DK-based gravity model (DKGM)
The
DK-based gravity model
(DKGM) is an extension of the local gravity model in which a node's
mass
is represented by its DK value, a metric that combines the node's degree and the results from \(k\)-core decomposition [2]. Let \(\mathcal{N}^{(\leq l)}(i)\) denote the set of neighbors of node \(i\) within \(l\) hops. Then the DKGM centrality of node \(i\) can be written as
\begin{equation*}
c_{DKGM}(i) = \sum_{j \in \mathcal{N}^{(\leq l)}(i)} \frac{DK(i) \, DK(j)}{d_{ij}^2},
\end{equation*}
where \(d_{ij}\) is the shortest path distance between nodes \(i\) and \(j\). Following [2], the truncated radius is typically set to \(l=2\).
The DK index of node \(i\), denoted \(DK(i)\), is given by
\begin{equation*}
DK(i) = d_i + k_s(i) + \frac{p(i)}{\max_k q(k) + 1},
\end{equation*}
where \(d_i\) is the degree of node \(i\), \(k_s(i)\) is the \(k\)-shell value of node \(i\), \(p(i)\) represents the iteration at which node \(i\) is removed during the \(k\)-core decomposition and \(q(k)\) denotes the total number of removal steps performed in that iteration. The DK index captures both local information (degree and \(k\)-shell) and global structural information (position within the \(k\)-core hierarchy) of nodes.
DKGM evaluates a node’s influence by considering both its own importance and the contributions of nearby nodes, giving less weight to nodes that are farther away.