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

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_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)