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# communicability¶

communicability(G)

Return communicability between all pairs of nodes in G.

The communicability between pairs of nodes in G is the sum of closed walks of different lengths starting at node u and ending at node v.

Parameters : G: graph comm: dictionary of dictionaries Dictionary of dictionaries keyed by nodes with communicability as the value. NetworkXError If the graph is not undirected and simple.

See also

communicability_centrality_exp
Communicability centrality for each node of G using matrix exponential.
communicability_centrality
Communicability centrality for each node in G using spectral decomposition.
communicability
Communicability between pairs of nodes in G.

Notes

This algorithm uses a spectral decomposition of the adjacency matrix. Let G=(V,E) be a simple undirected graph. Using the connection between the powers of the adjacency matrix and the number of walks in the graph, the communicability between nodes $$u$$ and $$v$$ based on the graph spectrum is [R172]

$C(u,v)=\sum_{j=1}^{n}\phi_{j}(u)\phi_{j}(v)e^{\lambda_{j}},$

where $$\phi_{j}(u)$$ is the $$u\rm{th}$$ element of the $$j\rm{th}$$ orthonormal eigenvector of the adjacency matrix associated with the eigenvalue $$\lambda_{j}$$.

References

 [R172] (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)])
>>> c = nx.communicability(G)