict_of_lists. If data and create_using are both multigraphs then create a multi- graph from a multigraph. 8.2 Dictionaries to_dict_of_dicts(G[, nodelist, edge_data]) Return adjacency representation of graph as a dictionary of dictionaries. from_dict_of_dicts(d[, create_using, ...]) Return a graph from a dictionary of dictionaries. 8.2.1 to_dict_of_dicts to_dict_of_dicts(G, nodelist=None, edge_data=None) Return adjacency representation of graph as a dictionary of dictionaries. Parameters G : graph A NetworkX graph nodelist : list Use only nodes speciﬁed in nodelist edge_data : list, optional If provided, the value of the dictionary will be set to edge_data for all edges. This is useful to make an adjacency matrix type representation with 1 as the edge data. If edgedata is None, the edgedata in G is used to ﬁll the values. If G is a multigraph, the edgedata is a dict for each pair (u,v). 8.2.2 from_dict_of_dicts from_dict_of_dicts(d, create_using=None, multigraph_input=False) Return a graph from a dictionary of dictionaries. Parameters d : dictionary of dictionaries A dictionary of dictionaries adjacency representation. create_using : NetworkX graph 342 Chapter 8. Converting to and from other data formats NetworkX Reference, Release 1.7 Use speciﬁed graph for result. Otherwise a new graph is created. multigraph_input : bool (default False) When True, the values of the inner dict are assumed to be containers of edge data for multiple edges. Otherwise this routine assumes the edge data are singletons. Examples >>> dod= {0:{1:{’weight’:1}}} # single edge (0,1) >>> G=nx.from_dict_of_dicts(dod) or >>> G=nx.Graph(dod) # use Graph constructor 8.3 Lists to_dict_of_lists(G[, nodelist]) Return adjacency representation of graph as a dictionary of lists. from_dict_of_lists(d[, create_using]) Return a graph from a dictionary of lists. to_edgelist(G[, nodelist]) Return a list of edges in the graph. from_edgelist(edgelist[, create_using]) Return a graph from a list of edges. 8.3.1 to_dict_of_lists to_dict_of_lists(G, nodelist=None) Return adjacency representation of graph as a dictionary of lists. Parameters G : graph A NetworkX graph nodelist : list Use only nodes speciﬁed in nodelist Notes Completely ignores edge data for MultiGraph and MultiDiGraph. 8.3.2 from_dict_of_lists from_dict_of_lists(d, create_using=None) Return a graph from a dictionary of lists. Parameters d : dictionary of lists A dictionary of lists adjacency representation. create_using : NetworkX graph Use speciﬁed graph for result. Otherwise a new graph is created. 8.3. Lists 343 NetworkX Reference, Release 1.7 Examples >>> dol= {0:[1]} # single edge (0,1) >>> G=nx.from_dict_of_lists(dol) or >>> G=nx.Graph(dol) # use Graph constructor 8.3.3 to_edgelist to_edgelist(G, nodelist=None) Return a list of edges in the graph. Parameters G : graph A NetworkX graph nodelist : list Use only nodes speciﬁed in nodelist 8.3.4 from_edgelist from_edgelist(edgelist, create_using=None) Return a graph from a list of edges. Parameters edgelist : list or iterator Edge tuples create_using : NetworkX graph Use speciﬁed graph for result. Otherwise a new graph is created. Examples >>> edgelist= [(0,1)] # single edge (0,1) >>> G=nx.from_edgelist(edgelist) or >>> G=nx.Graph(edgelist) # use Graph constructor 8.4 Numpy to_numpy_matrix(G[, nodelist, dtype, order, ...]) Return the graph adjacency matrix as a NumPy matrix. to_numpy_recarray(G[, nodelist, dtype, order]) Return the graph adjacency matrix as a NumPy recarray. from_numpy_matrix(A[, create_using]) Return a graph from numpy matrix. 8.4.1 to_numpy_matrix to_numpy_matrix(G, nodelist=None, dtype=None, order=None, multigraph_weight=, weight=’weight’) Return the graph adjacency matrix as a NumPy matrix. Parameters G : graph 344 Chapter 8. Converting to and from other data formats NetworkX Reference, Release 1.7 The NetworkX graph used to construct the NumPy matrix. nodelist : list, optional The rows and columns are ordered according to the nodes in nodelist. If nodelist is None, then the ordering is produced by G.nodes(). dtype : NumPy data type, optional A valid single NumPy data type used to initialize the array. This must be a simple type such as int or numpy.ﬂoat64 and not a compound data type (see to_numpy_recarray) If None, then the NumPy default is used. order : {‘C’, ‘F’}, optional Whether to store multidimensional data in C- or Fortran-contiguous (row- or column- wise) order in memory. If None, then the NumPy default is used. multigraph_weight : {sum, min, max}, optional An operator that determines how weights in multigraphs are handled. The default is to sum the weights of the multiple edges. weight : string or None optional (default=’weight’) The edge attribute that holds the numerical value used for the edge weight. If None then all edge weights are 1. Returns M : NumPy matrix Graph adjacency matrix. See Also: to_numpy_recarray, from_numpy_matrix Notes The matrix entries are assigned with weight edge attribute. When an edge does not have the weight attribute, the value of the entry is 1. For multiple edges, the values of the entries are the sums of the edge attributes for each edge. When nodelist does not contain every node in G, the matrix is built from the subgraph of G that is induced by the nodes in nodelist. Examples >>> G = nx.MultiDiGraph() >>> G.add_edge(0,1,weight=2) >>> G.add_edge(1,0) >>> G.add_edge(2,2,weight=3) >>> G.add_edge(2,2) >>> nx.to_numpy_matrix(G, nodelist=[0,1,2]) matrix([[ 0., 2., 0.], [ 1., 0., 0.], [ 0., 0., 4.]]) 8.4. Numpy 345 NetworkX Reference, Release 1.7 8.4.2 to_numpy_recarray to_numpy_recarray(G, nodelist=None, dtype=[(‘weight’, )], order=None) Return the graph adjacency matrix as a NumPy recarray. Parameters G : graph The NetworkX graph used to construct the NumPy matrix. nodelist : list, optional The rows and columns are ordered according to the nodes in nodelist. If nodelist is None, then the ordering is produced by G.nodes(). dtype : NumPy data-type, optional A valid NumPy named dtype used to initialize the NumPy recarray. The data type names are assumed to be keys in the graph edge attribute dictionary. order : {‘C’, ‘F’}, optional Whether to store multidimensional data in C- or Fortran-contiguous (row- or column- wise) order in memory. If None, then the NumPy default is used. Returns M : NumPy recarray The graph with speciﬁed edge data as a Numpy recarray Notes When nodelist does not contain every node in G, the matrix is built from the subgraph of G that is induced by the nodes in nodelist. Examples >>> G = nx.Graph() >>> G.add_edge(1,2,weight=7.0,cost=5) >>> A=nx.to_numpy_recarray(G,dtype=[(’weight’,float),(’cost’,int)]) >>> print(A.weight) [[ 0. 7.] [ 7. 0.]] >>> print(A.cost) [[0 5] [5 0]] 8.4.3 from_numpy_matrix from_numpy_matrix(A, create_using=None) Return a graph from numpy matrix. The numpy matrix is interpreted as an adjacency matrix for the graph. Parameters A : numpy matrix An adjacency matrix representation of a graph create_using : NetworkX graph Use speciﬁed graph for result. The default is Graph() 346 Chapter 8. Converting to and from other data formats NetworkX Reference, Release 1.7 See Also: to_numpy_matrix, to_numpy_recarray Notes If the numpy matrix has a single data type for each matrix entry it will be converted to an appropriate Python data type. If the numpy matrix has a user-speciﬁed compound data type the names of the data ﬁelds will be used as attribute keys in the resulting NetworkX graph. Examples Simple integer weights on edges: >>> import numpy >>> A=numpy.matrix([[1,1],[2,1]]) >>> G=nx.from_numpy_matrix(A) User deﬁned compound data type on edges: >>> import numpy >>> dt=[(’weight’,float),(’cost’,int)] >>> A=numpy.matrix([[(1.0,2)]],dtype=dt) >>> G=nx.from_numpy_matrix(A) >>> G.edges(data=True) [(0, 0, {’cost’: 2, ’weight’: 1.0})] 8.5 Scipy to_scipy_sparse_matrix(G[, nodelist, dtype, ...]) Return the graph adjacency matrix as a SciPy sparse matrix. from_scipy_sparse_matrix(A[, create_using]) Return a graph from scipy sparse matrix adjacency list. 8.5.1 to_scipy_sparse_matrix to_scipy_sparse_matrix(G, nodelist=None, dtype=None, weight=’weight’, format=’csr’) Return the graph adjacency matrix as a SciPy sparse matrix. Parameters G : graph The NetworkX graph used to construct the NumPy matrix. nodelist : list, optional The rows and columns are ordered according to the nodes in nodelist. If nodelist is None, then the ordering is produced by G.nodes(). dtype : NumPy data-type, optional A valid NumPy dtype used to initialize the array. If None, then the NumPy default is used. weight : string or None optional (default=’weight’) 8.5. Scipy 347 NetworkX Reference, Release 1.7 The edge attribute that holds the numerical value used for the edge weight. If None then all edge weights are 1. format : str in {‘bsr’, ‘csr’, ‘csc’, ‘coo’, ‘lil’, ‘dia’, ‘dok’} The type of the matrix to be returned (default ‘csr’). For some algorithms different implementations of sparse matrices can perform better. See [R204] for details. Returns M : SciPy sparse matrix Graph adjacency matrix. Notes The matrix entries are populated using the edge attribute held in parameter weight. When an edge does not have that attribute, the value of the entry is 1. For multiple edges the matrix values are the sums of the edge weights. When nodelist does not contain every node in G, the matrix is built from the subgraph of G that is induced by the nodes in nodelist. Uses coo_matrix format. To convert to other formats specify the format= keyword. References [R204] Examples >>> G = nx.MultiDiGraph() >>> G.add_edge(0,1,weight=2) >>> G.add_edge(1,0) >>> G.add_edge(2,2,weight=3) >>> G.add_edge(2,2) >>> S = nx.to_scipy_sparse_matrix(G, nodelist=[0,1,2]) >>> print(S.todense()) [[0 2 0] [1 0 0] [0 0 4]] 8.5.2 from_scipy_sparse_matrix from_scipy_sparse_matrix(A, create_using=None) Return a graph from scipy sparse matrix adjacency list. Parameters A : scipy sparse matrix An adjacency matrix representation of a graph create_using : NetworkX graph Use speciﬁed graph for result. The default is Graph() 348 Chapter 8. Converting to and from other data formats NetworkX Reference, Release 1.7 Examples >>> import scipy.sparse >>> A=scipy.sparse.eye(2,2,1) >>> G=nx.from_scipy_sparse_matrix(A) 8.5. Scipy 349 NetworkX Reference, Release 1.7 350 Chapter 8. Converting to and from other data formats CHAPTER NINE READING AND WRITING GRAPHS 9.1 Adjacency List Read and write NetworkX graphs as adjacency lists. Adjacency list format is useful for graphs without data associated with nodes or edges and for nodes that can be meaningfully represented as strings. 9.1.1 Format The adjacency list format consists of lines with node labels. The ﬁrst label in a line is the source node. Further labels in the line are considered target nodes and are added to the graph along with an edge between the source node and target node. The graph with edges a-b, a-c, d-e can be represented as the following adjacency list (anything following the # in a line is a comment): a b c # source target target de read_adjlist(path[, comments, delimiter, ...]) Read graph in adjacency list format from path. write_adjlist(G, path[, comments, ...]) Write graph G in single-line adjacency-list format to path. parse_adjlist(lines[, comments, delimiter, ...]) Parse lines of a graph adjacency list representation. generate_adjlist(G[, delimiter]) Generate a single line of the graph G in adjacency list format. 9.1.2 read_adjlist read_adjlist(path, comments=’#’, delimiter=None, create_using=None, nodetype=None, encoding=’utf- 8’) Read graph in adjacency list format from path. Parameters path : string or ﬁle Filename or ﬁle handle to read. Filenames ending in .gz or .bz2 will be uncompressed. create_using: NetworkX graph container : Use given NetworkX graph for holding nodes or edges. nodetype : Python type, optional Convert nodes to this type. 351 NetworkX Reference, Release 1.7 comments : string, optional Marker for comment lines delimiter : string, optional Separator for node labels. The default is whitespace. create_using: NetworkX graph container : Use given NetworkX graph for holding nodes or edges. Returns G: NetworkX graph : The graph corresponding to the lines in adjacency list format. See Also: write_adjlist Notes This format does not store graph or node data. Examples >>> G=nx.path_graph(4) >>> nx.write_adjlist(G, "test.adjlist") >>> G=nx.read_adjlist("test.adjlist") The path can be a ﬁlehandle or a string with the name of the ﬁle. If a ﬁlehandle is provided, it has to be opened in ‘rb’ mode. >>> fh=open("test.adjlist", ’rb’) >>> G=nx.read_adjlist(fh) Filenames ending in .gz or .bz2 will be compressed. >>> nx.write_adjlist(G,"test.adjlist.gz") >>> G=nx.read_adjlist("test.adjlist.gz") The optional nodetype is a function to convert node strings to nodetype. For example >>> G=nx.read_adjlist("test.adjlist", nodetype=int) will attempt to convert all nodes to integer type. Since nodes must be hashable, the function nodetype must return hashable types (e.g. int, ﬂoat, str, frozenset - or tuples of those, etc.) The optional create_using parameter is a NetworkX graph container. The default is Graph(), an undirected graph. To read the data as a directed graph use >>> G=nx.read_adjlist("test.adjlist", create_using=nx.DiGraph()) 352 Chapter 9. Reading and writing graphs NetworkX Reference, Release 1.7 9.1.3 write_adjlist write_adjlist(G, path, comments=’#’, delimiter=’ ‘, encoding=’utf-8’) Write graph G in single-line adjacency-list format to path. Parameters G : NetworkX graph path : string or ﬁle Filename or ﬁle handle for data output. Filenames ending in .gz or .bz2 will be com- pressed. comments : string, optional Marker for comment lines delimiter : string, optional Separator for node labels encoding : string, optional Text encoding. See Also: read_adjlist, generate_adjlist Notes This format does not store graph, node, or edge data. Examples >>> G=nx.path_graph(4) >>> nx.write_adjlist(G,"test.adjlist") The path can be a ﬁlehandle or a string with the name of the ﬁle. If a ﬁlehandle is provided, it has to be opened in ‘wb’ mode. >>> fh=open("test.adjlist",’wb’) >>> nx.write_adjlist(G, fh) 9.1.4 parse_adjlist parse_adjlist(lines, comments=’#’, delimiter=None, create_using=None, nodetype=None) Parse lines of a graph adjacency list representation. Parameters lines : list or iterator of strings Input data in adjlist format create_using: NetworkX graph container : Use given NetworkX graph for holding nodes or edges. nodetype : Python type, optional Convert nodes to this type. comments : string, optional 9.1. Adjacency List 353 NetworkX Reference, Release 1.7 Marker for comment lines delimiter : string, optional Separator for node labels. The default is whitespace. create_using: NetworkX graph container : Use given NetworkX graph for holding nodes or edges. Returns G: NetworkX graph : The graph corresponding to the lines in adjacency list format. See Also: read_adjlist Examples >>> lines = [’125’, ... ’234’, ... ’35’, ... ’4’, ... ’5’] >>> G = nx.parse_adjlist(lines, nodetype = int) >>> G.nodes() [1, 2, 3, 4, 5] >>> G.edges() [(1, 2), (1, 5), (2, 3), (2, 4), (3, 5)] 9.1.5 generate_adjlist generate_adjlist(G, delimiter=’ ‘) Generate a single line of the graph G in adjacency list format. Parameters G : NetworkX graph delimiter : string, optional Separator for node labels Returns lines : string Lines of data in adjlist format. See Also: write_adjlist, read_adjlist Examples >>> G = nx.lollipop_graph(4, 3) >>> for line in nx.generate_adjlist(G): ... print(line) 0123 123 23 34 354 Chapter 9. Reading and writing graphs NetworkX Reference, Release 1.7 45 56 6 9.2 Multiline Adjacency List Read and write NetworkX graphs as multi-line adjacency lists. The multi-line adjacency list format is useful for graphs with nodes that can be meaningfully represented as strings. With this format simple edge data can be stored but node or graph data is not. 9.2.1 Format The ﬁrst label in a line is the source node label followed by the node degree d. The next d lines are target node labels and optional edge data. That pattern repeats for all nodes in the graph. The graph with edges a-b, a-c, d-e can be represented as the following adjacency list (anything following the # in a line is a comment): # example.multiline-adjlist a2 b c d1 e read_multiline_adjlist(path[, comments, ...]) Read graph in multi-line adjacency list format from path. write_multiline_adjlist(G, path[, ...]) Write the graph G in multiline adjacency list format to path parse_multiline_adjlist(lines[, comments, ...]) Parse lines of a multiline adjacency list representation of a graph. generate_multiline_adjlist(G[, delimiter]) Generate a single line of the graph G in multiline adjacency list format. 9.2.2 read_multiline_adjlist read_multiline_adjlist(path, comments=’#’, delimiter=None, create_using=None, nodetype=None, edgetype=None, encoding=’utf-8’) Read graph in multi-line adjacency list format from path. Parameters path : string or ﬁle Filename or ﬁle handle to read. Filenames ending in .gz or .bz2 will be uncompressed. create_using: NetworkX graph container : Use given NetworkX graph for holding nodes or edges. nodetype : Python type, optional Convert nodes to this type. edgetype : Python type, optional Convert edge data to this type. comments : string, optional Marker for comment lines 9.2. Multiline Adjacency List 355 NetworkX Reference, Release 1.7 delimiter : string, optional Separator for node labels. The default is whitespace. create_using: NetworkX graph container : Use given NetworkX graph for holding nodes or edges. Returns G: NetworkX graph : See Also: write_multiline_adjlist Notes This format does not store graph, node, or edge data. Examples >>> G=nx.path_graph(4) >>> nx.write_multiline_adjlist(G,"test.adjlist") >>> G=nx.read_multiline_adjlist("test.adjlist") The path can be a ﬁle or a string with the name of the ﬁle. If a ﬁle s provided, it has to be opened in ‘rb’ mode. >>> fh=open("test.adjlist", ’rb’) >>> G=nx.read_multiline_adjlist(fh) Filenames ending in .gz or .bz2 will be compressed. >>> nx.write_multiline_adjlist(G,"test.adjlist.gz") >>> G=nx.read_multiline_adjlist("test.adjlist.gz") The optional nodetype is a function to convert node strings to nodetype. For example >>> G=nx.read_multiline_adjlist("test.adjlist", nodetype=int) will attempt to convert all nodes to integer type. The optional edgetype is a function to convert edge data strings to edgetype. >>> G=nx.read_multiline_adjlist("test.adjlist") The optional create_using parameter is a NetworkX graph container. The default is Graph(), an undirected graph. To read the data as a directed graph use >>> G=nx.read_multiline_adjlist("test.adjlist", create_using=nx.DiGraph()) 9.2.3 write_multiline_adjlist write_multiline_adjlist(G, path, delimiter=’ ‘, comments=’#’, encoding=’utf-8’) Write the graph G in multiline adjacency list format to path Parameters G : NetworkX graph comments : string, optional 356 Chapter 9. Reading and writing graphs NetworkX Reference, Release 1.7 Marker for comment lines delimiter : string, optional Separator for node labels encoding : string, optional Text encoding. See Also: read_multiline_adjlist Examples >>> G=nx.path_graph(4) >>> nx.write_multiline_adjlist(G,"test.adjlist") The path can be a ﬁle handle or a string with the name of the ﬁle. If a ﬁle handle is provided, it has to be opened in ‘wb’ mode. >>> fh=open("test.adjlist",’wb’) >>> nx.write_multiline_adjlist(G,fh) Filenames ending in .gz or .bz2 will be compressed. >>> nx.write_multiline_adjlist(G,"test.adjlist.gz") 9.2.4 parse_multiline_adjlist parse_multiline_adjlist(lines, comments=’#’, delimiter=None, create_using=None, node- type=None, edgetype=None) Parse lines of a multiline adjacency list representation of a graph. Parameters lines : list or iterator of strings Input data in multiline adjlist format create_using: NetworkX graph container : Use given NetworkX graph for holding nodes or edges. nodetype : Python type, optional Convert nodes to this type. comments : string, optional Marker for comment lines delimiter : string, optional Separator for node labels. The default is whitespace. create_using: NetworkX graph container : Use given NetworkX graph for holding nodes or edges. Returns G: NetworkX graph : The graph corresponding to the lines in multiline adjacency list format. 