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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) –
Returns:nodes – Dictionary of nodes with communicability betweenness as the value.
Return type:dictionary
Raises:NetworkXError – If the graph is not undirected and simple.

See also

Communicability between all pairs of nodes in G.
Communicability centrality for each node of G using matrix exponential.
Communicability centrality for each node in G using spectral decomposition.


Let G=(V,E) be a simple undirected graph with n nodes and m edges, and A denote the adjacency matrix of G.

Let G(r)=(V,E(r)) be the graph resulting from removing all edges connected to node r but not the node itself.

The adjacency matrix for G(r) is A+E(r), where E(r) has nonzeros only in row and column r.

The communicability betweenness of a node r is [1]

\omega_{r} = \frac{1}{C}\sum_{p}\sum_{q}\frac{G_{prq}}{G_{pq}},
p\neq q, q\neq r,

where G_{prq}=(e^{A}_{pq} - (e^{A+E(r)})_{pq} is the number of walks involving node r, G_{pq}=(e^{A})_{pq} is the number of closed walks starting at node p and ending at node q, and C=(n-1)^{2}-(n-1) is a normalization factor equal to the number of terms in the sum.

The resulting \omega_{r} 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.


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


>>> 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)