Hierarchical k-shell (HKS) centrality
The hierarchical \(k\)-shell (HKS) centrality is a hybrid extension of the \(k\)-shell centrality [2] that combines \(k\)-shell decomposition with node distances. The centrality of node \(i\) is defined as\begin{equation*}c_{HKS}(i) = \sum_{j \in \mathcal{N}(i)} \sum_{l \in \mathcal{N}(j)} s(l),\end{equation*}where \begin{equation*}s(l) = d_l (b_l + f_l).\end{equation*}Here, \(d_l\) is the degree of node \(l\), \(b_l\) is the iteration at which node \(l\) is removed during \(k\)-shell decomposition, and \(f_l\) captures the distance of node \(l\) to the nodes with the highest \(k\)-shell score. Specifically, let \(K\) denote the set of nodes with the highest \(k\)-shell index. Then\begin{equation*}f_l = \max_{u \in K} (b_u - d_{lu}),\end{equation*}where \(d_{lu}\) is the shortest path distance between nodes \(l\) and \(u\). This formulation integrates both local connectivity and hierarchical position to more accurately identify influential nodes.