communicability_exp

communicability_exp(G)

Returns communicability between all pairs of nodes in G.

Communicability between pair of node (u,v) of node 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

Communicability between pairs of nodes in G.

communicability_betweenness_centrality

Communicability betweeness centrality for each node in G.

Notes

This algorithm uses matrix exponentiation 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 is [1],

\[C(u,v) = (e^A)_{uv},\]

where A is the adjacency matrix of G.

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_exp(G)