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
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 [R177]

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

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

[R177]

(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)