communicability_betweenness_centrality

communicability_betweenness_centrality(G, normalized=True)

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

Dictionary of nodes with communicability betweenness as the value.

Raises :

NetworkXError

If the graph is not undirected and simple.

See also

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 \(G=(V,E)\) 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 [R174]

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

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.

References

[R174](1, 2) 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)