The \(k\)-path centrality , introduced by Sade [2], quantifies the influence of a node by counting all simple (cycle-free) paths of length at most \(k\) that originate from it.
Formally, for a node \(i\), the \(k\)-path centrality \(c_{k\text{-path}}(i)\) is the number of distinct simple paths of length \(1 \le \ell \le k\) starting at \(i\). For \(k = 1\), the \(k\)-path centrality equals the degree centrality.
The \(k\)-path centrality generalizes degree centrality by considering not only immediate neighbors but also nodes reachable within paths of length up to \(k\), capturing broader influence within the network.

References

[1] Shvydun, S. (2025). Zoo of Centralities: Encyclopedia of Node Metrics in Complex Networks. arXiv: 2511.05122 https://doi.org/10.48550/arXiv.2511.05122
[2] Sade, D. S. (1989). Sociometrics of macaca mulatta III: N-path centrality in grooming networks. Social Networks, 11(3), 273-292. doi: 10.1016/0378-8733(89)90006-3.