# -*- coding: utf-8 -*- """ ========================== Bipartite Graph Algorithms ========================== """ # Copyright (C) 2012 by # Aric Hagberg # Dan Schult # Pieter Swart # All rights reserved. # BSD license. import networkx as nx from itertools import count __author__ = """\n""".join(['Jordi Torrents ', 'Aric Hagberg ']) __all__ = [ 'is_bipartite', 'is_bipartite_node_set', 'color', 'sets', 'density', 'degrees', 'biadjacency_matrix'] def biadjacency_matrix(G, row_order, column_order=None, weight='weight', dtype=None): r"""Return the biadjacency matrix of the bipartite graph G. Let `G = (U, V, E)` be a bipartite graph with node sets `U = u_{1},...,u_{r}` and `V = v_{1},...,v_{s}`. The biadjacency matrix [1] is the `r` x `s` matrix `B` in which `b_{i,j} = 1` if, and only if, `(u_i, v_j) \in E`. If the parameter `weight` is not `None` and 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. Notes ----- No attempt is made to check that the input graph is bipartite. 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. See Also -------- to_numpy_matrix adjacency_matrix References ---------- [1] http://en.wikipedia.org/wiki/Adjacency_matrix#Adjacency_matrix_of_a_bipartite_graph """ try: import numpy as np except ImportError: raise ImportError('adjacency_matrix() requires numpy ', 'http://scipy.org/') if column_order is None: column_order = list(set(G) - set(row_order)) row = dict(zip(row_order,count())) col = dict(zip(column_order,count())) M = np.zeros((len(row),len(col)), dtype=dtype) for u in row_order: for v, d in G[u].items(): M[row[u],col[v]] = d.get(weight, 1) return np.asmatrix(M) def 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) >>> 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 """ if G.is_directed(): import itertools def neighbors(v): return itertools.chain.from_iterable([G.predecessors_iter(v), G.successors_iter(v)]) else: neighbors=G.neighbors_iter color = {} for n in G: # handle disconnected graphs if n in color or len(G[n])==0: # skip isolates continue queue = [n] color[n] = 1 # nodes seen with color (1 or 0) while queue: v = queue.pop() c = 1 - color[v] # opposite color of node v for w in neighbors(v): if w in color: if color[w] == color[v]: raise nx.NetworkXError("Graph is not bipartite.") else: color[w] = c queue.append(w) # color isolates with 0 color.update(dict.fromkeys(nx.isolates(G),0)) return color def is_bipartite(G): """ Returns True if graph G is bipartite, False if not. Parameters ---------- G : NetworkX graph Examples -------- >>> from networkx.algorithms import bipartite >>> G = nx.path_graph(4) >>> print(bipartite.is_bipartite(G)) True See Also -------- color, is_bipartite_node_set """ try: color(G) return True except nx.NetworkXError: return False def 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. Examples -------- >>> from networkx.algorithms import bipartite >>> G = nx.path_graph(4) >>> X = set([1,3]) >>> bipartite.is_bipartite_node_set(G,X) True Notes ----- For connected graphs the bipartite sets are unique. This function handles disconnected graphs. """ S=set(nodes) for CC in nx.connected_component_subgraphs(G): X,Y=sets(CC) if not ( (X.issubset(S) and Y.isdisjoint(S)) or (Y.issubset(S) and X.isdisjoint(S)) ): return False return True def 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. Examples -------- >>> from networkx.algorithms import bipartite >>> G = nx.path_graph(4) >>> X, Y = bipartite.sets(G) >>> list(X) [0, 2] >>> list(Y) [1, 3] See Also -------- color """ c = color(G) X = set(n for n in c if c[n]) # c[n] == 1 Y = set(n for n in c if not c[n]) # c[n] == 0 return (X, Y) def 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 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 See Also -------- color """ n=len(B) m=nx.number_of_edges(B) nb=len(nodes) nt=n-nb if m==0: # includes cases n==0 and n==1 d=0.0 else: if B.is_directed(): d=m/(2.0*float(nb*nt)) else: d= m/float(nb*nt) return d def 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 The degrees of the two bipartite sets as dictionaries keyed by node. 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} See Also -------- color, density """ bottom=set(nodes) top=set(B)-bottom return (B.degree(top,weight),B.degree(bottom,weight)) # fixture for nose tests def setup_module(module): from nose import SkipTest try: import numpy except: raise SkipTest("NumPy not available")