Functional centrality
Functional centrality
quantifies the importance of nodes according to their participation in closed walks of lengths up to \(k\), as proposed in [2]. The functional centrality of node \(i\) is defined as
\[
c_f(i) = \sum_{l=0}^{k} a_l (A^l)_{ii}
= \sum_{j=1}^{N} \left( v_j(i) \right)^2 \left( \sum_{l=0}^{k} a_l λ_j^l \right),
\]
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 \(λ_j\) of \(A\), and the coefficients \(a_l\) are defined as \(a_0 = 1\) and \(a_l = 1/l\) for any integer \(l>0\).
In functional centrality, the contribution of a node to walks of length \(l\) is weighted inversely by \(l\), so that shorter walks are given higher importance, while longer walks contribute progressively less. Functional centrality thus provides a way to quantify node importance with an emphasis on shorter, more localized interactions in the network.