Residual closeness centrality

Residual closeness centrality , also known as Dangalchev closeness centrality, evaluates node closeness after the removal of nodes or edges, with shortest-path lengths weighted by an exponential decay [2]. Let \(G_i\) denote the subgraph of \(G\) obtained by removing node \(i\). Then the residual closeness centrality of node \(i\) equals
\begin{equation*}
c_{residual}(i) = \sum_{j \neq k \neq i}\frac{1}{2^{d_{jk}(G_i)}},
\end{equation*}
where \(d_{jk}(G_i)\) denote the shortest-path distance between nodes \(j\) and \(k\) in \(G_i\). Thus, residual closeness reflects the distance-based importance of a node by quantifying the effect of its removal on the overall connectivity of 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] Dangalchev, C. (2006). Residual closeness in networks. Physica A: Statistical Mechanics and its Applications, 365(2), 556-564. doi: 10.1016/j.physa.2005.12.020.