Extended improved k-shell hybrid method
Maji et al. [2] introduced the extended improved \(k\)-shell hybrid (EIKH) centrality, in which the improved \(k\)-shell hybrid (IKH) centrality (\(c_{\mathrm{IKH}}\)) replaces the standard \(k\)-shell hybrid measure (\(c_{\mathrm{ksh}}\)) to enhance sensitivity to structural variations within the network. Specifically, the centrality \( c_{\mathrm{EIKH}}(i) \) of node \( i \) is defined as\begin{equation*}c_{\mathrm{ESKH}}(i) = \sum_{j \in \mathcal{N}(i)} c_{\mathrm{IKH}}(j) = \sum_{j \in \mathcal{N}(i)} \sum_{t \in \mathcal{N}^{(\leq l)}(j)} \frac{\sqrt{k_s(j) + k_s(t)} + μ(d_{jt}) \cdot k_t}{d_{jt}^2},\end{equation*}where \( \mathcal{N}(i) \) denotes the set of neighbors of node \( i \), \( \mathcal{N}^{(\leq l)}(j) \) represents the set of nodes within the \( l \)-hop neighborhood of node \( j \),\( d_{jt} \) is the shortest path distance between nodes \( j \) and \( t \),\( k_s(j) \) and \( k_s(t) \) are the \(k\)-shell indices of nodes \( j \) and \( t \), respectively,\( k_t \) is the degree of node \( t \), and \( μ\) is defined as\begin{equation*}μ (d_{jt}) = \frac{2(l - d_{jt} + 1)}{l(l + 1)}.\end{equation*}