Length-scaled betweenness centrality (LSBC), also known as distance-scaled betweenness, is a variant of betweenness centrality that weights shortest paths inversely proportional to their length [2, 3]. The centrality of node \(i\) is defined as
\begin{equation*}
c_{b_{dist}}(i) = \sum_{j=1}^{N} \sum_{k=1}^{N} \frac{1}{d_{jk}} \frac{σ_{jk}(i)}{σ_{jk}},
\end{equation*}
where \(d_{jk}\) is the length of the shortest path from \(j\) to \(k\), \(σ_{jk}\) is the total number of shortest paths between \(j\) and \(k\), and \(σ_{jk}(i)\) is the number of those paths passing through \(i\). This measure reflects the intuition that longer paths contribute less to a node's centrality.

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] Borgatti, S. P., & Everett, M. G. (2006). A graph-theoretic perspective on centrality. Social networks, 28(4), 466-484. doi: 10.1016/j.socnet.2005.11.005.
[3] Brandes, U. (2008). On variants of shortest-path betweenness centrality and their generic computation. Social networks, 30(2), 136-145. doi: 10.1016/j.socnet.2007.11.001.