The closeness vitality measures how the overall efficiency of a network changes when a given node is removed [2]. Specifically, it quantifies the variation in the total sum of shortest-path distances between all pairs of nodes after excluding node \(i\) from the graph \(G\). Let \(G_i\) denote the subgraph obtained by removing node \(i\) and its incident edges. The closeness vitality of node \(i\) is defined as\begin{equation*}c_{\text{vitality}}(i) = W(G) - W(G_i)= \sum_{j=1}^{N} d_G(i,j)+ \sum_{j \neq i} \sum_{k \neq i} \big( d_G(j,k) - d_{G_i}(j,k) \big),\end{equation*}where \(W(G)\) is the Wiener index of \(G\), defined as the total sum of shortest-path distances between all pairs of nodes in the network [3]. Here, \(d_G(i,j)\) denotes the shortest-path distance between nodes \(i\) and \(j\) in \(G\). A lower closeness vitality value indicates a more central node, since its removal causes a smaller increase in the total pairwise distances. However, if node \(i\) is a cut-vertex (or bridge endpoint), its removal disconnects the network, resulting in \(c_{\text{vitality}}(i) = -\infty\).

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] Brandes, U. (2005). Network analysis: methodological foundations (Vol. 3418). Springer Science & Business Media. doi: 10.1007/b106453.
[3] Wiener, H. (1947) Structural Determination of Paraffin Boiling Points. Journal of the American Chemical Society, 69, 17-20. doi: 10.1021/ja01193a005.