Gromov centrality
Gromov centrality
is a multi-scale measure of node centrality that quantifies the average triangle excess over all pairs of nodes in a given \(l\)-hop neighborhood [2]. Let \(\mathcal{N}^{(\leq l)}(i)\) denote
The set of nodes within the \(l\)-hop neighborhood of node \(i\), and let \(d_{ij}\) represent the shortest-path distance between nodes \(i\) and \(j\). The Gromov centrality of node \(i\), denoted \(c_{\text{Gromov}}(i)\), is defined as
\begin{equation*}
c_{\text{Gromov}}(i) = \frac{1}{|T(\mathcal{N}^{(\leq l)}(i))|} \sum_{(j,k) \in T(\mathcal{N}^{(\leq l)}(i))} \Delta_i(j,k),
\end{equation*}
where
\[
\Delta_i(j,k) = d_{jk} - d_{ij} - d_{ik} \le 0
\]
is the triangle inequality excess (equivalent to the Gromov product) between nodes \(j\) and \(k\) with respect to \(i\), and
\[
T(\mathcal{N}^{(\leq l)}(i)) = \{(j,k) \mid j,k \in \mathcal{N}^{(\leq l)}(i), \ j \neq k \}
\]
is the set of all unordered pairs of nodes in the \(l\)-neighborhood of \(i\).
The triangle inequality excess \(\Delta_i(j,k)\) equals zero if and only if node \(i\) lies on a geodesic (shortest path) between nodes \(j\) and \(k\). Very negative values of \(\Delta_i(j,k)\) indicate that passing through node \(i\) induces a significant detour between \(j\) and \(k\).
By definition, \(c_{\text{Gromov}}(i)\) is always non-positive. Gromov centrality thus characterizes the extent to which a node lies between other pairs of nodes in its \(l\)-neighborhood. When \(l = 1\), it reflects the proportion of triangles formed by a node's neighbors, and a locally central node exhibits a star-like structure. When \(l\) equals the diameter of the network, Gromov centrality becomes equivalent to closeness centrality.