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 :
|
Raises : | NetworkXError :
|
See also
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 [R158]
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}.
References
[R158] | (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)