9.2. Multiline Adjacency List 357 NetworkX Reference, Release 1.7 Examples >>> lines = [’12’, ... "2{’weight’:3, ’name’: ’Frodo’}", ... "3 {}", ... "21", ... "5{’weight’:6, ’name’: ’Saruman’}"] >>> G = nx.parse_multiline_adjlist(iter(lines), nodetype = int) >>> G.nodes() [1, 2, 3, 5] >>> G.edges(data = True) [(1, 2, {’name’: ’Frodo’, ’weight’: 3}), (1, 3, {}), (2, 5, {’name’: ’Saruman’, ’weight’: 6})] 9.2.5 generate_multiline_adjlist generate_multiline_adjlist(G, delimiter=’ ‘) Generate a single line of the graph G in multiline adjacency list format. Parameters G : NetworkX graph delimiter : string, optional Separator for node labels Returns lines : string Lines of data in multiline adjlist format. See Also: write_multiline_adjlist, read_multiline_adjlist Examples >>> G = nx.lollipop_graph(4, 3) >>> for line in nx.generate_multiline_adjlist(G): ... print(line) 03 1 {} 2 {} 3 {} 12 2 {} 3 {} 21 3 {} 31 4 {} 41 5 {} 51 6 {} 60 358 Chapter 9. Reading and writing graphs NetworkX Reference, Release 1.7 9.3 Edge List Read and write NetworkX graphs as edge lists. The multi-line adjacency list format is useful for graphs with nodes that can be meaningfully represented as strings. With the edgelist format simple edge data can be stored but node or graph data is not. There is no way of representing isolated nodes unless the node has a self-loop edge. 9.3.1 Format You can read or write three formats of edge lists with these functions. Node pairs with no data: 12 Python dictionary as data: 1 2 {’weight’:7, ’color’:’green’} Arbitrary data: 1 2 7 green read_edgelist(path[, comments, delimiter, ...]) Read a graph from a list of edges. write_edgelist(G, path[, comments, ...]) Write graph as a list of edges. read_weighted_edgelist(path[, comments, ...]) Read a graph as list of edges with numeric weights. write_weighted_edgelist(G, path[, comments, ...]) Write graph G as a list of edges with numeric weights. generate_edgelist(G[, delimiter, data]) Generate a single line of the graph G in edge list format. parse_edgelist(lines[, comments, delimiter, ...]) Parse lines of an edge list representation of a graph. 9.3.2 read_edgelist read_edgelist(path, comments=’#’, delimiter=None, create_using=None, nodetype=None, data=True, edgetype=None, encoding=’utf-8’) Read a graph from a list of edges. Parameters path : ﬁle or string File or ﬁlename to write. If a ﬁle is provided, it must be opened in ‘rb’ mode. Filenames ending in .gz or .bz2 will be uncompressed. comments : string, optional The character used to indicate the start of a comment. delimiter : string, optional The string used to separate values. The default is whitespace. create_using : Graph container, optional, Use speciﬁed container to build graph. The default is networkx.Graph, an undirected graph. nodetype : int, ﬂoat, str, Python type, optional Convert node data from strings to speciﬁed type 9.3. Edge List 359 NetworkX Reference, Release 1.7 data : bool or list of (label,type) tuples Tuples specifying dictionary key names and types for edge data edgetype : int, ﬂoat, str, Python type, optional OBSOLETE Convert edge data from strings to speciﬁed type and use as ‘weight’ encoding: string, optional : Specify which encoding to use when reading ﬁle. Returns G : graph A networkx Graph or other type speciﬁed with create_using See Also: parse_edgelist Notes Since nodes must be hashable, the function nodetype must return hashable types (e.g. int, ﬂoat, str, frozenset - or tuples of those, etc.) Examples >>> nx.write_edgelist(nx.path_graph(4), "test.edgelist") >>> G=nx.read_edgelist("test.edgelist") >>> fh=open("test.edgelist", ’rb’) >>> G=nx.read_edgelist(fh) >>> fh.close() >>> G=nx.read_edgelist("test.edgelist", nodetype=int) >>> G=nx.read_edgelist("test.edgelist",create_using=nx.DiGraph()) Edgelist with data in a list: >>> textline = ’123’ >>> fh = open(’test.edgelist’,’w’) >>> d = fh.write(textline) >>> fh.close() >>> G = nx.read_edgelist(’test.edgelist’, nodetype=int, data=((’weight’,float),)) >>> G.nodes() [1, 2] >>> G.edges(data = True) [(1, 2, {’weight’: 3.0})] See parse_edgelist() for more examples of formatting. 9.3.3 write_edgelist write_edgelist(G, path, comments=’#’, delimiter=’ ‘, data=True, encoding=’utf-8’) Write graph as a list of edges. Parameters G : graph A NetworkX graph 360 Chapter 9. Reading and writing graphs NetworkX Reference, Release 1.7 path : ﬁle or string File or ﬁlename to write. If a ﬁle is provided, it must be opened in ‘wb’ mode. Filenames ending in .gz or .bz2 will be compressed. comments : string, optional The character used to indicate the start of a comment delimiter : string, optional The string used to separate values. The default is whitespace. data : bool or list, optional If False write no edge data. If True write a string representation of the edge data dictio- nary.. If a list (or other iterable) is provided, write the keys speciﬁed in the list. encoding: string, optional : Specify which encoding to use when writing ﬁle. See Also: write_edgelist, write_weighted_edgelist Examples >>> G=nx.path_graph(4) >>> nx.write_edgelist(G, "test.edgelist") >>> G=nx.path_graph(4) >>> fh=open("test.edgelist",’wb’) >>> nx.write_edgelist(G, fh) >>> nx.write_edgelist(G, "test.edgelist.gz") >>> nx.write_edgelist(G, "test.edgelist.gz", data=False) >>> G=nx.Graph() >>> G.add_edge(1,2,weight=7,color=’red’) >>> nx.write_edgelist(G,’test.edgelist’,data=False) >>> nx.write_edgelist(G,’test.edgelist’,data=[’color’]) >>> nx.write_edgelist(G,’test.edgelist’,data=[’color’,’weight’]) 9.3.4 read_weighted_edgelist read_weighted_edgelist(path, comments=’#’, delimiter=None, create_using=None, nodetype=None, encoding=’utf-8’) Read a graph as list of edges with numeric weights. Parameters path : ﬁle or string File or ﬁlename to write. If a ﬁle is provided, it must be opened in ‘rb’ mode. Filenames ending in .gz or .bz2 will be uncompressed. comments : string, optional The character used to indicate the start of a comment. delimiter : string, optional The string used to separate values. The default is whitespace. create_using : Graph container, optional, 9.3. Edge List 361 NetworkX Reference, Release 1.7 Use speciﬁed container to build graph. The default is networkx.Graph, an undirected graph. nodetype : int, ﬂoat, str, Python type, optional Convert node data from strings to speciﬁed type encoding: string, optional : Specify which encoding to use when reading ﬁle. Returns G : graph A networkx Graph or other type speciﬁed with create_using Notes Since nodes must be hashable, the function nodetype must return hashable types (e.g. int, ﬂoat, str, frozenset - or tuples of those, etc.) Example edgelist ﬁle format. With numeric edge data: # read with # >>> G=nx.read_weighted_edgelist(fh) # source target data ab1 a c 3.14159 d e 42 9.3.5 write_weighted_edgelist write_weighted_edgelist(G, path, comments=’#’, delimiter=’ ‘, encoding=’utf-8’) Write graph G as a list of edges with numeric weights. Parameters G : graph A NetworkX graph path : ﬁle or string File or ﬁlename to write. If a ﬁle is provided, it must be opened in ‘wb’ mode. Filenames ending in .gz or .bz2 will be compressed. comments : string, optional The character used to indicate the start of a comment delimiter : string, optional The string used to separate values. The default is whitespace. encoding: string, optional : Specify which encoding to use when writing ﬁle. See Also: read_edgelist, write_edgelist, write_weighted_edgelist 362 Chapter 9. Reading and writing graphs NetworkX Reference, Release 1.7 Examples >>> G=nx.Graph() >>> G.add_edge(1,2,weight=7) >>> nx.write_weighted_edgelist(G, ’test.weighted.edgelist’) 9.3.6 generate_edgelist generate_edgelist(G, delimiter=’ ‘, data=True) Generate a single line of the graph G in edge list format. Parameters G : NetworkX graph delimiter : string, optional Separator for node labels data : bool or list of keys If False generate no edge data. If True use a dictionary representation of edge data. If a list of keys use a list of data values corresponding to the keys. Returns lines : string Lines of data in adjlist format. See Also: write_adjlist, read_adjlist Examples >>> G = nx.lollipop_graph(4, 3) >>> G[1][2][’weight’] = 3 >>> G[3][4][’capacity’] = 12 >>> for line in nx.generate_edgelist(G, data=False): ... print(line) 01 02 03 12 13 23 34 45 56 >>> for line in nx.generate_edgelist(G): ... print(line) 0 1 {} 0 2 {} 0 3 {} 1 2 {’weight’: 3} 1 3 {} 2 3 {} 3 4 {’capacity’: 12} 4 5 {} 5 6 {} 9.3. Edge List 363 NetworkX Reference, Release 1.7 >>> for line in nx.generate_edgelist(G,data=[’weight’]): ... print(line) 01 02 03 123 13 23 34 45 56 9.3.7 parse_edgelist parse_edgelist(lines, comments=’#’, delimiter=None, create_using=None, nodetype=None, data=True) Parse lines of an edge list representation of a graph. Returns G: NetworkX Graph : The graph corresponding to lines data : bool or list of (label,type) tuples If False generate no edge data or if True use a dictionary representation of edge data or a list tuples specifying dictionary key names and types for edge data. create_using: NetworkX graph container, optional : Use given NetworkX graph for holding nodes or edges. nodetype : Python type, optional Convert nodes to this type. comments : string, optional Marker for comment lines delimiter : string, optional Separator for node labels create_using: NetworkX graph container : Use given NetworkX graph for holding nodes or edges. See Also: read_weighted_edgelist Examples Edgelist with no data: >>> lines = ["12", ... "23", ... "34"] >>> G = nx.parse_edgelist(lines, nodetype = int) >>> G.nodes() [1, 2, 3, 4] 364 Chapter 9. Reading and writing graphs NetworkX Reference, Release 1.7 >>> G.edges() [(1, 2), (2, 3), (3, 4)] Edgelist with data in Python dictionary representation: >>> lines = ["12{’weight’:3}", ... "23{’weight’:27}", ... "34{’weight’:3.0}"] >>> G = nx.parse_edgelist(lines, nodetype = int) >>> G.nodes() [1, 2, 3, 4] >>> G.edges(data = True) [(1, 2, {’weight’: 3}), (2, 3, {’weight’: 27}), (3, 4, {’weight’: 3.0})] Edgelist with data in a list: >>> lines = ["123", ... "2 3 27", ... "3 4 3.0"] >>> G = nx.parse_edgelist(lines, nodetype = int, data=((’weight’,float),)) >>> G.nodes() [1, 2, 3, 4] >>> G.edges(data = True) [(1, 2, {’weight’: 3.0}), (2, 3, {’weight’: 27.0}), (3, 4, {’weight’: 3.0})] 9.4 GEXF Read and write graphs in GEXF format. GEXF (Graph Exchange XML Format) is a language for describing complex network structures, their associated data and dynamics. This implementation does not support mixed graphs (directed and unidirected edges together). 9.4.1 Format GEXF is an XML format. See http://gexf.net/format/schema.html for the speciﬁcation and http://gexf.net/format/basic.html for examples. read_gexf(path[, node_type, relabel, version]) Read graph in GEXF format from path. write_gexf(G, path[, encoding, prettyprint, ...]) Write G in GEXF format to path. relabel_gexf_graph(G) Relabel graph using “label” node keyword for node label. 9.4.2 read_gexf read_gexf(path, node_type=, relabel=False, version=‘1.1draft’) Read graph in GEXF format from path. “GEXF (Graph Exchange XML Format) is a language for describing complex networks structures, their associ- ated data and dynamics” [R248]. Parameters path : ﬁle or string File or ﬁlename to write. Filenames ending in .gz or .bz2 will be compressed. 9.4. GEXF 365 NetworkX Reference, Release 1.7 node_type: Python type (default: str) : Convert node ids to this type relabel : bool (default: False) If True relabel the nodes to use the GEXF node “label” attribute instead of the node “id” attribute as the NetworkX node label. Returns graph: NetworkX graph : If no parallel edges are found a Graph or DiGraph is returned. Otherwise a MultiGraph or MultiDiGraph is returned. Notes This implementation does not support mixed graphs (directed and unidirected edges together). References [R248] 9.4.3 write_gexf write_gexf(G, path, encoding=’utf-8’, prettyprint=True, version=‘1.1draft’) Write G in GEXF format to path. “GEXF (Graph Exchange XML Format) is a language for describing complex networks structures, their associ- ated data and dynamics” [R249]. Parameters G : graph A NetworkX graph path : ﬁle or string File or ﬁlename to write. Filenames ending in .gz or .bz2 will be compressed. encoding : string (optional) Encoding for text data. prettyprint : bool (optional) If True use line breaks and indenting in output XML. Notes This implementation does not support mixed graphs (directed and unidirected edges together). The node id attribute is set to be the string of the node label. If you want to specify an id use set it as node data, e.g. node[’a’][’id’]=1 to set the id of node ‘a’ to 1. References [R249] 366 Chapter 9. Reading and writing graphs NetworkX Reference, Release 1.7 Examples >>> G=nx.path_graph(4) >>> nx.write_gexf(G, "test.gexf") 9.4.4 relabel_gexf_graph relabel_gexf_graph(G) Relabel graph using “label” node keyword for node label. Parameters G : graph A NetworkX graph read from GEXF data Returns H : graph A NetworkX graph with relabed nodes Notes This function relabels the nodes in a NetworkX graph with the “label” attribute. It also handles relabeling the speciﬁc GEXF node attributes “parents”, and “pid”. 9.5 GML Read graphs in GML format. “GML, the G>raph Modelling Language, is our proposal for a portable ﬁle format for graphs. GML’s key features are portability, simple syntax, extensibility and ﬂexibility. A GML ﬁle consists of a hierarchical key-value lists. Graphs can be annotated with arbitrary data structures. The idea for a common ﬁle format was born at the GD‘95; this proposal is the outcome of many discussions. GML is the standard ﬁle format in the Graphlet graph editor system. It has been overtaken and adapted by several other systems for drawing graphs.” See http://www.infosun.ﬁm.uni-passau.de/Graphlet/GML/gml-tr.html Requires pyparsing: http://pyparsing.wikispaces.com/ 9.5.1 Format See http://www.infosun.ﬁm.uni-passau.de/Graphlet/GML/gml-tr.html for format speciﬁcation. Example graphs in GML format: http://www-personal.umich.edu/~mejn/netdata/ read_gml(path[, encoding, relabel]) Read graph in GML format from path. write_gml(G, path) Write the graph G in GML format to the ﬁle or ﬁle handle path. parse_gml(lines[, relabel]) Parse GML graph from a string or iterable. generate_gml(G) Generate a single entry of the graph G in GML format. 9.5.2 read_gml read_gml(path, encoding=’UTF-8’, relabel=False) Read graph in GML format from path. 9.5. GML 367 NetworkX Reference, Release 1.7 Parameters path : ﬁlename or ﬁlehandle The ﬁlename or ﬁlehandle to read from. encoding : string, optional Text encoding. relabel : bool, optional If True use the GML node label attribute for node names otherwise use the node id. Returns G : MultiGraph or MultiDiGraph Raises ImportError : If the pyparsing module is not available. See Also: write_gml, parse_gml Notes Requires pyparsing: http://pyparsing.wikispaces.com/ References GML speciﬁcation: http://www.infosun.ﬁm.uni-passau.de/Graphlet/GML/gml-tr.html Examples >>> G=nx.path_graph(4) >>> nx.write_gml(G,’test.gml’) >>> H=nx.read_gml(’test.gml’) 9.5.3 write_gml write_gml(G, path) Write the graph G in GML format to the ﬁle or ﬁle handle path. Parameters path : ﬁlename or ﬁlehandle The ﬁlename or ﬁlehandle to write. Filenames ending in .gz or .gz2 will be compressed. See Also: read_gml, parse_gml Notes GML speciﬁcations indicate that the ﬁle should only use 7bit ASCII text encoding.iso8859-1 (latin-1). This implementation does not support all Python data types as GML data. Nodes, node attributes, edge attributes, and graph attributes must be either dictionaries or single stings or numbers. If they are not an attempt is made to represent them as strings. For example, a list as edge data G[1][2][’somedata’]=[1,2,3], will be represented in the GML ﬁle as: 368 Chapter 9. Reading and writing graphs NetworkX Reference, Release 1.7 edge [ source 1 target 2 somedata "[1, 2, 3]" ] Examples >>> G=nx.path_graph(4) >>> nx.write_gml(G,"test.gml") Filenames ending in .gz or .bz2 will be compressed. >>> nx.write_gml(G,"test.gml.gz") 9.5.4 parse_gml parse_gml(lines, relabel=True) Parse GML graph from a string or iterable. Parameters lines : string or iterable Data in GML format. relabel : bool, optional If True use the GML node label attribute for node names otherwise use the node id. Returns G : MultiGraph or MultiDiGraph Raises ImportError : If the pyparsing module is not available. See Also: write_gml, read_gml Notes This stores nested GML attributes as dictionaries in the NetworkX graph, node, and edge attribute structures. Requires pyparsing: http://pyparsing.wikispaces.com/ References GML speciﬁcation: http://www.infosun.