7.10.1. networkx.algorithms.communicability_alg.communicability¶
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networkx.algorithms.communicability_alg.communicability(G)[source]¶ 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_exp- Communicability between all pairs of nodes in G using spectral decomposition.
communicability_betweenness_centrality- Communicability betweeness 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 [R569]
\[C(u,v)=\sum_{j=1}^{n}\phi_{j}(u)\phi_{j}(v)e^{\lambda_{j}},\]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
[R569] (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)