communicability

communicability(G)

Returns communicability between all pairs of nodes in G.

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

Parameters:

G: graph

Returns:

comm: dictionary of dictionaries

Dictionary of dictionaries keyed by nodes with communicability as the value.

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 betweenness 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)