ﬁm.uni-passau.de/Graphlet/GML/gml-tr.html 9.5.5 generate_gml generate_gml(G) Generate a single entry of the graph G in GML format. Parameters G : NetworkX graph 9.5. GML 369 NetworkX Reference, Release 1.7 Returns lines: string : Lines in GML format. Notes This implementation does not support all Python data types as GML data. Nodes, node attributes, edge attributes, and graph attributes must be either dictionaries or single stings or numbers. If they are not an attempt is made to represent them as strings. For example, a list as edge data G[1][2][’somedata’]=[1,2,3], will be represented in the GML ﬁle as: edge [ source 1 target 2 somedata "[1, 2, 3]" ] 9.6 Pickle Read and write NetworkX graphs as Python pickles. “The pickle module implements a fundamental, but powerful algorithm for serializing and de-serializing a Python object structure. “Pickling” is the process whereby a Python object hierarchy is converted into a byte stream, and “unpickling” is the inverse operation, whereby a byte stream is converted back into an object hierarchy.” Note that NetworkX graphs can contain any hashable Python object as node (not just integers and strings). For arbitrary data types it may be difﬁcult to represent the data as text. In that case using Python pickles to store the graph data can be used. 9.6.1 Format See http://docs.python.org/library/pickle.html read_gpickle(path) Read graph object in Python pickle format. write_gpickle(G, path) Write graph in Python pickle format. 9.6.2 read_gpickle read_gpickle(path) Read graph object in Python pickle format. Pickles are a serialized byte stream of a Python object [R250]. This format will preserve Python objects used as nodes or edges. Parameters path : ﬁle or string File or ﬁlename to write. Filenames ending in .gz or .bz2 will be uncompressed. Returns G : graph A NetworkX graph 370 Chapter 9. Reading and writing graphs NetworkX Reference, Release 1.7 References [R250] Examples >>> G=nx.path_graph(4) >>> nx.write_gpickle(G,"test.gpickle") >>> G=nx.read_gpickle("test.gpickle") 9.6.3 write_gpickle write_gpickle(G, path) Write graph in Python pickle format. Pickles are a serialized byte stream of a Python object [R251]. This format will preserve Python objects used as nodes or edges. Parameters G : graph A NetworkX graph path : ﬁle or string File or ﬁlename to write. Filenames ending in .gz or .bz2 will be compressed. References [R251] Examples >>> G=nx.path_graph(4) >>> nx.write_gpickle(G,"test.gpickle") 9.7 GraphML Read and write graphs in GraphML format. This implementation does not support mixed graphs (directed and unidirected edges together), hyperedges, nested graphs, or ports. “GraphML is a comprehensive and easy-to-use ﬁle format for graphs. It consists of a language core to describe the structural properties of a graph and a ﬂexible extension mechanism to add application-speciﬁc data. Its main features include support of • directed, undirected, and mixed graphs, • hypergraphs, • hierarchical graphs, • graphical representations, 9.7. GraphML 371 NetworkX Reference, Release 1.7 • references to external data, • application-speciﬁc attribute data, and • light-weight parsers. Unlike many other ﬁle formats for graphs, GraphML does not use a custom syntax. Instead, it is based on XML and hence ideally suited as a common denominator for all kinds of services generating, archiving, or processing graphs.” http://graphml.graphdrawing.org/ 9.7.1 Format GraphML is an XML format. See http://graphml.graphdrawing.org/speciﬁcation.html for the speciﬁcation and http://graphml.graphdrawing.org/primer/graphml-primer.html for examples. read_graphml(path[, node_type]) Read graph in GraphML format from path. write_graphml(G, path[, encoding, prettyprint]) Write G in GraphML XML format to path 9.7.2 read_graphml read_graphml(path, node_type=) Read graph in GraphML format from path. Parameters path : ﬁle or string File or ﬁlename to write. Filenames ending in .gz or .bz2 will be compressed. node_type: Python type (default: str) : Convert node ids to this type Returns graph: NetworkX graph : If no parallel edges are found a Graph or DiGraph is returned. Otherwise a MultiGraph or MultiDiGraph is returned. Notes This implementation does not support mixed graphs (directed and unidirected edges together), hypergraphs, nested graphs, or ports. For multigraphs the GraphML edge “id” will be used as the edge key. If not speciﬁed then they “key” attribute will be used. If there is no “key” attribute a default NetworkX multigraph edge key will be provided. Files with the yEd “yﬁles” extension will can be read but the graphics information is discarded. yEd compressed ﬁles (“ﬁle.graphmlz” extension) can be read by renaming the ﬁle to “ﬁle.graphml.gz”. 9.7.3 write_graphml write_graphml(G, path, encoding=’utf-8’, prettyprint=True) Write G in GraphML XML format to path Parameters G : graph A networkx graph 372 Chapter 9. Reading and writing graphs NetworkX Reference, Release 1.7 path : ﬁle or string File or ﬁlename to write. Filenames ending in .gz or .bz2 will be compressed. encoding : string (optional) Encoding for text data. prettyprint : bool (optional) If True use line breaks and indenting in output XML. Notes This implementation does not support mixed graphs (directed and unidirected edges together) hyperedges, nested graphs, or ports. Examples >>> G=nx.path_graph(4) >>> nx.write_graphml(G, "test.graphml") 9.8 JSON Generate and parse JSON serializable data for NetworkX graphs. node_link_data(G) Return data in node-link format that is suitable for JSON serialization node_link_graph(data[, directed, multigraph]) Return graph from node-link data format. adjacency_data(G) Return data in adjacency format that is suitable for JSON serialization adjacency_graph(data[, directed, multigraph]) Return graph from adjacency data format. tree_data(G, root) Return data in tree format that is suitable for JSON serialization tree_graph(data) Return graph from tree data format. dumps Serialize obj to a JSON formatted str. loads Deserialize s (a str or unicode instance containing a JSON dump Serialize obj as a JSON formatted stream to fp (a load Deserialize fp (a .read()-supporting ﬁle-like object containing 9.8.1 node_link_data node_link_data(G) Return data in node-link format that is suitable for JSON serialization and use in Javascript documents. Parameters G : NetworkX graph Returns data : dict A dictionary with node-link formatted data. See Also: node_link_graph, adjacency_data, tree_data 9.8. JSON 373 NetworkX Reference, Release 1.7 Notes Graph, node, and link attributes are stored in this format but keys for attributes must be strings if you want to serialize with JSON. Examples >>> from networkx.readwrite import json_graph >>> G = nx.Graph([(1,2)]) >>> data = json_graph.node_link_data(G) To serialize with json >>> import json >>> s = json.dumps(data) 9.8.2 node_link_graph node_link_graph(data, directed=False, multigraph=True) Return graph from node-link data format. Parameters data : dict node-link formatted graph data directed : bool If True, and direction not speciﬁed in data, return a directed graph. multigraph : bool If True, and multigraph not speciﬁed in data, return a multigraph. Returns G : NetworkX graph A NetworkX graph object See Also: node_link_data, adjacency_data, tree_data Examples >>> from networkx.readwrite import json_graph >>> G = nx.Graph([(1,2)]) >>> data = json_graph.node_link_data(G) >>> H = json_graph.node_link_graph(data) 9.8.3 adjacency_data adjacency_data(G) Return data in adjacency format that is suitable for JSON serialization and use in Javascript documents. Parameters G : NetworkX graph Returns data : dict 374 Chapter 9. Reading and writing graphs NetworkX Reference, Release 1.7 A dictionary with node-link formatted data. See Also: adjacency_graph, node_link_data, tree_data Notes Graph, node, and link attributes are stored in this format but keys for attributes must be strings if you want to serialize with JSON. Examples >>> from networkx.readwrite import json_graph >>> G = nx.Graph([(1,2)]) >>> data = json_graph.adjacency_data(G) To serialize with json >>> import json >>> s = json.dumps(data) 9.8.4 adjacency_graph adjacency_graph(data, directed=False, multigraph=True) Return graph from adjacency data format. Parameters data : dict Adjacency list formatted graph data Returns G : NetworkX graph A NetworkX graph object directed : bool If True, and direction not speciﬁed in data, return a directed graph. multigraph : bool If True, and multigraph not speciﬁed in data, return a multigraph. See Also: adjacency_graph, node_link_data, tree_data Examples >>> from networkx.readwrite import json_graph >>> G = nx.Graph([(1,2)]) >>> data = json_graph.adjacency_data(G) >>> H = json_graph.adjacency_graph(data) 9.8. JSON 375 NetworkX Reference, Release 1.7 9.8.5 tree_data tree_data(G, root) Return data in tree format that is suitable for JSON serialization and use in Javascript documents. Parameters G : NetworkX graph G must be an oriented tree root : node The root of the tree Returns data : dict A dictionary with node-link formatted data. See Also: tree_graph, node_link_data, node_link_data Notes Node attributes are stored in this format but keys for attributes must be strings if you want to serialize with JSON. Graph and edge attributes are not stored. Examples >>> from networkx.readwrite import json_graph >>> G = nx.DiGraph([(1,2)]) >>> data = json_graph.tree_data(G,root=1) To serialize with json >>> import json >>> s = json.dumps(data) 9.8.6 tree_graph tree_graph(data) Return graph from tree data format. Parameters data : dict Tree formatted graph data Returns G : NetworkX DiGraph See Also: tree_graph, node_link_data, adjacency_data Examples 376 Chapter 9. Reading and writing graphs NetworkX Reference, Release 1.7 >>> from networkx.readwrite import json_graph >>> G = nx.DiGraph([(1,2)]) >>> data = json_graph.tree_data(G,root=1) >>> H = json_graph.tree_graph(data) 9.8.7 dumps dumps = Serialize obj to a JSON formatted str. If skipkeys is false then dict keys that are not basic types (str, unicode, int, long, float, bool, None) will be skipped instead of raising a TypeError. If ensure_ascii is false, then the return value will be a unicode instance subject to normal Python str to unicode coercion rules instead of being escaped to an ASCII str. If check_circular is false, then the circular reference check for container types will be skipped and a circular reference will result in an OverflowError (or worse). If allow_nan is false, then it will be a ValueError to serialize out of range float values (nan, inf, -inf) in strict compliance of the JSON speciﬁcation, instead of using the JavaScript equivalents (NaN, Infinity, -Infinity). If indent is a non-negative integer, then JSON array elements and object members will be pretty-printed with that indent level. An indent level of 0 will only insert newlines. None is the most compact representation. If separators is an (item_separator, dict_separator) tuple then it will be used instead of the default (’, ’, ’: ’) separators. (’,’, ’:’) is the most compact JSON representation. encoding is the character encoding for str instances, default is UTF-8. default(obj) is a function that should return a serializable version of obj or raise TypeError. The default simply raises TypeError. To use a custom JSONEncoder subclass (e.g. one that overrides the .default() method to serialize addi- tional types), specify it with the cls kwarg; otherwise JSONEncoder is used. 9.8.8 loads loads = Deserialize s (a str or unicode instance containing a JSON document) to a Python object. If s is a str instance and is encoded with an ASCII based encoding other than utf-8 (e.g. latin-1) then an appropriate encoding name must be speciﬁed. Encodings that are not ASCII based (such as UCS-2) are not allowed and should be decoded to unicode ﬁrst. object_hook is an optional function that will be called with the result of any object literal decode (a dict). The return value of object_hook will be used instead of the dict. This feature can be used to implement custom decoders (e.g. JSON-RPC class hinting). object_pairs_hook is an optional function that will be called with the result of any object literal decoded with an ordered list of pairs. The return value of object_pairs_hook will be used instead of the dict. This feature can be used to implement custom decoders that rely on the order that the key and value pairs are decoded (for example, collections.OrderedDict will remember the order of insertion). If object_hook is also deﬁned, the object_pairs_hook takes priority. 9.8. JSON 377 NetworkX Reference, Release 1.7 parse_float, if speciﬁed, will be called with the string of every JSON ﬂoat to be decoded. By default this is equivalent to ﬂoat(num_str). This can be used to use another datatype or parser for JSON ﬂoats (e.g. decimal.Decimal). parse_int, if speciﬁed, will be called with the string of every JSON int to be decoded. By default this is equivalent to int(num_str). This can be used to use another datatype or parser for JSON integers (e.g. ﬂoat). parse_constant, if speciﬁed, will be called with one of the following strings: -Inﬁnity, Inﬁnity, NaN, null, true, false. This can be used to raise an exception if invalid JSON numbers are encountered. To use a custom JSONDecoder subclass, specify it with the cls kwarg; otherwise JSONDecoder is used. 9.8.9 dump dump = Serialize obj as a JSON formatted stream to fp (a .write()-supporting ﬁle-like object). If skipkeys is true then dict keys that are not basic types (str, unicode, int, long, float, bool, None) will be skipped instead of raising a TypeError. If ensure_ascii is false, then the some chunks written to fp may be unicode instances, subject to nor- mal Python str to unicode coercion rules. Unless fp.write() explicitly understands unicode (as in codecs.getwriter()) this is likely to cause an error. If check_circular is false, then the circular reference check for container types will be skipped and a circular reference will result in an OverflowError (or worse). If allow_nan is false, then it will be a ValueError to serialize out of range float values (nan, inf, -inf) in strict compliance of the JSON speciﬁcation, instead of using the JavaScript equivalents (NaN, Infinity, -Infinity). If indent is a non-negative integer, then JSON array elements and object members will be pretty-printed with that indent level. An indent level of 0 will only insert newlines. None is the most compact representation. If separators is an (item_separator, dict_separator) tuple then it will be used instead of the default (’, ’, ’: ’) separators. (’,’, ’:’) is the most compact JSON representation. encoding is the character encoding for str instances, default is UTF-8. default(obj) is a function that should return a serializable version of obj or raise TypeError. The default simply raises TypeError. To use a custom JSONEncoder subclass (e.g. one that overrides the .default() method to serialize addi- tional types), specify it with the cls kwarg; otherwise JSONEncoder is used. 9.8.10 load load = Deserialize fp (a .read()-supporting ﬁle-like object containing a JSON document) to a Python object. If the contents of fp is encoded with an ASCII based encoding other than utf-8 (e.g. latin-1), then an appropriate encoding name must be speciﬁed. Encodings that are not ASCII based (such as UCS-2) are not allowed, and should be wrapped with codecs.getreader(fp)(encoding), or simply decoded to a unicode object and passed to loads() object_hook is an optional function that will be called with the result of any object literal decode (a dict). The return value of object_hook will be used instead of the dict. This feature can be used to implement custom decoders (e.g. JSON-RPC class hinting). 378 Chapter 9. Reading and writing graphs NetworkX Reference, Release 1.7 object_pairs_hook is an optional function that will be called with the result of any object literal decoded with an ordered list of pairs. The return value of object_pairs_hook will be used instead of the dict. This feature can be used to implement custom decoders that rely on the order that the key and value pairs are decoded (for example, collections.OrderedDict will remember the order of insertion). If object_hook is also deﬁned, the object_pairs_hook takes priority. To use a custom JSONDecoder subclass, specify it with the cls kwarg; otherwise JSONDecoder is used. 9.9 LEDA Read graphs in LEDA format. LEDA is a C++ class library for efﬁcient data types and algorithms. 9.9.1 Format See http://www.algorithmic-solutions.info/leda_guide/graphs/leda_native_graph_ﬁleformat.html read_leda(path[, encoding]) Read graph in LEDA format from path. parse_leda(lines) Read graph in LEDA format from string or iterable. 9.9.2 read_leda read_leda(path, encoding=’UTF-8’) Read graph in LEDA format from path. Parameters path : ﬁle or string File or ﬁlename to read. Filenames ending in .gz or .bz2 will be uncompressed. Returns G : NetworkX graph References [R253] Examples G=nx.read_leda(‘ﬁle.leda’) 9.9.3 parse_leda parse_leda(lines) Read graph in LEDA format from string or iterable. Parameters lines : string or iterable Data in LEDA format. Returns G : NetworkX graph 9.9. LEDA 379 NetworkX Reference, Release 1.7 References [R252] Examples G=nx.parse_leda(string) 9.10 YAML Read and write NetworkX graphs in YAML format. “YAML is a data serialization format designed for human readability and interaction with scripting languages.” See http://www.yaml.org for documentation. 9.10.1 Format http://pyyaml.org/wiki/PyYAML read_yaml(path) Read graph in YAML format from path. write_yaml(G, path[, encoding]) Write graph G in YAML format to path. 9.10.2 read_yaml read_yaml(path) Read graph in YAML format from path. YAML is a data serialization format designed for human readability and interaction with scripting languages [R256]. Parameters path : ﬁle or string File or ﬁlename to read. Filenames ending in .gz or .bz2 will be uncompressed. Returns G : NetworkX graph References [R256] Examples >>> G=nx.path_graph(4) >>> nx.write_yaml(G,’test.yaml’) >>> G=nx.read_yaml(’test.yaml’) 380 Chapter 9. Reading and writing graphs NetworkX Reference, Release 1.7 9.10.3 write_yaml write_yaml(G, path, encoding=’UTF-8’, **kwds) Write graph G in YAML format to path. YAML is a data serialization format designed for human readability and interaction with scripting languages [R257]. Parameters G : graph A NetworkX graph path : ﬁle or string File or ﬁlename to write. Filenames ending in .gz or .bz2 will be compressed. encoding: string, optional : Specify which encoding to use when writing ﬁle. References [R257] Examples >>> G=nx.path_graph(4) >>> nx.write_yaml(G,’test.yaml’) 9.11 SparseGraph6 Read graphs in graph6 and sparse6 format. 