k-shell based on gravity centrality (KSGC)
The k -shell based on gravity centrality (KSGC) is an extension of the local gravity model that integrates both \(k\)-shell and degree centralities to identify influential nodes in networks [2]. Let \(\mathcal{N}^{(\leq k)}(i)\) denote set of nodes within the \(k\)-hop neighbourhood of node \(i\). The centrality \( c_{\textsc{KSGC}}(i) \) of node \( i \) is defined as\begin{equation*}c_{\textsc{KSGC}}(i) = \sum_{j \in \mathcal{N}^{(\leq k)}(i)}e^{\frac{k_s(i) - k_s(j)}{ks_{\max} - ks_{\min}}} \cdot \frac{d_i d_j}{d_{ij}^2},\end{equation*}where \( d_{ij} \) is the shortest-path distance between nodes \( i \) and \( j \), \( d_i \) is the degree of node \( i \) and \( k_s(i) \) is the \(k\)-shell index of node \( i \). The terms \( ks_{\max} \) and \( ks_{\min} \) represent the maximum and minimum \(k\)-shell values in the network, respectively. Yang and Xiao [2] consider \( l = 2 \) as the truncated radius.\vfill