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 Dictionary of nodes with communicability centrality 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.

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 [R175] [R176],

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

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

References

[R175]

(1, 2) Ernesto Estrada, Juan A. Rodriguez-Velazquez, “Subgraph centrality in complex networks”, Physical Review E 71, 056103 (2005). http://arxiv.org/abs/cond-mat/0504730

[R176]

(1, 2) Ernesto Estrada, Naomichi Hatano, “Communicability in complex networks”, Phys. Rev. E 77, 036111 (2008). http://arxiv.org/abs/0707.0756

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)