9.11.1 Format “graph6 and sparse6 are formats for storing undirected graphs in a compact manner, using only printable ASCII charac- ters. Files in these formats have text type and contain one line per graph.” http://cs.anu.edu.au/~bdm/data/formats.html See http://cs.anu.edu.au/~bdm/data/formats.txt for details. read_graph6(path) Read simple undirected graphs in graph6 format from path. parse_graph6(str) Read a simple undirected graph in graph6 format from string. read_graph6_list(path) Read simple undirected graphs in graph6 format from path. read_sparse6(path) Read simple undirected graphs in sparse6 format from path. parse_sparse6(string) Read undirected graph in sparse6 format from string. read_sparse6_list(path) Read undirected graphs in sparse6 format from path. 9.11.2 read_graph6 read_graph6(path) Read simple undirected graphs in graph6 format from path. 9.11. SparseGraph6 381 NetworkX Reference, Release 1.7 Returns a single Graph. 9.11.3 parse_graph6 parse_graph6(str) Read a simple undirected graph in graph6 format from string. Returns a single Graph. 9.11.4 read_graph6_list read_graph6_list(path) Read simple undirected graphs in graph6 format from path. Returns a list of Graphs, one for each line in ﬁle. 9.11.5 read_sparse6 read_sparse6(path) Read simple undirected graphs in sparse6 format from path. Returns a single MultiGraph. 9.11.6 parse_sparse6 parse_sparse6(string) Read undirected graph in sparse6 format from string. Returns a MultiGraph. 9.11.7 read_sparse6_list read_sparse6_list(path) Read undirected graphs in sparse6 format from path. Returns a list of MultiGraphs, one for each line in ﬁle. 9.12 Pajek Read graphs in Pajek format. This implementation handles directed and undirected graphs including those with self loops and parallel edges. 9.12.1 Format See http://vlado.fmf.uni-lj.si/pub/networks/pajek/doc/draweps.htm for format information. read_pajek(path[, encoding]) Read graph in Pajek format from path. write_pajek(G, path[, encoding]) Write graph in Pajek format to path. Continued on next page 382 Chapter 9. Reading and writing graphs NetworkX Reference, Release 1.7 Table 9.12 – continued from previous page parse_pajek(lines) Parse Pajek format graph from string or iterable. 9.12.2 read_pajek read_pajek(path, encoding=’UTF-8’) Read graph in Pajek format from path. Parameters path : ﬁle or string File or ﬁlename to write. Filenames ending in .gz or .bz2 will be uncompressed. Returns G : NetworkX MultiGraph or MultiDiGraph. References See http://vlado.fmf.uni-lj.si/pub/networks/pajek/doc/draweps.htm for format information. Examples >>> G=nx.path_graph(4) >>> nx.write_pajek(G, "test.net") >>> G=nx.read_pajek("test.net") To create a Graph instead of a MultiGraph use >>> G1=nx.Graph(G) 9.12.3 write_pajek write_pajek(G, path, encoding=’UTF-8’) Write graph in Pajek format to path. Parameters G : graph A Networkx graph path : ﬁle or string File or ﬁlename to write. Filenames ending in .gz or .bz2 will be compressed. References See http://vlado.fmf.uni-lj.si/pub/networks/pajek/doc/draweps.htm for format information. Examples >>> G=nx.path_graph(4) >>> nx.write_pajek(G, "test.net") 9.12. Pajek 383 NetworkX Reference, Release 1.7 9.12.4 parse_pajek parse_pajek(lines) Parse Pajek format graph from string or iterable. Parameters lines : string or iterable Data in Pajek format. Returns G : NetworkX graph See Also: read_pajek 9.13 GIS Shapeﬁle Generates a networkx.DiGraph from point and line shapeﬁles. “The Esri Shapeﬁle or simply a shapeﬁle is a popular geospatial vector data format for geographic information systems software. It is developed and regulated by Esri as a (mostly) open speciﬁcation for data interoperability among Esri and other software products.” See http://en.wikipedia.org/wiki/Shapeﬁle for additional information. read_shp(path) Generates a networkx.DiGraph from shapeﬁles. Point geometries are write_shp(G, outdir) Writes a networkx.DiGraph to two shapeﬁles, edges and nodes. 9.13.1 read_shp read_shp(path) Generates a networkx.DiGraph from shapeﬁles. Point geometries are translated into nodes, lines into edges. Coordinate tuples are used as keys. Attributes are preserved, line geometries are simpliﬁed into start and end coordinates. Accepts a single shapeﬁle or directory of many shapeﬁles. “The Esri Shapeﬁle or simply a shapeﬁle is a popular geospatial vector data format for geographic information systems software [R254].” Parameters path : ﬁle or string File, directory, or ﬁlename to read. Returns G : NetworkX graph References [R254] Examples >>> G=nx.read_shp(’test.shp’) 384 Chapter 9. Reading and writing graphs NetworkX Reference, Release 1.7 9.13.2 write_shp write_shp(G, outdir) Writes a networkx.DiGraph to two shapeﬁles, edges and nodes. Nodes and edges are expected to have a Well Known Binary (Wkb) or Well Known Text (Wkt) key in order to generate geometries. Also acceptable are nodes with a numeric tuple key (x,y). “The Esri Shapeﬁle or simply a shapeﬁle is a popular geospatial vector data format for geographic information systems software [R255].” Parameters outdir : directory path Output directory for the two shapeﬁles. Returns None : References [R255] Examples nx.write_shp(digraph, ‘/shapeﬁles’) # doctest +SKIP 9.13. GIS Shapeﬁle 385 NetworkX Reference, Release 1.7 386 Chapter 9. Reading and writing graphs CHAPTER TEN DRAWING 10.1 Matplotlib Draw networks with matplotlib (pylab). 10.1.1 See Also matplotlib: http://matplotlib.sourceforge.net/ pygraphviz: http://networkx.lanl.gov/pygraphviz/ draw(G[, pos, ax, hold]) Draw the graph G with Matplotlib (pylab). draw_networkx(G[, pos, with_labels]) Draw the graph G using Matplotlib. draw_networkx_nodes(G, pos[, nodelist, ...]) Draw the nodes of the graph G. draw_networkx_edges(G, pos[, edgelist, ...]) Draw the edges of the graph G. draw_networkx_labels(G, pos[, labels, ...]) Draw node labels on the graph G. draw_networkx_edge_labels(G, pos[, ...]) Draw edge labels. draw_circular(G, **kwargs) Draw the graph G with a circular layout. draw_random(G, **kwargs) Draw the graph G with a random layout. draw_spectral(G, **kwargs) Draw the graph G with a spectral layout. draw_spring(G, **kwargs) Draw the graph G with a spring layout. draw_shell(G, **kwargs) Draw networkx graph with shell layout. draw_graphviz(G[, prog]) Draw networkx graph with graphviz layout. 10.1.2 draw draw(G, pos=None, ax=None, hold=None, **kwds) Draw the graph G with Matplotlib (pylab). Draw the graph as a simple representation with no node labels or edge labels and using the full Matplotlib ﬁgure area and no axis labels by default. See draw_networkx() for more full-featured drawing that allows title, axis labels etc. Parameters G : graph A networkx graph pos : dictionary, optional A dictionary with nodes as keys and positions as values. If not speciﬁed a spring layout positioning will be computed. See networkx.layout for functions that compute node 387 NetworkX Reference, Release 1.7 positions. ax : Matplotlib Axes object, optional Draw the graph in speciﬁed Matplotlib axes. hold : bool, optional Set the Matplotlib hold state. If True subsequent draw commands will be added to the current axes. **kwds : optional keywords See networkx.draw_networkx() for a description of optional keywords. See Also: draw_networkx, draw_networkx_nodes, draw_networkx_edges, draw_networkx_labels, draw_networkx_edge_labels Notes This function has the same name as pylab.draw and pyplot.draw so beware when using >>> from networkx import * since you might overwrite the pylab.draw function. Good alternatives are: With pylab: >>> import pylab as P # >>> import networkx as nx >>> G=nx.dodecahedral_graph() >>> nx.draw(G) # networkx draw() >>> P.draw() # pylab draw() With pyplot >>> import matplotlib.pyplot as plt >>> import networkx as nx >>> G=nx.dodecahedral_graph() >>> nx.draw(G) # networkx draw() >>> plt.draw() # pyplot draw() Also see the NetworkX drawing examples at http://networkx.lanl.gov/gallery.html Examples >>> G=nx.dodecahedral_graph() >>> nx.draw(G) >>> nx.draw(G,pos=nx.spring_layout(G)) # use spring layout 10.1.3 draw_networkx draw_networkx(G, pos=None, with_labels=True, **kwds) Draw the graph G using Matplotlib. 388 Chapter 10. Drawing NetworkX Reference, Release 1.7 Draw the graph with Matplotlib with options for node positions, labeling, titles, and many other drawing fea- tures. See draw() for simple drawing without labels or axes. Parameters G : graph A networkx graph pos : dictionary, optional A dictionary with nodes as keys and positions as values. If not speciﬁed a spring layout positioning will be computed. See networkx.layout for functions that compute node positions. with_labels : bool, optional (default=True) Set to True to draw labels on the nodes. ax : Matplotlib Axes object, optional Draw the graph in the speciﬁed Matplotlib axes. nodelist : list, optional (default G.nodes()) Draw only speciﬁed nodes edgelist : list, optional (default=G.edges()) Draw only speciﬁed edges node_size : scalar or array, optional (default=300) Size of nodes. If an array is speciﬁed it must be the same length as nodelist. node_color : color string, or array of ﬂoats, (default=’r’) Node color. Can be a single color format string, or a sequence of colors with the same length as nodelist. If numeric values are speciﬁed they will be mapped to colors using the cmap and vmin,vmax parameters. See matplotlib.scatter for more details. node_shape : string, optional (default=’o’) The shape of the node. Speciﬁcation is as matplotlib.scatter marker, one of ‘so^>v>> G=nx.dodecahedral_graph() >>> nx.draw(G) >>> nx.draw(G,pos=nx.spring_layout(G)) # use spring layout >>> import pylab >>> limits=pylab.axis(’off’) # turn of axis Also see the NetworkX drawing examples at http://networkx.lanl.gov/gallery.html 10.1.4 draw_networkx_nodes draw_networkx_nodes(G, pos, nodelist=None, node_size=300, node_color=’r’, node_shape=’o’, al- pha=1.0, cmap=None, vmin=None, vmax=None, ax=None, linewidths=None, la- bel=None, **kwds) Draw the nodes of the graph G. This draws only the nodes of the graph G. Parameters G : graph A networkx graph 390 Chapter 10. Drawing NetworkX Reference, Release 1.7 pos : dictionary A dictionary with nodes as keys and positions as values. If not speciﬁed a spring layout positioning will be computed. See networkx.layout for functions that compute node positions. ax : Matplotlib Axes object, optional Draw the graph in the speciﬁed Matplotlib axes. nodelist : list, optional Draw only speciﬁed nodes (default G.nodes()) node_size : scalar or array Size of nodes (default=300). If an array is speciﬁed it must be the same length as nodelist. node_color : color string, or array of ﬂoats Node color. Can be a single color format string (default=’r’), or a sequence of colors with the same length as nodelist. If numeric values are speciﬁed they will be mapped to colors using the cmap and vmin,vmax parameters. See matplotlib.scatter for more details. node_shape : string The shape of the node. Speciﬁcation is as matplotlib.scatter marker, one of ‘so^>v>> G=nx.dodecahedral_graph() >>> nodes=nx.draw_networkx_nodes(G,pos=nx.spring_layout(G)) Also see the NetworkX drawing examples at http://networkx.lanl.gov/gallery.html 10.1. Matplotlib 391 NetworkX Reference, Release 1.7 10.1.5 draw_networkx_edges draw_networkx_edges(G, pos, edgelist=None, width=1.0, edge_color=’k’, style=’solid’, alpha=None, edge_cmap=None, edge_vmin=None, edge_vmax=None, ax=None, ar- rows=True, label=None, **kwds) Draw the edges of the graph G. This draws only the edges of the graph G. Parameters G : graph A networkx graph pos : dictionary A dictionary with nodes as keys and positions as values. If not speciﬁed a spring layout positioning will be computed. See networkx.layout for functions that compute node positions. edgelist : collection of edge tuples Draw only speciﬁed edges(default=G.edges()) width : ﬂoat Line width of edges (default =1.0) edge_color : color string, or array of ﬂoats Edge color. Can be a single color format string (default=’r’), or a sequence of colors with the same length as edgelist. If numeric values are speciﬁed they will be mapped to colors using the edge_cmap and edge_vmin,edge_vmax parameters. style : string Edge line style (default=’solid’) (solid|dashed|dotted,dashdot) alpha : ﬂoat The edge transparency (default=1.0) edge_ cmap : Matplotlib colormap Colormap for mapping intensities of edges (default=None) edge_vmin,edge_vmax : ﬂoats Minimum and maximum for edge colormap scaling (default=None) ax : Matplotlib Axes object, optional Draw the graph in the speciﬁed Matplotlib axes. arrows : bool, optional (default=True) For directed graphs, if True draw arrowheads. label : [None| string] Label for legend See Also: draw, draw_networkx, draw_networkx_nodes, draw_networkx_labels, draw_networkx_edge_labels 392 Chapter 10. Drawing NetworkX Reference, Release 1.7 Notes For directed graphs, “arrows” (actually just thicker stubs) are drawn at the head end. Arrows can be turned off with keyword arrows=False. Yes, it is ugly but drawing proper arrows with Matplotlib this way is tricky. Examples >>> G=nx.dodecahedral_graph() >>> edges=nx.draw_networkx_edges(G,pos=nx.spring_layout(G)) Also see the NetworkX drawing examples at http://networkx.lanl.gov/gallery.html 10.1.6 draw_networkx_labels draw_networkx_labels(G, pos, labels=None, font_size=12, font_color=’k’, font_family=’sans-serif’, font_weight=’normal’, alpha=1.0, ax=None, **kwds) Draw node labels on the graph G. Parameters G : graph A networkx graph pos : dictionary, optional A dictionary with nodes as keys and positions as values. If not speciﬁed a spring layout positioning will be computed. See networkx.layout for functions that compute node positions. font_size : int Font size for text labels (default=12) font_color : string Font color string (default=’k’ black) font_family : string Font family (default=’sans-serif’) font_weight : string Font weight (default=’normal’) alpha : ﬂoat The text transparency (default=1.0) ax : Matplotlib Axes object, optional Draw the graph in the speciﬁed Matplotlib axes. See Also: draw, draw_networkx, draw_networkx_nodes, draw_networkx_edges, draw_networkx_edge_labels 10.1. Matplotlib 393 NetworkX Reference, Release 1.7 Examples >>> G=nx.dodecahedral_graph() >>> labels=nx.draw_networkx_labels(G,pos=nx.spring_layout(G)) Also see the NetworkX drawing examples at http://networkx.lanl.gov/gallery.html 10.1.7 draw_networkx_edge_labels draw_networkx_edge_labels(G, pos, edge_labels=None, label_pos=0.5, font_size=10, font_color=’k’, font_family=’sans-serif’, font_weight=’normal’, alpha=1.0, bbox=None, ax=None, rotate=True, **kwds) Draw edge labels. Parameters G : graph A networkx graph pos : dictionary, optional A dictionary with nodes as keys and positions as values. If not speciﬁed a spring layout positioning will be computed. See networkx.layout for functions that compute node positions. ax : Matplotlib Axes object, optional Draw the graph in the speciﬁed Matplotlib axes. alpha : ﬂoat The text transparency (default=1.0) edge_labels : dictionary Edge labels in a dictionary keyed by edge two-tuple of text labels (default=None). Only labels for the keys in the dictionary are drawn. label_pos : ﬂoat Position of edge label along edge (0=head, 0.5=center, 1=tail) font_size : int Font size for text labels (default=12) font_color : string Font color string (default=’k’ black) font_weight : string Font weight (default=’normal’) font_family : string Font family (default=’sans-serif’) bbox : Matplotlib bbox Specify text box shape and colors. clip_on : bool Turn on clipping at axis boundaries (default=True) 394 Chapter 10. Drawing NetworkX Reference, Release 1.7 See Also: draw, draw_networkx, draw_networkx_nodes, draw_networkx_edges, draw_networkx_labels Examples >>> G=nx.dodecahedral_graph() >>> edge_labels=nx.draw_networkx_edge_labels(G,pos=nx.spring_layout(G)) Also see the NetworkX drawing examples at http://networkx.lanl.gov/gallery.html 10.1.8 draw_circular draw_circular(G, **kwargs) Draw the graph G with a circular layout. 10.1.9 draw_random draw_random(G, **kwargs) Draw the graph G with a random layout. 10.1.10 draw_spectral draw_spectral(G, **kwargs) Draw the graph G with a spectral layout. 10.1.11 draw_spring draw_spring(G, **kwargs) Draw the graph G with a spring layout. 10.1.12 draw_shell draw_shell(G, **kwargs) Draw networkx graph with shell layout. 10.1.13 draw_graphviz draw_graphviz(G, prog=’neato’, **kwargs) Draw networkx graph with graphviz layout. 10.2 Graphviz AGraph (dot) Interface to pygraphviz AGraph class. 10.2. Graphviz AGraph (dot) 395 NetworkX Reference, Release 1.7 10.2.1 Examples >>> G=nx.complete_graph(5) >>> A=nx.to_agraph(G) >>> H=nx.from_agraph(A) 10.2.2 See Also Pygraphviz: http://networkx.lanl.gov/pygraphviz from_agraph(A[, create_using]) Return a NetworkX Graph or DiGraph from a PyGraphviz graph. to_agraph(N) Return a pygraphviz graph from a NetworkX graph N. write_dot(G, path) Write NetworkX graph G to Graphviz dot format on path. read_dot(path) Return a NetworkX graph from a dot ﬁle on path. graphviz_layout(G[, prog, root, args]) Create node positions for G using Graphviz. pygraphviz_layout(G[, prog, root, args]) Create node positions for G using Graphviz. 