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Return the communicability centrality for each node of G

Communicability centrality, also called subgraph centrality, of a node n is the sum of closed walks of all lengths starting and ending at node n.

Parameters:G (graph) –
Returns:nodes – Dictionary of nodes with communicability centrality 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.


This version of the algorithm exponentiates the adjacency matrix. The communicability centrality of a node u in G can be found using the matrix exponential of the adjacency matrix of G [1] [2],

SC(u)=(e^A)_{uu} .


[1]Ernesto Estrada, Juan A. Rodriguez-Velazquez, “Subgraph centrality in complex networks”, Physical Review E 71, 056103 (2005). http://arxiv.org/abs/cond-mat/0504730
[2]Ernesto Estrada, Naomichi Hatano, “Communicability in complex networks”, Phys. Rev. E 77, 036111 (2008). http://arxiv.org/abs/0707.0756


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