# efficiency.py - functions for computing node, edge, and graph efficiency # # Copyright 2011, 2012, 2013, 2014, 2015 NetworkX developers # # This file is part of NetworkX. # # NetworkX is distributed under a BSD license; see LICENSE.txt for more # information. """Provides functions for computing the efficiency of nodes and graphs.""" from __future__ import division from itertools import permutations import networkx as nx from networkx.exception import NetworkXNoPath from ..utils import not_implemented_for __all__ = ['efficiency', 'local_efficiency', 'global_efficiency'] @not_implemented_for('directed') def efficiency(G, u, v): """Returns the efficiency of a pair of nodes in a graph. The *efficiency* of a pair of nodes is the multiplicative inverse of the shortest path distance between the nodes [1]_. Returns 0 if no path between nodes. Parameters ---------- G : :class:`networkx.Graph` An undirected graph for which to compute the average local efficiency. u, v : node Nodes in the graph ``G``. Returns ------- float Multiplicative inverse of the shortest path distance between the nodes. Notes ----- Edge weights are ignored when computing the shortest path distances. See also -------- local_efficiency global_efficiency References ---------- .. [1] Latora, Vito, and Massimo Marchiori. "Efficient behavior of small-world networks." *Physical Review Letters* 87.19 (2001): 198701. """ try: eff = 1 / nx.shortest_path_length(G, u, v) except NetworkXNoPath: eff = 0 return eff @not_implemented_for('directed') def global_efficiency(G): """Returns the average global efficiency of the graph. The *efficiency* of a pair of nodes in a graph is the multiplicative inverse of the shortest path distance between the nodes. The *average global efficiency* of a graph is the average efficiency of all pairs of nodes [1]_. Parameters ---------- G : :class:`networkx.Graph` An undirected graph for which to compute the average global efficiency. Returns ------- float The average global efficiency of the graph. Notes ----- Edge weights are ignored when computing the shortest path distances. See also -------- local_efficiency References ---------- .. [1] Latora, Vito, and Massimo Marchiori. "Efficient behavior of small-world networks." *Physical Review Letters* 87.19 (2001): 198701. """ n = len(G) denom = n * (n - 1) if denom != 0: g_eff = sum(efficiency(G, u, v) for u, v in permutations(G, 2)) / denom else: g_eff = 0 # TODO This can be made more efficient by computing all pairs shortest # path lengths in parallel. # # TODO This summation can be trivially parallelized. return g_eff @not_implemented_for('directed') def local_efficiency(G): """Returns the average local efficiency of the graph. The *efficiency* of a pair of nodes in a graph is the multiplicative inverse of the shortest path distance between the nodes. The *local efficiency* of a node in the graph is the average global efficiency of the subgraph induced by the neighbors of the node. The *average local efficiency* is the average of the local efficiencies of each node [1]_. Parameters ---------- G : :class:`networkx.Graph` An undirected graph for which to compute the average local efficiency. Returns ------- float The average local efficiency of the graph. Notes ----- Edge weights are ignored when computing the shortest path distances. See also -------- global_efficiency References ---------- .. [1] Latora, Vito, and Massimo Marchiori. "Efficient behavior of small-world networks." *Physical Review Letters* 87.19 (2001): 198701. """ # TODO This summation can be trivially parallelized. efficiency_list = (global_efficiency(G.subgraph(G[v])) for v in G) return sum(efficiency_list) / len(G)