10.2.3 from_agraph from_agraph(A, create_using=None) Return a NetworkX Graph or DiGraph from a PyGraphviz graph. Parameters A : PyGraphviz AGraph A graph created with PyGraphviz create_using : NetworkX graph class instance The output is created using the given graph class instance Notes The Graph G will have a dictionary G.graph_attr containing the default graphviz attributes for graphs, nodes and edges. Default node attributes will be in the dictionary G.node_attr which is keyed by node. Edge attributes will be returned as edge data in G. With edge_attr=False the edge data will be the Graphviz edge weight attribute or the value 1 if no edge weight attribute is found. Examples >>> K5=nx.complete_graph(5) >>> A=nx.to_agraph(K5) >>> G=nx.from_agraph(A) >>> G=nx.from_agraph(A) 10.2.4 to_agraph to_agraph(N) Return a pygraphviz graph from a NetworkX graph N. 396 Chapter 10. Drawing NetworkX Reference, Release 1.7 Parameters N : NetworkX graph A graph created with NetworkX Notes If N has an dict N.graph_attr an attempt will be made ﬁrst to copy properties attached to the graph (see from_agraph) and then updated with the calling arguments if any. Examples >>> K5=nx.complete_graph(5) >>> A=nx.to_agraph(K5) 10.2.5 write_dot write_dot(G, path) Write NetworkX graph G to Graphviz dot format on path. Parameters G : graph A networkx graph path : ﬁlename Filename or ﬁle handle to write 10.2.6 read_dot read_dot(path) Return a NetworkX graph from a dot ﬁle on path. Parameters path : ﬁle or string File name or ﬁle handle to read. 10.2.7 graphviz_layout graphviz_layout(G, prog=’neato’, root=None, args=’‘) Create node positions for G using Graphviz. Parameters G : NetworkX graph A graph created with NetworkX prog : string Name of Graphviz layout program root : string, optional Root node for twopi layout args : string, optional Extra arguments to Graphviz layout program 10.2. Graphviz AGraph (dot) 397 NetworkX Reference, Release 1.7 Returns : dictionary Dictionary of x,y, positions keyed by node. Notes This is a wrapper for pygraphviz_layout. Examples >>> G=nx.petersen_graph() >>> pos=nx.graphviz_layout(G) >>> pos=nx.graphviz_layout(G,prog=’dot’) 10.2.8 pygraphviz_layout pygraphviz_layout(G, prog=’neato’, root=None, args=’‘) Create node positions for G using Graphviz. Parameters G : NetworkX graph A graph created with NetworkX prog : string Name of Graphviz layout program root : string, optional Root node for twopi layout args : string, optional Extra arguments to Graphviz layout program Returns : dictionary Dictionary of x,y, positions keyed by node. Examples >>> G=nx.petersen_graph() >>> pos=nx.graphviz_layout(G) >>> pos=nx.graphviz_layout(G,prog=’dot’) 10.3 Graphviz with pydot Import and export NetworkX graphs in Graphviz dot format using pydot. Either this module or nx_pygraphviz can be used to interface with graphviz. 398 Chapter 10. Drawing NetworkX Reference, Release 1.7 10.3.1 See Also Pydot: http://code.google.com/p/pydot/ Graphviz: http://www.research.att.com/sw/tools/graphviz/ DOT Language: http://www.graphviz.org/doc/info/lang.html from_pydot(P) Return a NetworkX graph from a Pydot graph. to_pydot(N[, strict]) Return a pydot graph from a NetworkX graph N. write_dot(G, path) Write NetworkX graph G to Graphviz dot format on path. read_dot(path) Return a NetworkX MultiGraph or MultiDiGraph from a dot ﬁle on path. graphviz_layout(G[, prog, root]) Create node positions using Pydot and Graphviz. pydot_layout(G[, prog, root]) Create node positions using Pydot and Graphviz. 10.3.2 from_pydot from_pydot(P) Return a NetworkX graph from a Pydot graph. Parameters P : Pydot graph A graph created with Pydot Returns G : NetworkX multigraph A MultiGraph or MultiDiGraph. Examples >>> K5=nx.complete_graph(5) >>> A=nx.to_pydot(K5) >>> G=nx.from_pydot(A) # return MultiGraph >>> G=nx.Graph(nx.from_pydot(A)) # make a Graph instead of MultiGraph 10.3.3 to_pydot to_pydot(N, strict=True) Return a pydot graph from a NetworkX graph N. Parameters N : NetworkX graph A graph created with NetworkX Examples >>> K5=nx.complete_graph(5) >>> P=nx.to_pydot(K5) 10.3.4 write_dot write_dot(G, path) Write NetworkX graph G to Graphviz dot format on path. Path can be a string or a ﬁle handle. 10.3. Graphviz with pydot 399 NetworkX Reference, Release 1.7 10.3.5 read_dot read_dot(path) Return a NetworkX MultiGraph or MultiDiGraph from a dot ﬁle on path. Parameters path : ﬁlename or ﬁle handle Returns G : NetworkX multigraph A MultiGraph or MultiDiGraph. Notes Use G=nx.Graph(nx.read_dot(path)) to return a Graph instead of a MultiGraph. 10.3.6 graphviz_layout graphviz_layout(G, prog=’neato’, root=None, **kwds) Create node positions using Pydot and Graphviz. Returns a dictionary of positions keyed by node. Notes This is a wrapper for pydot_layout. Examples >>> G=nx.complete_graph(4) >>> pos=nx.graphviz_layout(G) >>> pos=nx.graphviz_layout(G,prog=’dot’) 10.3.7 pydot_layout pydot_layout(G, prog=’neato’, root=None, **kwds) Create node positions using Pydot and Graphviz. Returns a dictionary of positions keyed by node. Examples >>> G=nx.complete_graph(4) >>> pos=nx.pydot_layout(G) >>> pos=nx.pydot_layout(G,prog=’dot’) 10.4 Graph Layout Node positioning algorithms for graph drawing. 400 Chapter 10. Drawing NetworkX Reference, Release 1.7 circular_layout(G[, dim, scale]) Position nodes on a circle. random_layout(G[, dim]) Position nodes uniformly at random in the unit square. shell_layout(G[, nlist, dim, scale]) Position nodes in concentric circles. spring_layout(G[, dim, pos, ﬁxed, ...]) Position nodes using Fruchterman-Reingold force-directed algorithm. spectral_layout(G[, dim, weight, scale]) Position nodes using the eigenvectors of the graph Laplacian. 10.4.1 circular_layout circular_layout(G, dim=2, scale=1) Position nodes on a circle. Parameters G : NetworkX graph dim : int Dimension of layout, currently only dim=2 is supported scale : ﬂoat Scale factor for positions Returns dict : : A dictionary of positions keyed by node Notes This algorithm currently only works in two dimensions and does not try to minimize edge crossings. Examples >>> G=nx.path_graph(4) >>> pos=nx.circular_layout(G) 10.4.2 random_layout random_layout(G, dim=2) Position nodes uniformly at random in the unit square. For every node, a position is generated by choosing each of dim coordinates uniformly at random on the interval [0.0, 1.0). NumPy (http://scipy.org) is required for this function. Parameters G : NetworkX graph A position will be assigned to every node in G. dim : int Dimension of layout. Returns dict : : A dictionary of positions keyed by node 10.4. Graph Layout 401 NetworkX Reference, Release 1.7 Examples >>> G = nx.lollipop_graph(4, 3) >>> pos = nx.random_layout(G) 10.4.3 shell_layout shell_layout(G, nlist=None, dim=2, scale=1) Position nodes in concentric circles. Parameters G : NetworkX graph nlist : list of lists List of node lists for each shell. dim : int Dimension of layout, currently only dim=2 is supported scale : ﬂoat Scale factor for positions Returns dict : : A dictionary of positions keyed by node Notes This algorithm currently only works in two dimensions and does not try to minimize edge crossings. Examples >>> G=nx.path_graph(4) >>> shells=[[0],[1,2,3]] >>> pos=nx.shell_layout(G,shells) 10.4.4 spring_layout spring_layout(G, dim=2, pos=None, ﬁxed=None, iterations=50, weight=’weight’, scale=1) Position nodes using Fruchterman-Reingold force-directed algorithm. Parameters G : NetworkX graph dim : int Dimension of layout pos : dict or None optional (default=None) Initial positions for nodes as a dictionary with node as keys and values as a list or tuple. If None, then nuse random initial positions. ﬁxed : list or None optional (default=None) Nodes to keep ﬁxed at initial position. 402 Chapter 10. Drawing NetworkX Reference, Release 1.7 iterations : int optional (default=50) Number of iterations of spring-force relaxation weight : string or None optional (default=’weight’) The edge attribute that holds the numerical value used for the edge weight. If None, then all edge weights are 1. scale : ﬂoat Scale factor for positions Returns dict : : A dictionary of positions keyed by node Examples >>> G=nx.path_graph(4) >>> pos=nx.spring_layout(G) # The same using longer function name >>> pos=nx.fruchterman_reingold_layout(G) 10.4.5 spectral_layout spectral_layout(G, dim=2, weight=’weight’, scale=1) Position nodes using the eigenvectors of the graph Laplacian. Parameters G : NetworkX graph dim : int Dimension of layout weight : string or None optional (default=’weight’) The edge attribute that holds the numerical value used for the edge weight. If None, then all edge weights are 1. scale : ﬂoat Scale factor for positions Returns dict : : A dictionary of positions keyed by node Notes Directed graphs will be considered as unidrected graphs when positioning the nodes. For larger graphs (>500 nodes) this will use the SciPy sparse eigenvalue solver (ARPACK). Examples >>> G=nx.path_graph(4) >>> pos=nx.spectral_layout(G) 10.4. Graph Layout 403 NetworkX Reference, Release 1.7 404 Chapter 10. Drawing CHAPTER ELEVEN EXCEPTIONS Base exceptions and errors for NetworkX. class NetworkXException Base class for exceptions in NetworkX. class NetworkXError Exception for a serious error in NetworkX class NetworkXPointlessConcept Harary, F. and Read, R. “Is the Null Graph a Pointless Concept?” In Graphs and Combinatorics Conference, George Washington University. New York: Springer-Verlag, 1973. class NetworkXAlgorithmError Exception for unexpected termination of algorithms. class NetworkXUnfeasible Exception raised by algorithms trying to solve a problem instance that has no feasible solution. class NetworkXNoPath Exception for algorithms that should return a path when running on graphs where such a path does not exist. class NetworkXUnbounded Exception raised by algorithms trying to solve a maximization or a minimization problem instance that is un- bounded. 405 NetworkX Reference, Release 1.7 406 Chapter 11. Exceptions CHAPTER TWELVE UTILITIES 12.1 Helper functions Miscellaneous Helpers for NetworkX. These are not imported into the base networkx namespace but can be accessed, for example, as >>> import networkx >>> networkx.utils.is_string_like(’spam’) True is_string_like(obj) Check if obj is string. flatten(obj[, result]) Return ﬂattened version of (possibly nested) iterable object. iterable(obj) Return True if obj is iterable with a well-deﬁned len(). is_list_of_ints(intlist) Return True if list is a list of ints. make_str(t) Return the string representation of t. cumulative_sum(numbers) Yield cumulative sum of numbers. generate_unique_node() Generate a unique node label. default_opener(ﬁlename) Opens filename using system’s default program. 12.1.1 is_string_like is_string_like(obj) Check if obj is string. 12.1.2 ﬂatten flatten(obj, result=None) Return ﬂattened version of (possibly nested) iterable object. 12.1.3 iterable iterable(obj) Return True if obj is iterable with a well-deﬁned len(). 407 NetworkX Reference, Release 1.7 12.1.4 is_list_of_ints is_list_of_ints(intlist) Return True if list is a list of ints. 12.1.5 make_str make_str(t) Return the string representation of t. 12.1.6 cumulative_sum cumulative_sum(numbers) Yield cumulative sum of numbers. >>> import networkx.utils as utils >>> list(utils.cumulative_sum([1,2,3,4])) [1, 3, 6, 10] 12.1.7 generate_unique_node generate_unique_node() Generate a unique node label. 12.1.8 default_opener default_opener(ﬁlename) Opens filename using system’s default program. Parameters ﬁlename : str The path of the ﬁle to be opened. 12.2 Data structures and Algorithms Union-ﬁnd data structure. UnionFind.union(*objects) Find the sets containing the objects and merge them all. 12.2.1 union UnionFind.union(*objects) Find the sets containing the objects and merge them all. 12.3 Random sequence generators Utilities for generating random numbers, random sequences, and random selections. 408 Chapter 12. Utilities NetworkX Reference, Release 1.7 create_degree_sequence(n[, sfunction, max_tries]) Attempt to create a valid degree sequence of length n using speciﬁed function sfunction(n,**kwds). pareto_sequence(n[, exponent]) Return sample sequence of length n from a Pareto distribution. powerlaw_sequence(n[, exponent]) Return sample sequence of length n from a power law distribution. uniform_sequence(n) Return sample sequence of length n from a uniform distribution. cumulative_distribution(distribution) Return normalized cumulative distribution from discrete distribution. discrete_sequence(n[, distribution, ...]) Return sample sequence of length n from a given discrete distribution or discrete cumulative distribution. zipf_sequence(n[, alpha, xmin]) Return a sample sequence of length n from a Zipf distribution with zipf_rv(alpha[, xmin, seed]) Return a random value chosen from the Zipf distribution. random_weighted_sample(mapping, k) Return k items without replacement from a weighted sample. weighted_choice(mapping) Return a single element from a weighted sample. 12.3.1 create_degree_sequence create_degree_sequence(n, sfunction=None, max_tries=50, **kwds) Attempt to create a valid degree sequence of length n using speciﬁed function sfunction(n,**kwds). Parameters n : int Length of degree sequence = number of nodes sfunction: function : Function which returns a list of n real or integer values. Called as “sfunc- tion(n,**kwds)”. max_tries: int : Max number of attempts at creating valid degree sequence. Notes Repeatedly create a degree sequence by calling sfunction(n,**kwds) until achieving a valid degree sequence. If unsuccessful after max_tries attempts, raise an exception. For examples of sfunctions that return sequences of random numbers, see networkx.Utils. Examples >>> from networkx.utils import uniform_sequence, create_degree_sequence >>> seq=create_degree_sequence(10,uniform_sequence) 12.3.2 pareto_sequence pareto_sequence(n, exponent=1.0) Return sample sequence of length n from a Pareto distribution. 12.3.3 powerlaw_sequence powerlaw_sequence(n, exponent=2.0) Return sample sequence of length n from a power law distribution. 12.3. Random sequence generators 409 NetworkX Reference, Release 1.7 12.3.4 uniform_sequence uniform_sequence(n) Return sample sequence of length n from a uniform distribution. 12.3.5 cumulative_distribution cumulative_distribution(distribution) Return normalized cumulative distribution from discrete distribution. 12.3.6 discrete_sequence discrete_sequence(n, distribution=None, cdistribution=None) Return sample sequence of length n from a given discrete distribution or discrete cumulative distribution. One of the following must be speciﬁed. distribution = histogram of values, will be normalized cdistribution = normalized discrete cumulative distribution 12.3.7 zipf_sequence zipf_sequence(n, alpha=2.0, xmin=1) Return a sample sequence of length n from a Zipf distribution with exponent parameter alpha and minimum value xmin. See Also: zipf_rv 12.3.8 zipf_rv zipf_rv(alpha, xmin=1, seed=None) Return a random value chosen from the Zipf distribution. The return value is an integer drawn from the probability distribution ::math: p(x)=\frac{x^{-\alpha}}{\zeta(\alpha,x_{min})}, where ⇣(↵,xmin) is the Hurwitz zeta function. Parameters alpha : ﬂoat Exponent value of the distribution xmin : int Minimum value seed : int Seed value for random number generator Returns x : int Random value from Zipf distribution Raises ValueError: : 410 Chapter 12. Utilities NetworkX Reference, Release 1.7 If xmin < 1 or If alpha <= 1 Notes The rejection algorithm generates random values for a the power-law distribution in uniformly bounded expected time dependent on parameters. See [1] for details on its operation. References ..[1] Luc Devroye, Non-Uniform Random Variate Generation, Springer-Verlag, New York, 1986. http://cg.scs.carleton.ca/~luc/rnbookindex.html Examples >>> nx.zipf_rv(alpha=2, xmin=3, seed=42) 12.3.9 random_weighted_sample random_weighted_sample(mapping, k) Return k items without replacement from a weighted sample. The input is a dictionary of items with weights as values. 12.3.10 weighted_choice weighted_choice(mapping) Return a single element from a weighted sample. The input is a dictionary of items with weights as values. 12.4 Decorators open_file(path_arg[, mode]) Decorator to ensure clean opening and closing of ﬁles. require(*packages) Decorator to check whether speciﬁc packages can be imported. 12.4.1 open_ﬁle open_file(path_arg, mode=’r’) Decorator to ensure clean opening and closing of ﬁles. Parameters path_arg : int Location of the path argument in args. Even if the argument is a named positional argument (with a default value), you must specify its index as a positional argument. mode : str String for opening mode. 12.4. Decorators 411 NetworkX Reference, Release 1.7 Returns _open_ﬁle : function Function which cleanly executes the io. Examples Decorate functions like this: @open_file(0,’r’) def read_function(pathname): pass @open_file(1,’w’) def write_function(G,pathname): pass @open_file(1,’w’) def write_function(G, pathname=’graph.dot’) pass @open_file(’path’, ’w+’) def another_function(arg, **kwargs): path = kwargs[’path’] pass 12.4.2 require require(*packages) Decorator to check whether speciﬁc packages can be imported. If a package cannot be imported, then NetworkXError is raised. If all packages can be imported, then the original function is called. Parameters packages : container of strings Container of module names that will be imported. Returns _require : function The decorated function. Raises NetworkXError : If any of the packages cannot be imported : Examples Decorate functions like this: @require(’scipy’) def sp_function(): import scipy pass @require(’numpy’,’scipy’) def sp_np_function(): import numpy 412 Chapter 12. Utilities NetworkX Reference, Release 1.7 import scipy pass 12.4. Decorators 413 NetworkX Reference, Release 1.7 414 Chapter 12. Utilities CHAPTER THIRTEEN LICENSE NetworkX is distributed with the BSD license. Copyright (C) 2004-2012, NetworkX Developers Aric Hagberg Dan Schult Pieter Swart All rights reserved. Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: * Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. * Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. * Neither the name of the NetworkX Developers nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission. THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. 415 NetworkX Reference, Release 1.7 416 Chapter 13. License CHAPTER FOURTEEN CITING To cite NetworkX please use the following publication: Aric A. Hagberg, Daniel A. Schult and Pieter J. Swart, “Exploring network structure, dynamics, and function using NetworkX”, in Proceedings of the 7th Python in Science Conference (SciPy2008), Gäel Varoquaux, Travis Vaught, and Jarrod Millman (Eds), (Pasadena, CA USA), pp. 11–15, Aug 2008 417 NetworkX Reference, Release 1.7 418 Chapter 14. Citing CHAPTER FIFTEEN CREDITS NetworkX was originally written by Aric Hagberg, Dan Schult, and Pieter Swart, and has been developed with the help of many others. Thanks to Guido van Rossum for the idea of using Python for implementing a graph data structure http://www.python.org/doc/essays/graphs.html Thanks to David Eppstein for the idea of representing a graph G so that “for n in G” loops over the nodes in G and G[n] are node n’s neighbors. Thanks to everyone who has improved NetworkX by contributing code, bug reports (and ﬁxes), documentation, and input on design, featues, and the future of NetworkX. Thanks especially to the following contributors: • Katy Bold contributed the Karate Club graph. • Hernan Rozenfeld added dorogovtsev_goltsev_mendes_graph and did stress testing. • Brendt Wohlberg added examples from the Stanford GraphBase. • Jim Bagrow reported bugs in the search methods. • Holly Johnsen helped ﬁx the path based centrality measures. • Arnar Flatberg ﬁxed the graph laplacian routines. • Chris Myers suggested using None as a default datatype, suggested improvements for the IO routines, added grid generator index tuple labeling and associated routines, and reported bugs. • Joel Miller tested and improved the connected components methods ﬁxed bugs and typos in the graph generators, and contributed the random clustered graph generator. • Keith Briggs sorted out naming issues for random graphs and wrote dense_gnm_random_graph. • Ignacio Rozada provided the Krapivsky-Redner graph generator. • Phillipp Pagel helped ﬁx eccentricity etc. for disconnected graphs. • Sverre Sundsdal contributed bidirectional shortest path and Dijkstra routines, s-metric computation and graph generation • Ross M. Richardson contributed the expected degree graph generator and helped test the pygraphviz interface. • Christopher Ellison implemented the VF2 isomorphism algorithm and is a core developer. • Eben Kenah contributed the strongly connected components and DFS functions. • Sasha Gutfriend contributed edge betweenness algorithms. • Udi Weinsberg helped develop intersection and difference operators. 