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Source code for networkx.algorithms.centrality.percolation

# coding=utf8
#    Copyright (C) 2018 by
#    Pranay Kanwar <pranay.kanwar@gmail.com>
#    All rights reserved.
#    BSD license.
"""Percolation centrality measures."""
from __future__ import division

__author__ = """\n""".join(['Pranay Kanwar <pranay.kanwar@gmail.com>'])

import networkx as nx

from networkx.algorithms.centrality.betweenness import\
    _single_source_dijkstra_path_basic as dijkstra
from networkx.algorithms.centrality.betweenness import\
    _single_source_shortest_path_basic as shortest_path

__all__ = ['percolation_centrality']

[docs]def percolation_centrality(G, attribute='percolation', states=None, weight=None): r"""Compute the percolation centrality for nodes. Percolation centrality of a node $v$, at a given time, is defined as the proportion of ‘percolated paths’ that go through that node. This measure quantifies relative impact of nodes based on their topological connectivity, as well as their percolation states. Percolation states of nodes are used to depict network percolation scenarios (such as during infection transmission in a social network of individuals, spreading of computer viruses on computer networks, or transmission of disease over a network of towns) over time. In this measure usually the percolation state is expressed as a decimal between 0.0 and 1.0. When all nodes are in the same percolated state this measure is equivalent to betweenness centrality. Parameters ---------- G : graph A NetworkX graph. attribute : None or string, optional (default='percolation') Name of the node attribute to use for percolation state, used if `states` is None. states : None or dict, optional (default=None) Specify percolation states for the nodes, nodes as keys states as values. weight : None or string, optional (default=None) If None, all edge weights are considered equal. Otherwise holds the name of the edge attribute used as weight. Returns ------- nodes : dictionary Dictionary of nodes with percolation centrality as the value. See Also -------- betweenness_centrality Notes ----- The algorithm is from Mahendra Piraveenan, Mikhail Prokopenko, and Liaquat Hossain [1]_ Pair dependecies are calculated and accumulated using [2]_ For weighted graphs the edge weights must be greater than zero. Zero edge weights can produce an infinite number of equal length paths between pairs of nodes. References ---------- .. [1] Mahendra Piraveenan, Mikhail Prokopenko, Liaquat Hossain Percolation Centrality: Quantifying Graph-Theoretic Impact of Nodes during Percolation in Networks http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0053095 .. [2] Ulrik Brandes: A Faster Algorithm for Betweenness Centrality. Journal of Mathematical Sociology 25(2):163-177, 2001. http://www.inf.uni-konstanz.de/algo/publications/b-fabc-01.pdf """ percolation = dict.fromkeys(G, 0.0) # b[v]=0 for v in G nodes = G if states is None: states = nx.get_node_attributes(nodes, attribute) # sum of all percolation states p_sigma_x_t = 0.0 for v in states.values(): p_sigma_x_t += v for s in nodes: # single source shortest paths if weight is None: # use BFS S, P, sigma = shortest_path(G, s) else: # use Dijkstra's algorithm S, P, sigma = dijkstra(G, s, weight) # accumulation percolation = _accumulate_percolation(percolation, G, S, P, sigma, s, states, p_sigma_x_t) n = len(G) for v in percolation: percolation[v] *= 1 / (n - 2) return percolation
def _accumulate_percolation(percolation, G, S, P, sigma, s, states, p_sigma_x_t): delta = dict.fromkeys(S, 0) while S: w = S.pop() coeff = (1 + delta[w]) / sigma[w] for v in P[w]: delta[v] += sigma[v] * coeff if w != s: # percolation weight pw_s_w = states[s] / (p_sigma_x_t - states[w]) percolation[w] += delta[w] * pw_s_w return percolation