communicability_centrality¶

communicability_centrality
(G)[source]¶ Return communicability centrality for each node in G.
Communicability centrality, also called subgraph centrality, of a node \(n\) is the sum of closed walks of all lengths starting and ending at node \(n\).
Parameters: G (graph) – Returns: nodes – Dictionary of nodes with communicability centrality as the value. Return type: dictionary Raises: NetworkXError
– If the graph is not undirected and simple.See also
communicability()
 Communicability between all pairs of nodes in G.
communicability_centrality()
 Communicability centrality for each node of G.
Notes
This version of the algorithm computes eigenvalues and eigenvectors of the adjacency matrix.
Communicability centrality of a node \(u\) in G can be found using a spectral decomposition of the adjacency matrix [1] [2],
\[SC(u)=\sum_{j=1}^{N}(v_{j}^{u})^2 e^{\lambda_{j}},\]where \(v_j\) is an eigenvector of the adjacency matrix \(A\) of G corresponding corresponding to the eigenvalue \(\lambda_j\).
Examples
>>> G = nx.Graph([(0,1),(1,2),(1,5),(5,4),(2,4),(2,3),(4,3),(3,6)]) >>> sc = nx.communicability_centrality(G)
References
[1] Ernesto Estrada, Juan A. RodriguezVelazquez, “Subgraph centrality in complex networks”, Physical Review E 71, 056103 (2005). http://arxiv.org/abs/condmat/0504730 [2] Ernesto Estrada, Naomichi Hatano, “Communicability in complex networks”, Phys. Rev. E 77, 036111 (2008). http://arxiv.org/abs/0707.0756