419 NetworkX Reference, Release 1.7 • Matteo Dell’Amico wrote the random regular graph generator. • Andrew Conway contributed ego_graph, eigenvector centrality, line graph and much more. • Raf Guns wrote the GraphML writer. • Salim Fadhley and Matteo Dell’Amico contributed the A* algorithm. • Fabrice Desclaux contributed the Matplotlib edge labeling code. • Arpad Horvath ﬁxed the barabasi_albert_graph() generator. • Minh Van Nguyen contributed the connected_watts_strogatz_graph() and documentation for the Graph and MultiGraph classes. • Willem Ligtenberg contributed the directed scale free graph generator. • Loïc Séguin-C. contributed the Ford-Fulkerson max ﬂow and min cut algorithms, and ported all of NetworkX to Python3. He is a NetworkX core developer. • Paul McGuire improved the performance of the GML data parser. • Jesus Cerquides contributed the chordal graph algorithms. • Ben Edwards contributed tree generating functions, the rich club coefﬁcient algorithm, the graph product func- tions, and a whole lot of other useful nuts and bolts. • Jon Olav Vik contributed cycle ﬁnding algorithms. • Hugh Brown improved the words.py example from the n^2 algorithm. • Ben Reilly contributed the shapeﬁle reader and writer. • Leo Lopes contributed the maximal independent set algorithm. • Jordi Torrents contributed the bipartite clustering, bipartite node redundancy, square clustering, other bipartite and articulation point algorithms. • Dheeraj M R contributed the distance-regular testing algorithm • Franck Kalala contributed the subgraph_centrality and communicability algorithms • Simon Knight improved the GraphML functions to handle yEd/yﬁles data, and to handle types correctly. • Conrad Lee contributed the k-clique community ﬁnding algorithm. • Sérgio Nery Simões wrote the function for ﬁnding all simple paths, and all shortest paths. • Robert King contributed union, disjoint union, compose, and intersection operators that work on lists of graphs. • Nick Mancuso wrote the approximation algorithms for dominating set, edge dominating set, independent set, max clique, and min-weighted vertex cover. 420 Chapter 15. Credits CHAPTER SIXTEEN GLOSSARY dictionary A Python dictionary maps keys to values. Also known as “hashes”, or “associative arrays”. See http://docs.python.org/tutorial/datastructures.html#dictionaries ebunch An iteratable container of edge tuples like a list, iterator, or ﬁle. edge Edges are either two-tuples of nodes (u,v) or three tuples of nodes with an edge attribute dictionary (u,v,dict). edge attribute Edges can have arbitrary Python objects assigned as attributes by using keyword/value pairs when adding an edge assigning to the G.edge[u][v] attribute dictionary for the speciﬁed edge u-v. hashable An object is hashable if it has a hash value which never changes during its lifetime (it needs a __hash__() method), and can be compared to other objects (it needs an __eq__() or __cmp__() method). Hashable objects which compare equal must have the same hash value. Hashability makes an object usable as a dictionary key and a set member, because these data structures use the hash value internally. All of Python’s immutable built-in objects are hashable, while no mutable containers (such as lists or dictionar- ies) are. Objects which are instances of user-deﬁned classes are hashable by default; they all compare unequal, and their hash value is their id(). Deﬁnition from http://docs.python.org/glossary.html nbunch An nbunch is any iterable container of nodes that is not itself a node in the graph. It can be an iterable or an iterator, e.g. a list, set, graph, ﬁle, etc.. node A node can be any hashable Python object except None. node attribute Nodes can have arbitrary Python objects assigned as attributes by using keyword/value pairs when adding a node or assigning to the G.node[n] attribute dictionary for the speciﬁed node n. 421 NetworkX Reference, Release 1.7 422 Chapter 16. Glossary BIBLIOGRAPHY [R104] Boppana, R., & Halldórsson, M. M. (1992). 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References, “Sparse Matrices”, http://docs.scipy.org/doc/scipy/reference/sparse.html [R248] GEXF graph format, http://gexf.net/format/ [R249] GEXF graph format, http://gexf.net/format/ [R250] http://docs.python.org/library/pickle.html [R251] http://docs.python.org/library/pickle.html [R253] http://www.algorithmic-solutions.info/leda_guide/graphs/leda_native_graph_ﬁleformat.html [R252] http://www.algorithmic-solutions.info/leda_guide/graphs/leda_native_graph_ﬁleformat.html Bibliography 429 NetworkX Reference, Release 1.7 [R256] http://www.yaml.org [R257] http://www.yaml.org [R254] http://en.wikipedia.org/wiki/Shapeﬁle [R255] http://en.wikipedia.org/wiki/Shapeﬁle 430 Bibliography PYTHON MODULE INDEX a networkx.algorithms.approximation, 127 networkx.algorithms.approximation.clique, 127 networkx.algorithms.approximation.dominating_set, 128 networkx.algorithms.approximation.independent_set, 129 networkx.algorithms.approximation.matching, 130 networkx.algorithms.approximation.ramsey, 130 networkx.algorithms.approximation.vertex_cover, 130 networkx.algorithms.assortativity, 131 networkx.algorithms.bipartite, 140 networkx.algorithms.bipartite.basic, 142 networkx.algorithms.bipartite.centrality, 155 networkx.algorithms.bipartite.cluster, 152 networkx.algorithms.bipartite.projection, 146 networkx.algorithms.bipartite.redundancy, 154 networkx.algorithms.bipartite.spectral, 151 networkx.algorithms.block, 158 networkx.algorithms.boundary, 159 networkx.algorithms.centrality, 160 networkx.algorithms.chordal.chordal_alg, 176 networkx.algorithms.clique, 179 networkx.algorithms.cluster, 181 networkx.algorithms.community, 185 networkx.algorithms.community.kclique, 185 networkx.algorithms.components, 186 networkx.algorithms.components.attracting, 192 networkx.algorithms.components.biconnected, 194 networkx.algorithms.components.connected, 186 networkx.algorithms.components.strongly_connected, 188 networkx.algorithms.components.weakly_connected, 192 networkx.algorithms.core, 199 networkx.algorithms.cycles, 203 networkx.algorithms.dag, 204 networkx.algorithms.distance_measures, 206 networkx.algorithms.distance_regular, 208 networkx.algorithms.euler, 210 networkx.algorithms.flow, 212 networkx.algorithms.graphical, 223 networkx.algorithms.hierarchy, 224 networkx.algorithms.isolate, 225 networkx.algorithms.isomorphism, 226 networkx.algorithms.isomorphism.isomorphvf2, 229 networkx.algorithms.link_analysis.hits_alg, 243 networkx.algorithms.link_analysis.pagerank_alg, 239 networkx.algorithms.matching, 245 networkx.algorithms.mis, 246 networkx.algorithms.mst, 247 networkx.algorithms.operators.all, 252 networkx.algorithms.operators.binary, 249 networkx.algorithms.operators.product, 254 networkx.algorithms.operators.unary, 249 networkx.algorithms.richclub, 256 networkx.algorithms.shortest_paths.astar, 274 networkx.algorithms.shortest_paths.dense, 272 networkx.algorithms.shortest_paths.generic, 431 NetworkX Reference, Release 1.7 257 networkx.algorithms.shortest_paths.unweighted, 261 networkx.algorithms.shortest_paths.weighted, 264 networkx.algorithms.simple_paths, 275 networkx.algorithms.swap, 276 networkx.algorithms.traversal.breadth_first_search, 279 networkx.algorithms.traversal.depth_first_search, 278 networkx.algorithms.vitality, 279 c networkx.classes.function, 281 networkx.convert, 341 d networkx.drawing.layout, 400 networkx.drawing.nx_agraph, 395 networkx.drawing.nx_pydot, 398 networkx.drawing.nx_pylab, 387 e networkx.exception, 405 g networkx.generators.atlas, 289 networkx.generators.bipartite, 321 networkx.generators.classic, 289 networkx.generators.degree_seq, 307 networkx.generators.directed, 314 networkx.generators.ego, 326 networkx.generators.geometric, 317 networkx.generators.hybrid, 320 networkx.generators.intersection, 327 networkx.generators.line, 325 networkx.generators.random_clustered, 312 networkx.generators.random_graphs, 298 networkx.generators.small, 294 networkx.generators.social, 329 networkx.generators.stochastic, 326 l networkx.linalg.attrmatrix, 335 networkx.linalg.graphmatrix, 331 networkx.linalg.laplacianmatrix, 332 networkx.linalg.spectrum, 334 r networkx.readwrite.adjlist, 351 networkx.readwrite.edgelist, 359 networkx.readwrite.gexf, 365 networkx.readwrite.gml, 367 networkx.readwrite.gpickle, 370 networkx.readwrite.graphml, 371 networkx.readwrite.json_graph, 373 networkx.readwrite.leda, 379 networkx.readwrite.multiline_adjlist, 355 networkx.readwrite.nx_shp, 384 networkx.readwrite.nx_yaml, 380 networkx.readwrite.pajek, 382 networkx.readwrite.sparsegraph6, 381 u networkx.utils, 407 networkx.utils.decorators, 411 networkx.utils.misc, 407 networkx.utils.random_sequence, 408 networkx.utils.union_find, 408 432 Python Module Index INDEX Symbols __contains__() (DiGraph method), 56 __contains__() (Graph method), 28 __contains__() (MultiDiGraph method), 116 __contains__() (MultiGraph method), 86 __getitem__() (DiGraph method), 52 __getitem__() (Graph method), 25 __getitem__() (MultiDiGraph method), 113 __getitem__() (MultiGraph method), 84 __init__() (DiGraph method), 38 __init__() (DiGraphMatcher method), 232 __init__() (Graph method), 12 __init__() (GraphMatcher method), 231 __init__() (MultiDiGraph method), 97 __init__() (MultiGraph method), 69 __iter__() (DiGraph method), 48 __iter__() (Graph method), 21 __iter__() (MultiDiGraph method), 108 __iter__() (MultiGraph method), 80 __len__() (DiGraph method), 57 __len__() (Graph method), 29 __len__() (MultiDiGraph method), 118 __len__() (MultiGraph method), 88 A add_cycle() (DiGraph method), 45 add_cycle() (Graph method), 19 add_cycle() (MultiDiGraph method), 105 add_cycle() (MultiGraph method), 77 add_edge() (DiGraph method), 41 add_edge() (Graph method), 15 add_edge() (MultiDiGraph method), 101 add_edge() (MultiGraph method), 72 add_edges_from() (DiGraph method), 42 add_edges_from() (Graph method), 16 add_edges_from() (MultiDiGraph method), 102 add_edges_from() (MultiGraph method), 73 add_node() (DiGraph method), 39 add_node() (Graph method), 13 add_node() (MultiDiGraph method), 98 add_node() (MultiGraph method), 69 add_nodes_from() (DiGraph method), 40 add_nodes_from() (Graph method), 13 add_nodes_from() (MultiDiGraph method), 99 add_nodes_from() (MultiGraph method), 70 add_path() (DiGraph method), 45 add_path() (Graph method), 19 add_path() (MultiDiGraph method), 105 add_path() (MultiGraph method), 76 add_star() (DiGraph method), 44 add_star() (Graph method), 18 add_star() (MultiDiGraph method), 104 add_star() (MultiGraph method), 76 add_weighted_edges_from() (DiGraph method), 43 add_weighted_edges_from() (Graph method), 17 add_weighted_edges_from() (MultiDiGraph method), 102 add_weighted_edges_from() (MultiGraph method), 74 adjacency_data() (in module net- workx.readwrite.json_graph), 374 adjacency_graph() (in module net- workx.readwrite.json_graph), 375 adjacency_iter() (DiGraph method), 54 adjacency_iter() (Graph method), 26 adjacency_iter() (MultiDiGraph method), 114 adjacency_iter() (MultiGraph method), 85 adjacency_list() (DiGraph method), 53 adjacency_list() (Graph method), 25 adjacency_list() (MultiDiGraph method), 114 adjacency_list() (MultiGraph method), 84 adjacency_matrix() (in module net- workx.linalg.graphmatrix), 331 adjacency_spectrum() (in module net- workx.linalg.spectrum), 334 all_neighbors() (in module networkx.classes.function), 283 all_pairs_dijkstra_path() (in module net- workx.algorithms.shortest_paths.weighted), 267 all_pairs_dijkstra_path_length() (in module net- workx.algorithms.shortest_paths.weighted), 268 all_pairs_shortest_path() (in module net- 433 NetworkX Reference, Release 1.7 workx.algorithms.shortest_paths.unweighted), 262 all_pairs_shortest_path_length() (in module net- workx.algorithms.shortest_paths.unweighted), 263 all_shortest_paths() (in module net- workx.algorithms.shortest_paths.generic), 258 all_simple_paths() (in module net- workx.algorithms.simple_paths), 275 approximate_current_ﬂow_betweenness_centrality() (in module networkx.algorithms.centrality), 167 articulation_points() (in module net- workx.algorithms.components.biconnected), 198 astar_path() (in module net- workx.algorithms.shortest_paths.astar), 274 astar_path_length() (in module net- workx.algorithms.shortest_paths.astar), 275 attr_matrix() (in module networkx.linalg.attrmatrix), 335 attr_sparse_matrix() (in module net- workx.linalg.attrmatrix), 337 attracting_component_subgraphs() (in module net- workx.algorithms.components.attracting), 194 attracting_components() (in module net- workx.algorithms.components.attracting), 193 attribute_assortativity_coefﬁcient() (in module net- workx.algorithms.assortativity), 132 attribute_mixing_dict() (in module net- workx.algorithms.assortativity), 140 attribute_mixing_matrix() (in module net- workx.algorithms.assortativity), 138 authority_matrix() (in module net- workx.algorithms.link_analysis.hits_alg), 245 average_clustering() (in module net- workx.algorithms.bipartite.cluster), 153 average_clustering() (in module net- workx.algorithms.cluster), 183 average_degree_connectivity() (in module net- workx.algorithms.assortativity), 136 average_neighbor_degree() (in module net- workx.algorithms.assortativity), 135 average_shortest_path_length() (in module net- workx.algorithms.shortest_paths.generic), 260 B balanced_tree() (in module networkx.generators.classic), 290 barabasi_albert_graph() (in module net- workx.generators.random_graphs), 304 barbell_graph() (in module networkx.generators.classic), 290 bellman_ford() (in module net- workx.algorithms.shortest_paths.weighted), 271 betweenness_centrality() (in module net- workx.algorithms.bipartite.centrality), 157 betweenness_centrality() (in module net- workx.algorithms.centrality), 163 bfs_edges() (in module net- workx.algorithms.traversal.breadth_ﬁrst_search), 279 bfs_predecessors() (in module net- workx.algorithms.traversal.breadth_ﬁrst_search), 279 bfs_successors() (in module net- workx.algorithms.traversal.breadth_ﬁrst_search), 279 bfs_tree() (in module net- workx.algorithms.traversal.breadth_ﬁrst_search), 279 biadjacency_matrix() (in module net- workx.algorithms.bipartite.basic), 145 biconnected_component_edges() (in module net- workx.algorithms.components.biconnected), 196 biconnected_component_subgraphs() (in module net- workx.algorithms.components.biconnected), 197 biconnected_components() (in module net- workx.algorithms.components.biconnected), 195 bidirectional_dijkstra() (in module net- workx.algorithms.shortest_paths.weighted), 269 binomial_graph() (in module net- workx.generators.random_graphs), 301 bipartite_alternating_havel_hakimi_graph() (in module networkx.generators.bipartite), 323 bipartite_conﬁguration_model() (in module net- workx.generators.bipartite), 321 bipartite_gnmk_random_graph() (in module net- workx.generators.bipartite), 324 bipartite_havel_hakimi_graph() (in module net- workx.generators.bipartite), 322 bipartite_preferential_attachment_graph() (in module networkx.generators.bipartite), 323 bipartite_random_graph() (in module net- workx.generators.bipartite), 324 bipartite_reverse_havel_hakimi_graph() (in module net- workx.generators.bipartite), 322 blockmodel() (in module networkx.algorithms.block), 158 bull_graph() (in module networkx.generators.small), 295 434 Index NetworkX Reference, Release 1.7 C candidate_pairs_iter() (DiGraphMatcher method), 233 candidate_pairs_iter() (GraphMatcher method), 232 cartesian_product() (in module net- workx.algorithms.operators.product), 254 categorical_edge_match() (in module net- workx.algorithms.isomorphism), 234 categorical_multiedge_match() (in module net- workx.algorithms.isomorphism), 235 categorical_node_match() (in module net- workx.algorithms.isomorphism), 234 center() (in module net- workx.algorithms.distance_measures), 207 chordal_graph_cliques() (in module net- workx.algorithms.chordal.chordal_alg), 177 chordal_graph_treewidth() (in module net- workx.algorithms.chordal.chordal_alg), 177 chvatal_graph() (in module networkx.generators.small), 295 circular_ladder_graph() (in module net- workx.generators.classic), 291 circular_layout() (in module networkx.drawing.layout), 401 clear() (DiGraph method), 46 clear() (Graph method), 20 clear() (MultiDiGraph method), 106 clear() (MultiGraph method), 77 clique_removal() (in module net- workx.algorithms.approximation.clique), 128 cliques_containing_node() (in module net- workx.algorithms.clique), 181 closeness_centrality() (in module net- workx.algorithms.bipartite.centrality), 155 closeness_centrality() (in module net- workx.algorithms.centrality), 162 closeness_vitality() (in module net- workx.algorithms.vitality), 279 clustering() (in module net- workx.algorithms.bipartite.cluster), 152 clustering() (in module networkx.algorithms.cluster), 182 collaboration_weighted_projected_graph() (in module networkx.algorithms.bipartite.projection), 148 color() (in module networkx.algorithms.bipartite.basic), 143 communicability() (in module net- workx.algorithms.centrality), 170 communicability_betweenness_centrality() (in module networkx.algorithms.centrality), 173 communicability_centrality() (in module net- workx.algorithms.centrality), 172 communicability_centrality_exp() (in module net- workx.algorithms.centrality), 173 communicability_exp() (in module net- workx.algorithms.centrality), 171 complement() (in module net- workx.algorithms.operators.unary), 249 complete_bipartite_graph() (in module net- workx.generators.classic), 291 complete_graph() (in module net- workx.generators.classic), 291 compose() (in module net- workx.algorithms.operators.binary), 249 compose_all() (in module net- workx.algorithms.operators.all), 252 condensation() (in module net- workx.algorithms.components.strongly_connected), 191 conﬁguration_model() (in module net- workx.generators.degree_seq), 307 connected_component_subgraphs() (in module net- workx.algorithms.components.connected), 187 connected_components() (in module net- workx.algorithms.components.connected), 187 connected_double_edge_swap() (in module net- workx.algorithms.swap), 277 connected_watts_strogatz_graph() (in module net- workx.generators.random_graphs), 303 copy() (DiGraph method), 63 copy() (Graph method), 33 copy() (MultiDiGraph method), 124 copy() (MultiGraph method), 92 core_number() (in module networkx.algorithms.core), 199 cost_of_ﬂow() (in module networkx.algorithms.