Extended k-shell hybrid method
The
extended \(k\)-shell hybrid method
(ESKH), proposed by Namtirtha et al. [2], is an extension of the \(k\)-shell hybrid method (ksh) in which the centrality of a node depends on the hybrid centralities of its neighbors. The centrality \( c_{\mathrm{ESKH}}(i) \) of node \( i \) is defined as
\begin{equation*}
c_{\mathrm{ESKH}}(i)
= \sum_{j \in \mathcal{N}(i)} c_{\mathrm{ksh}}(j)
= \sum_{j \in \mathcal{N}(i)} \sum_{t \in \mathcal{N}^{(\leq l)}(j)}
\frac{\sqrt{k_s(j) + k_s(t)} + μ\,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 \( μ \in (0,1) \) is a tunable parameter that balances the relative influence of the two components.