Warning

This documents an unmaintained version of NetworkX. Please upgrade to a maintained version and see the current NetworkX documentation.

# Source code for networkx.algorithms.bipartite.cluster

#-*- coding: utf-8 -*-
#    Jordi Torrents <jtorrents@milnou.net>
#    Aric Hagberg <hagberg@lanl.gov>
import itertools
import networkx as nx
__author__ = """\n""".join(['Jordi Torrents <jtorrents@milnou.net>',
'Aric Hagberg (hagberg@lanl.gov)'])
__all__ = ['clustering',
'average_clustering',
'latapy_clustering',
'robins_alexander_clustering']

# functions for computing clustering of pairs

def cc_dot(nu, nv):
return float(len(nu & nv)) / len(nu | nv)

def cc_max(nu, nv):
return float(len(nu & nv)) / max(len(nu), len(nv))

def cc_min(nu, nv):
return float(len(nu & nv)) / min(len(nu), len(nv))

modes = {'dot': cc_dot,
'min': cc_min,
'max': cc_max}

[docs]def latapy_clustering(G, nodes=None, mode='dot'): r"""Compute a bipartite clustering coefficient for nodes. The bipartie clustering coefficient is a measure of local density of connections defined as [1]_: .. math:: c_u = \frac{\sum_{v \in N(N(u))} c_{uv} }{|N(N(u))|} where N(N(u)) are the second order neighbors of u in G excluding u, and c_{uv} is the pairwise clustering coefficient between nodes u and v. The mode selects the function for c_{uv} which can be: dot: .. math:: c_{uv}=\frac{|N(u)\cap N(v)|}{|N(u) \cup N(v)|} min: .. math:: c_{uv}=\frac{|N(u)\cap N(v)|}{min(|N(u)|,|N(v)|)} max: .. math:: c_{uv}=\frac{|N(u)\cap 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. Examples -------- >>> from networkx.algorithms import bipartite >>> G = nx.path_graph(4) # path graphs are bipartite >>> c = bipartite.clustering(G) >>> c[0] 0.5 >>> c = bipartite.clustering(G,mode='min') >>> c[0] 1.0 See Also -------- robins_alexander_clustering square_clustering average_clustering References ---------- .. [1] Latapy, Matthieu, Clémence Magnien, and Nathalie Del Vecchio (2008). Basic notions for the analysis of large two-mode networks. Social Networks 30(1), 31--48. """ if not nx.algorithms.bipartite.is_bipartite(G): raise nx.NetworkXError("Graph is not bipartite") try: cc_func = modes[mode] except KeyError: raise nx.NetworkXError( "Mode for bipartite clustering must be: dot, min or max") if nodes is None: nodes = G ccs = {} for v in nodes: cc = 0.0 nbrs2 = set([u for nbr in G[v] for u in G[nbr]]) - set([v]) for u in nbrs2: cc += cc_func(set(G[u]), set(G[v])) if cc > 0.0: # len(nbrs2)>0 cc /= len(nbrs2) ccs[v] = cc return ccs
clustering = latapy_clustering
[docs]def average_clustering(G, nodes=None, mode='dot'): r"""Compute the average bipartite clustering coefficient. A clustering coefficient for the whole graph is the average, .. math:: C = \frac{1}{n}\sum_{v \in G} c_v, where n is the number of nodes in G. Similar measures for the two bipartite sets can be defined [1]_ .. math:: C_X = \frac{1}{|X|}\sum_{v \in X} c_v, where X is a bipartite set of G. 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. Examples -------- >>> from networkx.algorithms import bipartite >>> G=nx.star_graph(3) # star graphs are 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 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. See :mod:bipartite documentation <networkx.algorithms.bipartite> for further details on how bipartite graphs are handled in NetworkX. References ---------- .. [1] Latapy, Matthieu, Clémence Magnien, and Nathalie Del Vecchio (2008). Basic notions for the analysis of large two-mode networks. Social Networks 30(1), 31--48. """ if nodes is None: nodes = G ccs = latapy_clustering(G, nodes=nodes, mode=mode) return float(sum(ccs[v] for v in nodes)) / len(nodes)
[docs]def robins_alexander_clustering(G): r"""Compute the bipartite clustering of G. Robins and Alexander [1]_ defined bipartite clustering coefficient as four times the number of four cycles C_4 divided by the number of three paths L_3 in a bipartite graph: .. math:: CC_4 = \frac{4 * C_4}{L_3} Parameters ---------- G : graph a bipartite graph Returns ------- clustering : float The Robins and Alexander bipartite clustering for the input graph. Examples -------- >>> from networkx.algorithms import bipartite >>> G = nx.davis_southern_women_graph() >>> print(round(bipartite.robins_alexander_clustering(G), 3)) 0.468 See Also -------- latapy_clustering square_clustering References ---------- .. [1] Robins, G. and M. Alexander (2004). Small worlds among interlocking directors: Network structure and distance in bipartite graphs. Computational & Mathematical Organization Theory 10(1), 69–94. """ if G.order() < 4 or G.size() < 3: return 0 L_3 = _threepaths(G) if L_3 == 0: return 0 C_4 = _four_cycles(G) return (4. * C_4) / L_3
def _four_cycles(G): cycles = 0 for v in G: for u, w in itertools.combinations(G[v], 2): cycles += len((set(G[u]) & set(G[w])) - set([v])) return cycles / 4 def _threepaths(G): paths = 0 for v in G: for u in G[v]: for w in set(G[u]) - set([v]): paths += len(set(G[w]) - set([v, u])) # Divide by two because we count each three path twice # one for each possible starting point return paths / 2