networkx.algorithms.communicability_alg.communicability¶

communicability(G)[source]¶

Returns 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)
Returns: comm – Dictionary of dictionaries keyed by nodes with communicability as the value.
Return type: dictionary of dictionaries
Raises: NetworkXError – If the graph is not undirected and simple.

See also

communicability_exp(): Communicability between all pairs of nodes in G using spectral decomposition.
communicability_betweenness_centrality(): Communicability betweeness centrality for each node 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 1

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

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

References

1: Ernesto Estrada, Naomichi Hatano, “Communicability in complex networks”, Phys. Rev. E 77, 036111 (2008). https://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)