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 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 centrality for each node of G using matrix exponential.
Communicability centrality for each node in G using spectral decomposition.
Communicability between pairs of nodes in G.


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]


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


>>> G = nx.Graph([(0,1),(1,2),(1,5),(5,4),(2,4),(2,3),(4,3),(3,6)])
>>> c = nx.communicability(G)