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communicability_centrality

communicability_centrality(G)

Return communicability centrality for each node in 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()

Communicability between all pairs of nodes in G.

communicability_centrality()

Communicability centrality for each node of G.

Notes

This version of the algorithm computes eigenvalues and eigenvectors of the adjacency matrix.

Communicability centrality of a node \(u\) in G can be found using a spectral decomposition of the adjacency matrix [1] [2],

\[SC(u)=\sum_{j=1}^{N}(v_{j}^{u})^2 e^{\lambda_{j}},\]

where \(v_j\) is an eigenvector of the adjacency matrix \(A\) of G corresponding corresponding to the eigenvalue \(\lambda_j\).

Examples

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

References

[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