k-betweenness centrality
The
\(k\)-betweenness centrality
, also known as bounded-distance betweenness (BDBC) [2] or range-limited betweenness centrality [3], is a variant of the standard betweenness centrality in which only paths of length at most \(k\) are considered [4]. It is formally defined as
\begin{equation*}
c_{k-betw}(i) = \sum_{\substack{j\neq l \neq i \\ d_{jl} \leq k}}{\frac{σ_{jl}(i)}{σ_{jl}}},
\end{equation*}
where \(σ_{jl}\) denotes the number of shortest paths from node \(j\) to node \(l\), and \(σ_{jl}(i)\) represents the number of paths that pass through node \(i\). The rationale behind the \(k\)-betweenness centrality is that very long paths are less likely to be used in real processes and should therefore contribute less, or not at all, to a node’s centrality. By restricting attention to shorter paths, \(k\)-betweenness provides a more localized measure of a node’s role in network flow.