156 lines
4.5 KiB
Python
156 lines
4.5 KiB
Python
# efficiency.py - functions for computing node, edge, and graph efficiency
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#
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# Copyright 2011, 2012, 2013, 2014, 2015 NetworkX developers
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#
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# This file is part of NetworkX.
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#
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# NetworkX is distributed under a BSD license; see LICENSE.txt for more
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# information.
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"""Provides functions for computing the efficiency of nodes and graphs."""
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from itertools import permutations
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import networkx as nx
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from networkx.exception import NetworkXNoPath
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from ..utils import not_implemented_for
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__all__ = ['efficiency', 'local_efficiency', 'global_efficiency']
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@not_implemented_for('directed')
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def efficiency(G, u, v):
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"""Returns the efficiency of a pair of nodes in a graph.
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The *efficiency* of a pair of nodes is the multiplicative inverse of the
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shortest path distance between the nodes [1]_. Returns 0 if no path
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between nodes.
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Parameters
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----------
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G : :class:`networkx.Graph`
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An undirected graph for which to compute the average local efficiency.
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u, v : node
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Nodes in the graph ``G``.
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Returns
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-------
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float
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Multiplicative inverse of the shortest path distance between the nodes.
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Notes
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-----
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Edge weights are ignored when computing the shortest path distances.
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See also
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--------
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local_efficiency
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global_efficiency
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References
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----------
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.. [1] Latora, Vito, and Massimo Marchiori.
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"Efficient behavior of small-world networks."
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*Physical Review Letters* 87.19 (2001): 198701.
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<https://doi.org/10.1103/PhysRevLett.87.198701>
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"""
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try:
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eff = 1 / nx.shortest_path_length(G, u, v)
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except NetworkXNoPath:
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eff = 0
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return eff
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@not_implemented_for('directed')
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def global_efficiency(G):
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"""Returns the average global efficiency of the graph.
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The *efficiency* of a pair of nodes in a graph is the multiplicative
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inverse of the shortest path distance between the nodes. The *average
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global efficiency* of a graph is the average efficiency of all pairs of
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nodes [1]_.
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Parameters
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----------
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G : :class:`networkx.Graph`
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An undirected graph for which to compute the average global efficiency.
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Returns
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-------
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float
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The average global efficiency of the graph.
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Notes
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-----
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Edge weights are ignored when computing the shortest path distances.
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See also
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--------
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local_efficiency
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References
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----------
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.. [1] Latora, Vito, and Massimo Marchiori.
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"Efficient behavior of small-world networks."
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*Physical Review Letters* 87.19 (2001): 198701.
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<https://doi.org/10.1103/PhysRevLett.87.198701>
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"""
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n = len(G)
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denom = n * (n - 1)
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if denom != 0:
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lengths = nx.all_pairs_shortest_path_length(G)
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g_eff = 0
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for source, targets in lengths:
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for target, distance in targets.items():
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if distance > 0:
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g_eff += 1 / distance
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g_eff /= denom
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# g_eff = sum(1 / d for s, tgts in lengths
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# for t, d in tgts.items() if d > 0) / denom
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else:
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g_eff = 0
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# TODO This can be made more efficient by computing all pairs shortest
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# path lengths in parallel.
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return g_eff
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@not_implemented_for('directed')
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def local_efficiency(G):
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"""Returns the average local efficiency of the graph.
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The *efficiency* of a pair of nodes in a graph is the multiplicative
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inverse of the shortest path distance between the nodes. The *local
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efficiency* of a node in the graph is the average global efficiency of the
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subgraph induced by the neighbors of the node. The *average local
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efficiency* is the average of the local efficiencies of each node [1]_.
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Parameters
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----------
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G : :class:`networkx.Graph`
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An undirected graph for which to compute the average local efficiency.
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Returns
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-------
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float
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The average local efficiency of the graph.
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Notes
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-----
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Edge weights are ignored when computing the shortest path distances.
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See also
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--------
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global_efficiency
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References
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----------
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.. [1] Latora, Vito, and Massimo Marchiori.
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"Efficient behavior of small-world networks."
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*Physical Review Letters* 87.19 (2001): 198701.
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<https://doi.org/10.1103/PhysRevLett.87.198701>
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"""
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# TODO This summation can be trivially parallelized.
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efficiency_list = (global_efficiency(G.subgraph(G[v])) for v in G)
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return sum(efficiency_list) / len(G)
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