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# communicability_betweenness_centrality¶

communicability_betweenness_centrality(G, normalized=True)[source]

Return communicability betweenness for all pairs of nodes in G.

Communicability betweenness measure makes use of the number of walks connecting every pair of nodes as the basis of a betweenness centrality measure.

Parameters: G (graph) – nodes – Dictionary of nodes with communicability betweenness as the value. dictionary NetworkXError – If the graph is not undirected and simple.

communicability()
Communicability between all pairs of nodes in G.
communicability_centrality()
Communicability centrality for each node of G using matrix exponential.
communicability_centrality_exp()
Communicability centrality for each node in G using spectral decomposition.

Notes

Let be a simple undirected graph with nodes and edges, and denote the adjacency matrix of .

Let be the graph resulting from removing all edges connected to node but not the node itself.

The adjacency matrix for is , where has nonzeros only in row and column .

The communicability betweenness of a node is where is the number of walks involving node r, is the number of closed walks starting at node and ending at node , and is a normalization factor equal to the number of terms in the sum.

The resulting takes values between zero and one. The lower bound cannot be attained for a connected graph, and the upper bound is attained in the star graph.

References

  Ernesto Estrada, Desmond J. Higham, Naomichi Hatano, “Communicability Betweenness in Complex Networks” Physica A 388 (2009) 764-774. http://arxiv.org/abs/0905.4102

Examples

>>> G = nx.Graph([(0,1),(1,2),(1,5),(5,4),(2,4),(2,3),(4,3),(3,6)])
>>> cbc = nx.communicability_betweenness_centrality(G)