Odd subgraph centrality
Odd subgraph centrality
is a variant of subgraph centrality that counts the number of closed walks of
odd
length in a network [2]. Focusing on odd-length walks highlights genuine cycles, since even-length walks can arise from trivial back-and-forth movements in acyclic subgraphs. The odd subgraph centrality of node \(i\), denoted \(c_{odd}(i)\), is defined as
\[
c_{odd}(i) = \sum_{k=0}^{\infty} \frac{(A^{2k+1})_{ii}}{(2k+1)!}
= \sum_{j=1}^{N} \left( v_j(i) \right)^2 \sinh(λ_j),
\]
where \(A\) is the adjacency matrix of the network, \(v_j(i)\) is the \(i\)-th component of the eigenvector \(v_j\) corresponding to eigenvalue \(λ_j\), and \(N\) is the number of nodes. Odd subgraph centrality has been applied to empirical food web networks to identify keystone species involved in cyclic trophic interactions.