communicability_exp¶
- communicability_exp(G)¶
Return communicability between all pairs of nodes in G.
Communicability between pair of node (u,v) of node 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_exp
- Communicability between all pairs of nodes in G using spectral decomposition.
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 [R179],
\[C(u,v) = (e^A)_{uv},\]where \(A\) is the adjacency matrix of G.
References
[R179] (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_exp(G)