Subgraph centrality , also known as communicability centrality , quantifies the extent to which a node participates in all subgraphs of a network. It is computed based on the number of closed walks of different lengths, where a walk is considered closed if its starting and ending nodes coincide [2]. The centrality \(c_s(i)\) of node \(i\) is the sum of weighted closed walks of all lengths starting and ending at node \(i\)\begin{equation*}c_{s}(i) = \sum_{k=0}^{\infty}{\frac{(A^k)_{ii}}{k!}}=[e^A]_{ii}=\sum_{j=1}^{N}{\left( v_j(i) \right)^2e^{λ_j}},\end{equation*}where \(A\) denotes the adjacency matrix of the graph \(G\), \(v_j(i)\) is the \(i\)-th component of the eigenvector \(v_j\) associated with the eigenvalue \(λ_j\) of \(A\).

References

[1] Shvydun, S. (2025). Zoo of Centralities: Encyclopedia of Node Metrics in Complex Networks. arXiv: 2511.05122 https://doi.org/10.48550/arXiv.2511.05122
[2] Estrada, E., & Rodríguez-Velázquez, J. A. (2005). Subgraph centrality in complex networks. Physical Review E, 71(5). doi: 10.1103/PhysRevE.71.056103.