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.