Extended gravity centrality (EGC)
Inspired by the LocalRank centrality [2] and the extended neighborhood coreness [3], Ma et al. [4] proposed the
extended gravity centrality
(EGC) of node \(i\), denoted by \(c_{\text{EGC}}(i)\), as
\begin{equation*}
c_{\text{EGC}}(i) = \sum_{j \in \mathcal{N}(i)} c_{\text{Gravity}}(j)
= \sum_{j \in \mathcal{N}(i)} \sum_{l \in \mathcal{N}^{(\leq l)}(j)} \frac{k_s(j)\,k_s(l)}{d_{jl}^2},
\end{equation*}
where \(\mathcal{N}(i)\) denotes the set of neighbors of node \(i\), \(d_{jl}\) is the shortest path distance between nodes \(j\) and \(l\), \(k_s(j)\) represents the \(k\)-shell value of node \(j\) and \(\mathcal{N}^{(\leq l)}(j)\) denotes the set of nodes within the \(l\)-hop neighborhood of node \(j\).
The EGC thus integrates the gravity centralities of a node’s immediate neighbors, capturing both local and higher-order topological influences in the network.