networkx.algorithms.centrality.subgraph_centrality

subgraph_centrality(G)

Returns subgraph centrality for each node in G.

Subgraph centrality of a node n is the sum of weighted closed walks of all lengths starting and ending at node n. The weights decrease with path length. Each closed walk is associated with a connected subgraph (1).

Parameters

G (graph)

Returns

nodes – Dictionary of nodes with subgraph centrality as the value.

Return type

dictionary

Raises

NetworkXError – If the graph is not undirected and simple.

See also

subgraph_centrality_exp()

Alternative algorithm of the subgraph centrality for each node of G.

Notes

This version of the algorithm computes eigenvalues and eigenvectors of the adjacency matrix.

Subgraph centrality of a node u in G can be found using a spectral decomposition of the adjacency matrix 1,

\[SC(u)=\sum_{j=1}^{N}(v_{j}^{u})^2 e^{\lambda_{j}},\]

where v_j is an eigenvector of the adjacency matrix A of G corresponding corresponding to the eigenvalue lambda_j.

Examples

(Example from 1) >>> G = nx.Graph( … [ … (1, 2), … (1, 5), … (1, 8), … (2, 3), … (2, 8), … (3, 4), … (3, 6), … (4, 5), … (4, 7), … (5, 6), … (6, 7), … (7, 8), … ] … ) >>> sc = nx.subgraph_centrality(G) >>> print([f”{node} {sc[node]:0.2f}” for node in sorted(sc)]) [‘1 3.90’, ‘2 3.90’, ‘3 3.64’, ‘4 3.71’, ‘5 3.64’, ‘6 3.71’, ‘7 3.64’, ‘8 3.90’]

References

1(1,2,3)

Ernesto Estrada, Juan A. Rodriguez-Velazquez, “Subgraph centrality in complex networks”, Physical Review E 71, 056103 (2005). https://arxiv.org/abs/cond-mat/0504730