Even subgraph centrality

Even subgraph centrality counts the number of closed walks of even length in a network [2]. Even-length walks include contributions from both cyclic and acyclic structures, reflecting back-and-forth movements that capture redundancy, potential signal propagation, and indirect interactions. The even subgraph centrality of node \(i\), denoted \(c_{even}(i)\), is defined as
\[
c_{even}(i) = \sum_{k=0}^{\infty} \frac{(A^{2k})_{ii}}{(2k)!}
= \sum_{j=1}^{N} \left( v_j(i) \right)^2 \cosh(λ_j),
\]
where \(A\) is the adjacency matrix of the network, and \(v_j(i)\) is the \(i\)-th component of the eigenvector \(v_j\) corresponding to eigenvalue \(λ_j\).

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] Rodríguez, J. A., Estrada, E., & Gutiérrez, A. (2007). Functional centrality in graphs. Linear and Multilinear Algebra, 55(3), 293-302. doi: 10.1080/03081080601002221.