Closeness vitality
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 \(c_{\text{vitality}}(i) = -\infty\).