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 walks of different lengths starting at node u and ending at node v.

Parameters:

Ggraph

Returns:

commdictionary 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 matrix exponentiation
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:

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

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 \(\lambda_{j}\) [1].

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)