δ-betweenness centrality
Agneessens et al. [2] propose a generalized version of betweenness centrality referred to as
\(δ\)-betweenness
, which incorporates a tuning parameter \(δ \in \mathbb{R}\), reflecting the relative importance of geodesic distances in the network. The \(δ\)-betweenness of a node \(i\) can be expressed as
\begin{equation*}
c_{δ-betw}(i) = \sum_{j=1}^{N}{\sum_{k=1}^{N}{\frac{σ_{jk}(i)}{σ_{jk}}(d_{jk}-1)^{-δ}}},
\end{equation*}
where \(σ_{jk}\) is the number of shortest paths between nodes \(j\) and \(k\), \(σ_{jk}(i)\) is the number of paths that pass through node \(i\), and \(d_{jk}\) is the length of the shortest path from \(j\) to \(k\). Note that for \(δ = 0\), the \(δ\)-betweenness centrality reduces to the standard betweenness centrality.