PathRank
PathRank
is a centrality measure that quantifies a node's importance based on all simple paths terminating at it [2]. Let \(P^{(1)}_{ij}, P^{(2)}_{ij}, \dots, P^{(s_{ij})}_{ij}\) denote the set of simple paths from node \(i\) to node \(j\) of length at most \(K\). Each path contributes to the centrality of node \(j\) with a weight inversely proportional to the square of its length:
\[
b_{ij} = \sum_{s=1}^{s_{ij}} \frac{1}{|P^{(s)}_{ij}|^2},
\]
where \(|P^{(s)}_{ij}|\) is the length of the \(s\)-th path. This weighting emphasizes shorter paths, reflecting their greater significance in connectivity. The PathRank centrality of node \(j\) is obtained by summing these contributions over all source nodes:
\[
c_{PathRank}(j) = \sum_{i=1}^{N} b_{ij}.
\]
Thus, PathRank accumulates the influence of all paths terminating at a node, giving higher weight to shorter paths. In practice, a maximum path length of \(K = 5\) is typically used [2].