The \(m\)-reach centrality , also referred to as \(m\)-step reach centrality or the \(K\)-order propagation number [2], quantifies the extent of a node’s influence by counting the number of nodes that can be reached within \(m\) steps from it [3]. Formally, for a node \(i\), the \(m\)-reach centrality \(c_{m\text{-reach}}(i)\) is defined as
\[
c_{m\text{-reach}}(i) = |\{ j \in \mathcal{N} : d_{ij} \le m \}| = |N^{(\le m)}(i)|,
\]
where \(\mathcal{N}\) is the set of all nodes in the network and \(d_{ij}\) denotes the shortest-path distance between nodes \(i\) and \(j\). The \(m\)-reach centrality generalizes several well-known measures:

  • For \(m = 1\), it coincides with the degree centrality.
  • For \(m = 2\), it is equivalent to the reachability measure [4].

The parameter \(m\) should not exceed the diameter of the network, as values larger than the diameter would include all nodes and thus provide no further discrimination between nodes.

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] Li-Ya, H., Ping-Chuan, T., You-Liang, H., Yi, Z., & Xie-Feng, C. (2019). Node importance based on the weighted K-order propagation number algorithm. Acta Physica Sinica, 68(12). doi: 10.3390/e22030364.
[3] Borgatti, S. P. (2003). The key player problem. Washington, D.C.: National Academy of Sciences Press. pp. 241-252. doi: 10.2139/ssrn.1149843.
[4] Higley, J., Hoffmann-Lange, U., Kadushin, C., & Moore, G. (1991). Elite integration in stable democracies: a reconsideration. European Sociological Review, 7(1), 35-53. doi: 10.1093/oxfordjournals.esr.a036576.