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Source code for networkx.algorithms.bipartite.redundancy

# -*- coding: utf-8 -*-
"""Node redundancy for bipartite graphs."""
#    Copyright (C) 2011 by
#    Jordi Torrents <jtorrents@milnou.net>
#    Aric Hagberg <hagberg@lanl.gov>
#    All rights reserved.
#    BSD license.
from __future__ import division

from itertools import combinations

from networkx import NetworkXError

__author__ = """\n""".join(['Jordi Torrents <jtorrents@milnou.net>',
                            'Aric Hagberg (hagberg@lanl.gov)'])
__all__ = ['node_redundancy']


[docs]def node_redundancy(G, nodes=None): r"""Computes the node redundancy coefficients for the nodes in the bipartite graph ``G``. 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. More formally, for any vertex `v`, the *redundancy coefficient of `v`* is defined by .. math:: rc(v) = \frac{|\{\{u, w\} \subseteq N(v), \: \exists v' \neq v,\: (v',u) \in E\: \mathrm{and}\: (v',w) \in E\}|}{ \frac{|N(v)|(|N(v)|-1)}{2}}, where `N(v)` is the set of 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. Examples -------- Compute the redundancy coefficient of each node in a graph:: >>> import networkx as nx >>> 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:: >>> import networkx as nx >>> from networkx.algorithms import bipartite >>> G = nx.cycle_graph(4) >>> rc = bipartite.node_redundancy(G) >>> sum(rc.values()) / len(G) 1.0 Compute the average redundancy for a set of nodes:: >>> import networkx as nx >>> from networkx.algorithms import bipartite >>> G = nx.cycle_graph(4) >>> rc = bipartite.node_redundancy(G) >>> nodes = [0, 2] >>> sum(rc[n] for n in nodes) / len(nodes) 1.0 Raises ------ NetworkXError If any of the nodes in the graph (or in ``nodes``, if specified) has (out-)degree less than two (which would result in division by zero, according to the definition of the redundancy coefficient). 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 if any(len(G[v]) < 2 for v in nodes): raise NetworkXError('Cannot compute redundancy coefficient for a node' ' that has fewer than two neighbors.') # TODO This can be trivially parallelized. return {v: _node_redundancy(G, v) for v in nodes}
def _node_redundancy(G, v): """Returns the redundancy of the node ``v`` in the bipartite graph ``G``. If ``G`` is a graph with ``n`` nodes, the redundancy of a node is the ratio of the "overlap" of ``v`` to the maximum possible overlap of ``v`` according to its degree. The overlap of ``v`` is the number of pairs of neighbors that have mutual neighbors themselves, other than ``v``. ``v`` must have at least two neighbors in ``G``. """ n = len(G[v]) # TODO On Python 3, we could just use `G[u].keys() & G[w].keys()` instead # of instantiating the entire sets. overlap = sum(1 for (u, w) in combinations(G[v], 2) if (set(G[u]) & set(G[w])) - {v}) return (2 * overlap) / (n * (n - 1))