ﬂow), 221 could_be_isomorphic() (in module net- workx.algorithms.isomorphism), 228 create_degree_sequence() (in module net- workx.utils.random_sequence), 409 create_empty_copy() (in module net- workx.classes.function), 282 cubical_graph() (in module networkx.generators.small), 295 cumulative_distribution() (in module net- workx.utils.random_sequence), 410 cumulative_sum() (in module networkx.utils.misc), 408 current_ﬂow_betweenness_centrality() (in module net- workx.algorithms.centrality), 165 current_ﬂow_closeness_centrality() (in module net- workx.algorithms.centrality), 164 cycle_basis() (in module networkx.algorithms.cycles), 203 cycle_graph() (in module networkx.generators.classic), 291 Index 435 NetworkX Reference, Release 1.7 D davis_southern_women_graph() (in module net- workx.generators.social), 329 default_opener() (in module networkx.utils.misc), 408 degree() (DiGraph method), 57 degree() (Graph method), 29 degree() (in module networkx.classes.function), 281 degree() (MultiDiGraph method), 118 degree() (MultiGraph method), 88 degree_assortativity_coefﬁcient() (in module net- workx.algorithms.assortativity), 131 degree_centrality() (in module net- workx.algorithms.bipartite.centrality), 156 degree_centrality() (in module net- workx.algorithms.centrality), 160 degree_histogram() (in module net- workx.classes.function), 281 degree_iter() (DiGraph method), 58 degree_iter() (Graph method), 30 degree_iter() (MultiDiGraph method), 118 degree_iter() (MultiGraph method), 89 degree_mixing_dict() (in module net- workx.algorithms.assortativity), 139 degree_mixing_matrix() (in module net- workx.algorithms.assortativity), 139 degree_pearson_correlation_coefﬁcient() (in module net- workx.algorithms.assortativity), 134 degree_sequence_tree() (in module net- workx.generators.degree_seq), 311 degrees() (in module net- workx.algorithms.bipartite.basic), 144 dense_gnm_random_graph() (in module net- workx.generators.random_graphs), 300 density() (in module net- workx.algorithms.bipartite.basic), 144 density() (in module networkx.classes.function), 282 desargues_graph() (in module net- workx.generators.small), 296 dfs_edges() (in module net- workx.algorithms.traversal.depth_ﬁrst_search), 278 dfs_labeled_edges() (in module net- workx.algorithms.traversal.depth_ﬁrst_search), 279 dfs_postorder_nodes() (in module net- workx.algorithms.traversal.depth_ﬁrst_search), 278 dfs_predecessors() (in module net- workx.algorithms.traversal.depth_ﬁrst_search), 278 dfs_preorder_nodes() (in module net- workx.algorithms.traversal.depth_ﬁrst_search), 278 dfs_successors() (in module net- workx.algorithms.traversal.depth_ﬁrst_search), 278 dfs_tree() (in module net- workx.algorithms.traversal.depth_ﬁrst_search), 278 diameter() (in module net- workx.algorithms.distance_measures), 207 diamond_graph() (in module networkx.generators.small), 296 dictionary, 421 difference() (in module net- workx.algorithms.operators.binary), 251 DiGraph() (in module networkx), 36 dijkstra_path() (in module net- workx.algorithms.shortest_paths.weighted), 264 dijkstra_path_length() (in module net- workx.algorithms.shortest_paths.weighted), 265 dijkstra_predecessor_and_distance() (in module net- workx.algorithms.shortest_paths.weighted), 270 directed_conﬁguration_model() (in module net- workx.generators.degree_seq), 309 discrete_sequence() (in module net- workx.utils.random_sequence), 410 disjoint_union() (in module net- workx.algorithms.operators.binary), 250 disjoint_union_all() (in module net- workx.algorithms.operators.all), 253 dodecahedral_graph() (in module net- workx.generators.small), 296 dorogovtsev_goltsev_mendes_graph() (in module net- workx.generators.classic), 291 double_edge_swap() (in module net- workx.algorithms.swap), 276 draw() (in module networkx.drawing.nx_pylab), 387 draw_circular() (in module networkx.drawing.nx_pylab), 395 draw_graphviz() (in module net- workx.drawing.nx_pylab), 395 draw_networkx() (in module net- workx.drawing.nx_pylab), 388 draw_networkx_edge_labels() (in module net- workx.drawing.nx_pylab), 394 draw_networkx_edges() (in module net- workx.drawing.nx_pylab), 392 draw_networkx_labels() (in module net- workx.drawing.nx_pylab), 393 draw_networkx_nodes() (in module net- workx.drawing.nx_pylab), 390 draw_random() (in module networkx.drawing.nx_pylab), 395 draw_shell() (in module networkx.drawing.nx_pylab), 436 Index NetworkX Reference, Release 1.7 395 draw_spectral() (in module networkx.drawing.nx_pylab), 395 draw_spring() (in module networkx.drawing.nx_pylab), 395 dump (in module networkx.readwrite.json_graph), 378 dumps (in module networkx.readwrite.json_graph), 377 E ebunch, 421 eccentricity() (in module net- workx.algorithms.distance_measures), 207 edge, 421 edge attribute, 421 edge_betweenness_centrality() (in module net- workx.algorithms.centrality), 164 edge_boundary() (in module net- workx.algorithms.boundary), 159 edge_current_ﬂow_betweenness_centrality() (in module networkx.algorithms.centrality), 166 edge_load() (in module networkx.algorithms.centrality), 176 edges() (DiGraph method), 48 edges() (Graph method), 22 edges() (in module networkx.classes.function), 284 edges() (MultiDiGraph method), 108 edges() (MultiGraph method), 80 edges_iter() (DiGraph method), 49 edges_iter() (Graph method), 22 edges_iter() (in module networkx.classes.function), 284 edges_iter() (MultiDiGraph method), 109 edges_iter() (MultiGraph method), 81 ego_graph() (in module networkx.generators.ego), 326 eigenvector_centrality() (in module net- workx.algorithms.centrality), 168 eigenvector_centrality_numpy() (in module net- workx.algorithms.centrality), 169 empty_graph() (in module networkx.generators.classic), 292 erdos_renyi_graph() (in module net- workx.generators.random_graphs), 301 estrada_index() (in module net- workx.algorithms.centrality), 174 eulerian_circuit() (in module networkx.algorithms.euler), 211 expected_degree_graph() (in module net- workx.generators.degree_seq), 310 F fast_could_be_isomorphic() (in module net- workx.algorithms.isomorphism), 228 fast_gnp_random_graph() (in module net- workx.generators.random_graphs), 298 faster_could_be_isomorphic() (in module net- workx.algorithms.isomorphism), 228 ﬁnd_cliques() (in module networkx.algorithms.clique), 179 ﬁnd_induced_nodes() (in module net- workx.algorithms.chordal.chordal_alg), 178 ﬂatten() (in module networkx.utils.misc), 407 ﬂorentine_families_graph() (in module net- workx.generators.social), 329 ﬂow_hierarchy() (in module net- workx.algorithms.hierarchy), 224 ﬂoyd_warshall() (in module net- workx.algorithms.shortest_paths.dense), 272 ﬂoyd_warshall_numpy() (in module net- workx.algorithms.shortest_paths.dense), 273 ﬂoyd_warshall_predecessor_and_distance() (in module networkx.algorithms.shortest_paths.dense), 273 ford_fulkerson() (in module networkx.algorithms.ﬂow), 214 ford_fulkerson_ﬂow() (in module net- workx.algorithms.ﬂow), 215 freeze() (in module networkx.classes.function), 286 from_agraph() (in module networkx.drawing.nx_agraph), 396 from_dict_of_dicts() (in module networkx.convert), 342 from_dict_of_lists() (in module networkx.convert), 343 from_edgelist() (in module networkx.convert), 344 from_numpy_matrix() (in module networkx.convert), 346 from_pydot() (in module networkx.drawing.nx_pydot), 399 from_scipy_sparse_matrix() (in module net- workx.convert), 348 frucht_graph() (in module networkx.generators.small), 296 G general_random_intersection_graph() (in module net- workx.generators.intersection), 328 generate_adjlist() (in module networkx.readwrite.adjlist), 354 generate_edgelist() (in module net- workx.readwrite.edgelist), 363 generate_gml() (in module networkx.readwrite.gml), 369 generate_multiline_adjlist() (in module net- workx.readwrite.multiline_adjlist), 358 generate_unique_node() (in module networkx.utils.misc), 408 generic_edge_match() (in module net- workx.algorithms.isomorphism), 238 generic_multiedge_match() (in module net- workx.algorithms.isomorphism), 238 Index 437 NetworkX Reference, Release 1.7 generic_node_match() (in module net- workx.algorithms.isomorphism), 237 generic_weighted_projected_graph() (in module net- workx.algorithms.bipartite.projection), 150 geographical_threshold_graph() (in module net- workx.generators.geometric), 318 get_edge_attributes() (in module net- workx.classes.function), 286 get_edge_data() (DiGraph method), 51 get_edge_data() (Graph method), 23 get_edge_data() (MultiDiGraph method), 112 get_edge_data() (MultiGraph method), 82 get_node_attributes() (in module net- workx.classes.function), 285 global_parameters() (in module net- workx.algorithms.distance_regular), 210 gn_graph() (in module networkx.generators.directed), 314 gnc_graph() (in module networkx.generators.directed), 315 gnm_random_graph() (in module net- workx.generators.random_graphs), 300 gnp_random_graph() (in module net- workx.generators.random_graphs), 299 gnr_graph() (in module networkx.generators.directed), 315 google_matrix() (in module net- workx.algorithms.link_analysis.pagerank_alg), 242 Graph() (in module networkx), 9 graph_atlas_g() (in module networkx.generators.atlas), 289 graph_clique_number() (in module net- workx.algorithms.clique), 180 graph_number_of_cliques() (in module net- workx.algorithms.clique), 181 graphviz_layout() (in module net- workx.drawing.nx_agraph), 397 graphviz_layout() (in module net- workx.drawing.nx_pydot), 400 grid_2d_graph() (in module networkx.generators.classic), 292 grid_graph() (in module networkx.generators.classic), 292 H has_edge() (DiGraph method), 56 has_edge() (Graph method), 28 has_edge() (MultiDiGraph method), 116 has_edge() (MultiGraph method), 87 has_node() (DiGraph method), 55 has_node() (Graph method), 27 has_node() (MultiDiGraph method), 116 has_node() (MultiGraph method), 86 has_path() (in module net- workx.algorithms.shortest_paths.generic), 261 hashable, 421 havel_hakimi_graph() (in module net- workx.generators.degree_seq), 311 heawood_graph() (in module networkx.generators.small), 296 hits() (in module net- workx.algorithms.link_analysis.hits_alg), 243 hits_numpy() (in module net- workx.algorithms.link_analysis.hits_alg), 244 hits_scipy() (in module net- workx.algorithms.link_analysis.hits_alg), 244 house_graph() (in module networkx.generators.small), 296 house_x_graph() (in module networkx.generators.small), 296 hub_matrix() (in module net- workx.algorithms.link_analysis.hits_alg), 245 hypercube_graph() (in module net- workx.generators.classic), 292 I icosahedral_graph() (in module net- workx.generators.small), 296 in_degree() (DiGraph method), 58 in_degree() (MultiDiGraph method), 119 in_degree_centrality() (in module net- workx.algorithms.centrality), 161 in_degree_iter() (DiGraph method), 59 in_degree_iter() (MultiDiGraph method), 120 in_edges() (DiGraph method), 51 in_edges() (MultiDiGraph method), 111 in_edges_iter() (DiGraph method), 51 in_edges_iter() (MultiDiGraph method), 111 incidence_matrix() (in module net- workx.linalg.graphmatrix), 332 info() (in module networkx.classes.function), 282 initialize() (DiGraphMatcher method), 233 initialize() (GraphMatcher method), 231 intersection() (in module net- workx.algorithms.operators.binary), 251 intersection_all() (in module net- workx.algorithms.operators.all), 254 intersection_array() (in module net- workx.algorithms.distance_regular), 209 is_aperiodic() (in module networkx.algorithms.dag), 206 is_attracting_component() (in module net- workx.algorithms.components.attracting), 438 Index NetworkX Reference, Release 1.7 193 is_biconnected() (in module net- workx.algorithms.components.biconnected), 194 is_bipartite() (in module net- workx.algorithms.bipartite.basic), 142 is_bipartite_node_set() (in module net- workx.algorithms.bipartite.basic), 142 is_chordal() (in module net- workx.algorithms.chordal.chordal_alg), 176 is_connected() (in module net- workx.algorithms.components.connected), 186 is_directed() (in module networkx.classes.function), 282 is_directed_acyclic_graph() (in module net- workx.algorithms.dag), 206 is_distance_regular() (in module net- workx.algorithms.distance_regular), 208 is_eulerian() (in module networkx.algorithms.euler), 210 is_frozen() (in module networkx.classes.function), 287 is_isolate() (in module networkx.algorithms.isolate), 225 is_isomorphic() (DiGraphMatcher method), 233 is_isomorphic() (GraphMatcher method), 231 is_isomorphic() (in module net- workx.algorithms.isomorphism), 226 is_kl_connected() (in module net- workx.generators.hybrid), 321 is_list_of_ints() (in module networkx.utils.misc), 408 is_string_like() (in module networkx.utils.misc), 407 is_strongly_connected() (in module net- workx.algorithms.components.strongly_connected), 189 is_valid_degree_sequence() (in module net- workx.algorithms.graphical), 223 is_valid_degree_sequence_erdos_gallai() (in module net- workx.algorithms.graphical), 224 is_valid_degree_sequence_havel_hakimi() (in module networkx.algorithms.graphical), 223 is_weakly_connected() (in module net- workx.algorithms.components.weakly_connected), 192 isolates() (in module networkx.algorithms.isolate), 225 isomorphisms_iter() (DiGraphMatcher method), 233 isomorphisms_iter() (GraphMatcher method), 231 iterable() (in module networkx.utils.misc), 407 K k_clique_communities() (in module net- workx.algorithms.community.kclique), 185 k_core() (in module networkx.algorithms.core), 200 k_corona() (in module networkx.algorithms.core), 202 k_crust() (in module networkx.algorithms.core), 201 k_nearest_neighbors() (in module net- workx.algorithms.assortativity), 137 k_random_intersection_graph() (in module net- workx.generators.intersection), 328 k_shell() (in module networkx.algorithms.core), 201 karate_club_graph() (in module net- workx.generators.social), 329 kl_connected_subgraph() (in module net- workx.generators.hybrid), 320 kosaraju_strongly_connected_components() (in module net- workx.algorithms.components.strongly_connected), 191 krackhardt_kite_graph() (in module net- workx.generators.small), 297 L ladder_graph() (in module networkx.generators.classic), 293 laplacian_matrix() (in module net- workx.linalg.laplacianmatrix), 333 laplacian_spectrum() (in module net- workx.linalg.spectrum), 334 LCF_graph() (in module networkx.generators.small), 295 lexicographic_product() (in module net- workx.algorithms.operators.product), 255 line_graph() (in module networkx.generators.line), 325 load (in module networkx.readwrite.json_graph), 378 load_centrality() (in module net- workx.algorithms.centrality), 175 loads (in module networkx.readwrite.json_graph), 377 lollipop_graph() (in module networkx.generators.classic), 293 M make_clique_bipartite() (in module net- workx.algorithms.clique), 180 make_max_clique_graph() (in module net- workx.algorithms.clique), 180 make_small_graph() (in module net- workx.generators.small), 294 make_str() (in module networkx.utils.misc), 408 match() (DiGraphMatcher method), 233 match() (GraphMatcher method), 232 max_clique() (in module net- workx.algorithms.approximation.clique), 127 max_ﬂow() (in module networkx.algorithms.ﬂow), 212 max_ﬂow_min_cost() (in module net- workx.algorithms.ﬂow), 221 max_weight_matching() (in module net- workx.algorithms.matching), 246 maximal_independent_set() (in module net- workx.algorithms.mis), 247 maximal_matching() (in module net- workx.algorithms.matching), 245 Index 439 NetworkX Reference, Release 1.7 maximum_independent_set() (in module net- workx.algorithms.approximation.independent_set), 129 min_cost_ﬂow() (in module networkx.algorithms.ﬂow), 220 min_cost_ﬂow_cost() (in module net- workx.algorithms.ﬂow), 218 min_cut() (in module networkx.algorithms.ﬂow), 213 min_edge_dominating_set() (in module net- workx.algorithms.approximation.dominating_set), 129 min_maximal_matching() (in module net- workx.algorithms.approximation.matching), 130 min_weighted_dominating_set() (in module net- workx.algorithms.approximation.dominating_set), 128 min_weighted_vertex_cover() (in module net- workx.algorithms.approximation.vertex_cover), 131 minimum_spanning_edges() (in module net- workx.algorithms.mst), 248 minimum_spanning_tree() (in module net- workx.algorithms.mst), 247 moebius_kantor_graph() (in module net- workx.generators.small), 297 MultiDiGraph() (in module networkx), 95 MultiGraph() (in module networkx), 66 N navigable_small_world_graph() (in module net- workx.generators.geometric), 319 nbunch, 421 nbunch_iter() (DiGraph method), 54 nbunch_iter() (Graph method), 26 nbunch_iter() (MultiDiGraph method), 115 nbunch_iter() (MultiGraph method), 85 negative_edge_cycle() (in module net- workx.algorithms.shortest_paths.weighted), 271 neighbors() (DiGraph method), 52 neighbors() (Graph method), 24 neighbors() (MultiDiGraph method), 112 neighbors() (MultiGraph method), 83 neighbors_iter() (DiGraph method), 52 neighbors_iter() (Graph method), 25 neighbors_iter() (MultiDiGraph method), 113 neighbors_iter() (MultiGraph method), 83 network_simplex() (in module net- workx.algorithms.ﬂow), 216 networkx.algorithms.approximation (module), 127 networkx.algorithms.approximation.clique (module), 127 networkx.algorithms.approximation.dominating_set (module), 128 networkx.algorithms.approximation.independent_set (module), 129 networkx.algorithms.approximation.matching (module), 130 networkx.algorithms.approximation.ramsey (module), 130 networkx.algorithms.approximation.vertex_cover (mod- ule), 130 networkx.algorithms.assortativity (module), 131 networkx.algorithms.bipartite (module), 140 networkx.algorithms.bipartite.basic (module), 142 networkx.algorithms.bipartite.centrality (module), 155 networkx.algorithms.bipartite.cluster (module), 152 networkx.algorithms.bipartite.projection (module), 146 networkx.algorithms.bipartite.redundancy (module), 154 networkx.algorithms.bipartite.spectral (module), 151 networkx.algorithms.block (module), 158 networkx.algorithms.boundary (module), 159 networkx.algorithms.centrality (module), 160 networkx.algorithms.chordal.chordal_alg (module), 176 networkx.algorithms.clique (module), 179 networkx.algorithms.cluster (module), 181 networkx.algorithms.community (module), 185 networkx.algorithms.community.kclique (module), 185 networkx.algorithms.components (module), 186 networkx.algorithms.components.attracting (module), 192 networkx.algorithms.components.biconnected (module), 194 networkx.algorithms.components.connected (module), 186 networkx.algorithms.components.strongly_connected (module), 188 networkx.algorithms.components.weakly_connected (module), 192 networkx.algorithms.core (module), 199 networkx.algorithms.cycles (module), 203 networkx.algorithms.dag (module), 204 networkx.algorithms.distance_measures (module), 206 networkx.algorithms.distance_regular (module), 208 networkx.algorithms.euler (module), 210 networkx.algorithms.