Local gravity model
The
local gravity model
is a variant of the gravity model in which a node’s centrality depends only on its \(l\)-hop neighborhood [2]. Let \(\mathcal{N}^{(\geq l)}(i)\) denote the set of nodes within \(l\)-hop neighborhood of node \(i\). The centrality \(c_{\text{Local-Gravity}}(i)\) of node \(i\) is defined as
\begin{equation*}
c_{\text{Local-Gravity}}(i) = \sum_{j \in \mathcal{N}^{(\geq l)}(i)} \frac{d_i\,d_j}{d_{ij}^2},
\end{equation*}
where \(d_{ij}\) represents the shortest path distance between nodes \(i\) and \(j\), and \(d_i\) is the degree of node \(i\).
Thus, the local gravity model incorporates only local structural information within an \(l\)-hop neighborhood. When \(l\) equals the diameter of the network, the local gravity model becomes equivalent to the original gravity model.