"""Percolation centrality measures."""
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