The BottleNeck centrality identifies nodes that act as bottlenecks in a network by inspecting shortest-path trees [2]. For each source node \(s\) let \(T_s\) denote a shortest-path tree rooted at \(s\) and let \(V(T_s)\) denote the set of nodes in \(T_s\) reachable from \(s\), excluding the root \(s\). The BottleNeck centrality of node \(i\) is defined as
\begin{equation*}
c_{\text{BottleNeck}}(i) = \sum_{s=1}^{N} P_s(i),
\end{equation*}
where
\begin{equation*}
P_s(i) =
\begin{cases}
1, & \text{if more than } |V(T_s)|/4 \text{ shortest paths from } s \text{ to other nodes in } T_s \text{ pass through } i, \\
0, & \text{otherwise}.
\end{cases}
\end{equation*}
The total score \(c_{\text{BottleNeck}}(i)\) thus counts how often node \(i\) serves as a major intermediary across all sources. Intuitively, nodes with high BottleNeck centrality are critical intermediaries through which a significant fraction of shortest paths pass, making them important for the connectivity and flow 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] Pržulj, N., Wigle, D. A., & Jurisica, I. (2004). Functional topology in a network of protein interactions. Bioinformatics, 20(3), 340-348. doi: 10.1093/bioinformatics/btg415.