networkx.algorithms.centrality.communicability_betweenness_centrality¶

communicability_betweenness_centrality
(G, normalized=True)[source]¶ Returns subgraph communicability 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.
Notes
Let
G=(V,E)
be a simple undirected graph withn
nodes andm
edges, andA
denote the adjacency matrix ofG
.Let
G(r)=(V,E(r))
be the graph resulting from removing all edges connected to noder
but not the node itself.The adjacency matrix for
G(r)
isA+E(r)
, whereE(r)
has nonzeros only in row and columnr
.The subraph 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 nodep
and ending at nodeq
, andC=(n1)^{2}(n1)
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
 1
Ernesto Estrada, Desmond J. Higham, Naomichi Hatano, “Communicability Betweenness in Complex Networks” Physica A 388 (2009) 764774. https://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)