ﬂow (module), 212 networkx.algorithms.graphical (module), 223 networkx.algorithms.hierarchy (module), 224 networkx.algorithms.isolate (module), 225 networkx.algorithms.isomorphism (module), 226 networkx.algorithms.isomorphism.isomorphvf2 (mod- ule), 229 networkx.algorithms.link_analysis.hits_alg (module), 243 networkx.algorithms.link_analysis.pagerank_alg (mod- ule), 239 networkx.algorithms.matching (module), 245 networkx.algorithms.mis (module), 246 440 Index NetworkX Reference, Release 1.7 networkx.algorithms.mst (module), 247 networkx.algorithms.operators.all (module), 252 networkx.algorithms.operators.binary (module), 249 networkx.algorithms.operators.product (module), 254 networkx.algorithms.operators.unary (module), 249 networkx.algorithms.richclub (module), 256 networkx.algorithms.shortest_paths.astar (module), 274 networkx.algorithms.shortest_paths.dense (module), 272 networkx.algorithms.shortest_paths.generic (module), 257 networkx.algorithms.shortest_paths.unweighted (mod- ule), 261 networkx.algorithms.shortest_paths.weighted (module), 264 networkx.algorithms.simple_paths (module), 275 networkx.algorithms.swap (module), 276 networkx.algorithms.traversal.breadth_ﬁrst_search (mod- ule), 279 networkx.algorithms.traversal.depth_ﬁrst_search (mod- ule), 278 networkx.algorithms.vitality (module), 279 networkx.classes.function (module), 281 networkx.convert (module), 341 networkx.drawing.layout (module), 400 networkx.drawing.nx_agraph (module), 395 networkx.drawing.nx_pydot (module), 398 networkx.drawing.nx_pylab (module), 387 networkx.exception (module), 405 networkx.generators.atlas (module), 289 networkx.generators.bipartite (module), 321 networkx.generators.classic (module), 289 networkx.generators.degree_seq (module), 307 networkx.generators.directed (module), 314 networkx.generators.ego (module), 326 networkx.generators.geometric (module), 317 networkx.generators.hybrid (module), 320 networkx.generators.intersection (module), 327 networkx.generators.line (module), 325 networkx.generators.random_clustered (module), 312 networkx.generators.random_graphs (module), 298 networkx.generators.small (module), 294 networkx.generators.social (module), 329 networkx.generators.stochastic (module), 326 networkx.linalg.attrmatrix (module), 335 networkx.linalg.graphmatrix (module), 331 networkx.linalg.laplacianmatrix (module), 332 networkx.linalg.spectrum (module), 334 networkx.readwrite.adjlist (module), 351 networkx.readwrite.edgelist (module), 359 networkx.readwrite.gexf (module), 365 networkx.readwrite.gml (module), 367 networkx.readwrite.gpickle (module), 370 networkx.readwrite.graphml (module), 371 networkx.readwrite.json_graph (module), 373 networkx.readwrite.leda (module), 379 networkx.readwrite.multiline_adjlist (module), 355 networkx.readwrite.nx_shp (module), 384 networkx.readwrite.nx_yaml (module), 380 networkx.readwrite.pajek (module), 382 networkx.readwrite.sparsegraph6 (module), 381 networkx.utils (module), 407 networkx.utils.decorators (module), 411 networkx.utils.misc (module), 407 networkx.utils.random_sequence (module), 408 networkx.utils.union_ﬁnd (module), 408 NetworkXAlgorithmError (class in networkx), 405 NetworkXError (class in networkx), 405 NetworkXException (class in networkx), 405 NetworkXNoPath (class in networkx), 405 NetworkXPointlessConcept (class in networkx), 405 NetworkXUnbounded (class in networkx), 405 NetworkXUnfeasible (class in networkx), 405 newman_watts_strogatz_graph() (in module net- workx.generators.random_graphs), 302 node, 421 node attribute, 421 node_boundary() (in module net- workx.algorithms.boundary), 160 node_clique_number() (in module net- workx.algorithms.clique), 181 node_connected_component() (in module net- workx.algorithms.components.connected), 188 node_link_data() (in module net- workx.readwrite.json_graph), 373 node_link_graph() (in module net- workx.readwrite.json_graph), 374 node_redundancy() (in module net- workx.algorithms.bipartite.redundancy), 154 nodes() (DiGraph method), 47 nodes() (Graph method), 20 nodes() (in module networkx.classes.function), 283 nodes() (MultiDiGraph method), 107 nodes() (MultiGraph method), 79 nodes_iter() (DiGraph method), 47 nodes_iter() (Graph method), 21 nodes_iter() (in module networkx.classes.function), 283 nodes_iter() (MultiDiGraph method), 107 nodes_iter() (MultiGraph method), 79 nodes_with_selﬂoops() (DiGraph method), 62 nodes_with_selﬂoops() (Graph method), 32 nodes_with_selﬂoops() (MultiDiGraph method), 122 nodes_with_selﬂoops() (MultiGraph method), 91 non_neighbors() (in module networkx.classes.function), 283 normalized_laplacian_matrix() (in module net- workx.linalg.laplacianmatrix), 333 Index 441 NetworkX Reference, Release 1.7 null_graph() (in module networkx.generators.classic), 293 number_attracting_components() (in module net- workx.algorithms.components.attracting), 193 number_connected_components() (in module net- workx.algorithms.components.connected), 187 number_of_cliques() (in module net- workx.algorithms.clique), 181 number_of_edges() (DiGraph method), 61 number_of_edges() (Graph method), 31 number_of_edges() (in module net- workx.classes.function), 284 number_of_edges() (MultiDiGraph method), 122 number_of_edges() (MultiGraph method), 90 number_of_nodes() (DiGraph method), 57 number_of_nodes() (Graph method), 29 number_of_nodes() (in module net- workx.classes.function), 283 number_of_nodes() (MultiDiGraph method), 117 number_of_nodes() (MultiGraph method), 88 number_of_selﬂoops() (DiGraph method), 63 number_of_selﬂoops() (Graph method), 33 number_of_selﬂoops() (MultiDiGraph method), 123 number_of_selﬂoops() (MultiGraph method), 92 number_strongly_connected_components() (in module net- workx.algorithms.components.strongly_connected), 189 number_weakly_connected_components() (in module net- workx.algorithms.components.weakly_connected), 192 numeric_assortativity_coefﬁcient() (in module net- workx.algorithms.assortativity), 133 numerical_edge_match() (in module net- workx.algorithms.isomorphism), 236 numerical_multiedge_match() (in module net- workx.algorithms.isomorphism), 237 numerical_node_match() (in module net- workx.algorithms.isomorphism), 235 O octahedral_graph() (in module net- workx.generators.small), 297 open_ﬁle() (in module networkx.utils.decorators), 411 order() (DiGraph method), 56 order() (Graph method), 28 order() (MultiDiGraph method), 117 order() (MultiGraph method), 87 out_degree() (DiGraph method), 60 out_degree() (MultiDiGraph method), 120 out_degree_centrality() (in module net- workx.algorithms.centrality), 161 out_degree_iter() (DiGraph method), 60 out_degree_iter() (MultiDiGraph method), 121 out_edges() (DiGraph method), 49 out_edges() (MultiDiGraph method), 109 out_edges_iter() (DiGraph method), 50 out_edges_iter() (MultiDiGraph method), 110 overlap_weighted_projected_graph() (in module net- workx.algorithms.bipartite.projection), 149 P pagerank() (in module net- workx.algorithms.link_analysis.pagerank_alg), 239 pagerank_numpy() (in module net- workx.algorithms.link_analysis.pagerank_alg), 240 pagerank_scipy() (in module net- workx.algorithms.link_analysis.pagerank_alg), 241 pappus_graph() (in module networkx.generators.small), 297 pareto_sequence() (in module net- workx.utils.random_sequence), 409 parse_adjlist() (in module networkx.readwrite.adjlist), 353 parse_edgelist() (in module networkx.readwrite.edgelist), 364 parse_gml() (in module networkx.readwrite.gml), 369 parse_graph6() (in module net- workx.readwrite.sparsegraph6), 382 parse_leda() (in module networkx.readwrite.leda), 379 parse_multiline_adjlist() (in module net- workx.readwrite.multiline_adjlist), 357 parse_pajek() (in module networkx.readwrite.pajek), 384 parse_sparse6() (in module net- workx.readwrite.sparsegraph6), 382 path_graph() (in module networkx.generators.classic), 293 periphery() (in module net- workx.algorithms.distance_measures), 208 petersen_graph() (in module networkx.generators.small), 297 powerlaw_cluster_graph() (in module net- workx.generators.random_graphs), 305 powerlaw_sequence() (in module net- workx.utils.random_sequence), 409 predecessor() (in module net- workx.algorithms.shortest_paths.unweighted), 263 predecessors() (DiGraph method), 53 predecessors() (MultiDiGraph method), 114 predecessors_iter() (DiGraph method), 53 442 Index NetworkX Reference, Release 1.7 predecessors_iter() (MultiDiGraph method), 114 projected_graph() (in module net- workx.algorithms.bipartite.projection), 146 pydot_layout() (in module networkx.drawing.nx_pydot), 400 pygraphviz_layout() (in module net- workx.drawing.nx_agraph), 398 R radius() (in module net- workx.algorithms.distance_measures), 208 ramsey_R2() (in module net- workx.algorithms.approximation.ramsey), 130 random_clustered_graph() (in module net- workx.generators.random_clustered), 313 random_degree_sequence_graph() (in module net- workx.generators.degree_seq), 312 random_geometric_graph() (in module net- workx.generators.geometric), 317 random_layout() (in module networkx.drawing.layout), 401 random_lobster() (in module net- workx.generators.random_graphs), 305 random_powerlaw_tree() (in module net- workx.generators.random_graphs), 306 random_powerlaw_tree_sequence() (in module net- workx.generators.random_graphs), 307 random_regular_graph() (in module net- workx.generators.random_graphs), 304 random_shell_graph() (in module net- workx.generators.random_graphs), 306 random_weighted_sample() (in module net- workx.utils.random_sequence), 411 read_adjlist() (in module networkx.readwrite.adjlist), 351 read_dot() (in module networkx.drawing.nx_agraph), 397 read_dot() (in module networkx.drawing.nx_pydot), 400 read_edgelist() (in module networkx.readwrite.edgelist), 359 read_gexf() (in module networkx.readwrite.gexf), 365 read_gml() (in module networkx.readwrite.gml), 367 read_gpickle() (in module networkx.readwrite.gpickle), 370 read_graph6() (in module net- workx.readwrite.sparsegraph6), 381 read_graph6_list() (in module net- workx.readwrite.sparsegraph6), 382 read_graphml() (in module networkx.readwrite.graphml), 372 read_leda() (in module networkx.readwrite.leda), 379 read_multiline_adjlist() (in module net- workx.readwrite.multiline_adjlist), 355 read_pajek() (in module networkx.readwrite.pajek), 383 read_shp() (in module networkx.readwrite.nx_shp), 384 read_sparse6() (in module net- workx.readwrite.sparsegraph6), 382 read_sparse6_list() (in module net- workx.readwrite.sparsegraph6), 382 read_weighted_edgelist() (in module net- workx.readwrite.edgelist), 361 read_yaml() (in module networkx.readwrite.nx_yaml), 380 relabel_gexf_graph() (in module net- workx.readwrite.gexf), 367 remove_edge() (DiGraph method), 43 remove_edge() (Graph method), 17 remove_edge() (MultiDiGraph method), 103 remove_edge() (MultiGraph method), 74 remove_edges_from() (DiGraph method), 44 remove_edges_from() (Graph method), 18 remove_edges_from() (MultiDiGraph method), 104 remove_edges_from() (MultiGraph method), 75 remove_node() (DiGraph method), 40 remove_node() (Graph method), 14 remove_node() (MultiDiGraph method), 100 remove_node() (MultiGraph method), 71 remove_nodes_from() (DiGraph method), 41 remove_nodes_from() (Graph method), 15 remove_nodes_from() (MultiDiGraph method), 100 remove_nodes_from() (MultiGraph method), 71 require() (in module networkx.utils.decorators), 412 reverse() (DiGraph method), 66 reverse() (MultiDiGraph method), 126 rich_club_coefﬁcient() (in module net- workx.algorithms.richclub), 256 S scale_free_graph() (in module net- workx.generators.directed), 316 sedgewick_maze_graph() (in module net- workx.generators.small), 297 selﬂoop_edges() (DiGraph method), 62 selﬂoop_edges() (Graph method), 32 selﬂoop_edges() (MultiDiGraph method), 123 selﬂoop_edges() (MultiGraph method), 91 semantic_feasibility() (DiGraphMatcher method), 234 semantic_feasibility() (GraphMatcher method), 232 set_edge_attributes() (in module net- workx.classes.function), 285 set_node_attributes() (in module net- workx.classes.function), 284 sets() (in module networkx.algorithms.bipartite.basic), 143 shell_layout() (in module networkx.drawing.layout), 402 shortest_path() (in module net- workx.algorithms.shortest_paths.generic), 258 Index 443 NetworkX Reference, Release 1.7 shortest_path_length() (in module net- workx.algorithms.shortest_paths.generic), 259 simple_cycles() (in module networkx.algorithms.cycles), 204 single_source_dijkstra() (in module net- workx.algorithms.shortest_paths.weighted), 268 single_source_dijkstra_path() (in module net- workx.algorithms.shortest_paths.weighted), 266 single_source_dijkstra_path_length() (in module net- workx.algorithms.shortest_paths.weighted), 266 single_source_shortest_path() (in module net- workx.algorithms.shortest_paths.unweighted), 261 single_source_shortest_path_length() (in module net- workx.algorithms.shortest_paths.unweighted), 262 size() (DiGraph method), 61 size() (Graph method), 30 size() (MultiDiGraph method), 121 size() (MultiGraph method), 89 spectral_bipartivity() (in module net- workx.algorithms.bipartite.spectral), 151 spectral_layout() (in module networkx.drawing.layout), 403 spring_layout() (in module networkx.drawing.layout), 402 square_clustering() (in module net- workx.algorithms.cluster), 184 star_graph() (in module networkx.generators.classic), 293 stochastic_graph() (in module net- workx.generators.stochastic), 327 strong_product() (in module net- workx.algorithms.operators.product), 255 strongly_connected_component_subgraphs() (in module net- workx.algorithms.components.strongly_connected), 190 strongly_connected_components() (in module net- workx.algorithms.components.strongly_connected), 189 strongly_connected_components_recursive() (in module net- workx.algorithms.components.strongly_connected), 190 subgraph() (DiGraph method), 65 subgraph() (Graph method), 35 subgraph() (MultiDiGraph method), 126 subgraph() (MultiGraph method), 94 subgraph_is_isomorphic() (DiGraphMatcher method), 233 subgraph_is_isomorphic() (GraphMatcher method), 231 subgraph_isomorphisms_iter() (DiGraphMatcher method), 233 subgraph_isomorphisms_iter() (GraphMatcher method), 232 successors() (DiGraph method), 53 successors() (MultiDiGraph method), 113 successors_iter() (DiGraph method), 53 successors_iter() (MultiDiGraph method), 113 symmetric_difference() (in module net- workx.algorithms.operators.binary), 252 syntactic_feasibility() (DiGraphMatcher method), 234 syntactic_feasibility() (GraphMatcher method), 232 T tensor_product() (in module net- workx.algorithms.operators.product), 256 tetrahedral_graph() (in module net- workx.generators.small), 297 to_agraph() (in module networkx.drawing.nx_agraph), 396 to_dict_of_dicts() (in module networkx.convert), 342 to_dict_of_lists() (in module networkx.convert), 343 to_directed() (DiGraph method), 64 to_directed() (Graph method), 34 to_directed() (MultiDiGraph method), 125 to_directed() (MultiGraph method), 93 to_edgelist() (in module networkx.convert), 344 to_networkx_graph() (in module networkx.convert), 341 to_numpy_matrix() (in module networkx.convert), 344 to_numpy_recarray() (in module networkx.convert), 346 to_pydot() (in module networkx.drawing.nx_pydot), 399 to_scipy_sparse_matrix() (in module networkx.convert), 347 to_undirected() (DiGraph method), 64 to_undirected() (Graph method), 34 to_undirected() (MultiDiGraph method), 124 to_undirected() (MultiGraph method), 93 topological_sort() (in module networkx.algorithms.dag), 205 topological_sort_recursive() (in module net- workx.algorithms.dag), 205 transitivity() (in module networkx.algorithms.cluster), 182 tree_data() (in module networkx.readwrite.json_graph), 376 tree_graph() (in module networkx.readwrite.json_graph), 376 triangles() (in module networkx.algorithms.cluster), 181 trivial_graph() (in module networkx.generators.classic), 293 truncated_cube_graph() (in module net- workx.generators.small), 297 444 Index NetworkX Reference, Release 1.7 truncated_tetrahedron_graph() (in module net- workx.generators.small), 298 tutte_graph() (in module networkx.generators.small), 298 U uniform_random_intersection_graph() (in module net- workx.generators.intersection), 327 uniform_sequence() (in module net- workx.utils.random_sequence), 410 union() (in module net- workx.algorithms.operators.binary), 250 union() (UnionFind method), 408 union_all() (in module net- workx.algorithms.operators.all), 253 W watts_strogatz_graph() (in module net- workx.generators.random_graphs), 302 waxman_graph() (in module net- workx.generators.geometric), 319 weakly_connected_component_subgraphs() (in module net- workx.algorithms.components.weakly_connected), 192 weakly_connected_components() (in module net- workx.algorithms.components.weakly_connected), 192 weighted_choice() (in module net- workx.utils.random_sequence), 411 weighted_projected_graph() (in module net- workx.algorithms.bipartite.projection), 147 wheel_graph() (in module networkx.generators.classic), 294 write_adjlist() (in module networkx.readwrite.adjlist), 353 write_dot() (in module networkx.drawing.nx_agraph), 397 write_dot() (in module networkx.drawing.nx_pydot), 399 write_edgelist() (in module networkx.readwrite.edgelist), 360 write_gexf() (in module networkx.readwrite.gexf), 366 write_gml() (in module networkx.readwrite.gml), 368 write_gpickle() (in module networkx.readwrite.gpickle), 371 write_graphml() (in module net- workx.readwrite.graphml), 372 write_multiline_adjlist() (in module net- workx.readwrite.multiline_adjlist), 356 write_pajek() (in module networkx.readwrite.pajek), 383 write_shp() (in module networkx.readwrite.nx_shp), 385 write_weighted_edgelist() (in module net- workx.readwrite.edgelist), 362 write_yaml() (in module networkx.readwrite.nx_yaml), 381 Z zipf_rv() (in module networkx.utils.random_sequence), 410 zipf_sequence() (in module net- workx.utils.random_sequence), 410 Index 445