Proximal betweenness centrality
Proximal betweenness centrality
is a variant of traditional betweenness centrality that quantifies how frequently a node appears in a
penultimate
position along a shortest path [2, 3]. The proximal betweenness centrality of a node \(i\) is defined as
\begin{equation*}
c_{\mathrm{pr.betw}}(i) = \sum_{j=1}^{N} \sum_{k=1}^{N} \frac{b_{jk}(i)}{σ_{jk}},
\end{equation*}
where \(σ_{jk}\) denotes the total number of shortest paths from node \(j\) to node \(k\), and \(b_{jk}(i)\) is the number of those paths in which node \(i\) occupies the
penultimate
position. In this context, a node \(i\) is considered penultimate if it lies either directly before the destination node \(k\) (proximal to the target) or directly after the source node \(j\) (proximal to the source) on a shortest path. Following Borgatti
et al.
[2], two variants of proximal betweenness centrality can be distinguished:
- Proximal source betweenness: counts node \(i\) when it occurs immediately before the destination node \(k\) on shortest paths originating from node \(j\);
- Proximal target betweenness: counts node \(i\) when it occurs immediately after the source node \(j\) on shortest paths terminating at node \(k\).