LocalRank centrality
LocalRank centrality
(also called semi-local or local centrality) quantifies a node’s importance by considering both its nearest and next-nearest neighbors [2]. For a node \(i\), it is defined as
\begin{equation*}
c_{\mathrm{LR}}(i) = \sum_{j \in \mathcal{N}(i)} \sum_{k \in \mathcal{N}(j)} n(k),
\end{equation*}
where \(\mathcal{N}(i)\) is the set of neighbors of \(i\), and \(n(k)=|N^{(\le 2)}(k)|\) is the number of nearest and next-nearest neighbors of node \(k\).
Intuitively, LocalRank captures both the direct connectivity of a node and the connectivity of its neighbors, allowing nodes connected to highly connected neighborhoods to achieve higher centrality. This makes it more discriminative than degree centrality while remaining computationally efficient.