125 lines
4.0 KiB
Python
125 lines
4.0 KiB
Python
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"""Percolation centrality measures."""
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import networkx as nx
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from networkx.algorithms.centrality.betweenness import (
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_single_source_dijkstra_path_basic as dijkstra,
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)
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from networkx.algorithms.centrality.betweenness import (
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_single_source_shortest_path_basic as shortest_path,
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)
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__all__ = ["percolation_centrality"]
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def percolation_centrality(G, attribute="percolation", states=None, weight=None):
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r"""Compute the percolation centrality for nodes.
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Percolation centrality of a node $v$, at a given time, is defined
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as the proportion of ‘percolated paths’ that go through that node.
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This measure quantifies relative impact of nodes based on their
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topological connectivity, as well as their percolation states.
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Percolation states of nodes are used to depict network percolation
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scenarios (such as during infection transmission in a social network
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of individuals, spreading of computer viruses on computer networks, or
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transmission of disease over a network of towns) over time. In this
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measure usually the percolation state is expressed as a decimal
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between 0.0 and 1.0.
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When all nodes are in the same percolated state this measure is
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equivalent to betweenness centrality.
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Parameters
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----------
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G : graph
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A NetworkX graph.
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attribute : None or string, optional (default='percolation')
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Name of the node attribute to use for percolation state, used
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if `states` is None.
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states : None or dict, optional (default=None)
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Specify percolation states for the nodes, nodes as keys states
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as values.
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weight : None or string, optional (default=None)
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If None, all edge weights are considered equal.
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Otherwise holds the name of the edge attribute used as weight.
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The weight of an edge is treated as the length or distance between the two sides.
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Returns
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-------
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nodes : dictionary
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Dictionary of nodes with percolation centrality as the value.
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See Also
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--------
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betweenness_centrality
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Notes
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-----
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The algorithm is from Mahendra Piraveenan, Mikhail Prokopenko, and
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Liaquat Hossain [1]_
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Pair dependecies are calculated and accumulated using [2]_
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For weighted graphs the edge weights must be greater than zero.
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Zero edge weights can produce an infinite number of equal length
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paths between pairs of nodes.
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References
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----------
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.. [1] Mahendra Piraveenan, Mikhail Prokopenko, Liaquat Hossain
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Percolation Centrality: Quantifying Graph-Theoretic Impact of Nodes
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during Percolation in Networks
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http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0053095
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.. [2] Ulrik Brandes:
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A Faster Algorithm for Betweenness Centrality.
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Journal of Mathematical Sociology 25(2):163-177, 2001.
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https://doi.org/10.1080/0022250X.2001.9990249
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"""
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percolation = dict.fromkeys(G, 0.0) # b[v]=0 for v in G
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nodes = G
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if states is None:
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states = nx.get_node_attributes(nodes, attribute)
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# sum of all percolation states
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p_sigma_x_t = 0.0
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for v in states.values():
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p_sigma_x_t += v
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for s in nodes:
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# single source shortest paths
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if weight is None: # use BFS
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S, P, sigma, _ = shortest_path(G, s)
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else: # use Dijkstra's algorithm
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S, P, sigma, _ = dijkstra(G, s, weight)
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# accumulation
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percolation = _accumulate_percolation(
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percolation, S, P, sigma, s, states, p_sigma_x_t
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)
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n = len(G)
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for v in percolation:
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percolation[v] *= 1 / (n - 2)
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return percolation
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def _accumulate_percolation(percolation, S, P, sigma, s, states, p_sigma_x_t):
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delta = dict.fromkeys(S, 0)
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while S:
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w = S.pop()
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coeff = (1 + delta[w]) / sigma[w]
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for v in P[w]:
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delta[v] += sigma[v] * coeff
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if w != s:
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# percolation weight
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pw_s_w = states[s] / (p_sigma_x_t - states[w])
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percolation[w] += delta[w] * pw_s_w
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return percolation
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