δ-closeness centrality
Agneessens et al. [2] propose a generalized version of closeness centrality referred to as \(δ\)-closeness , which incorporates a tuning parameter \(δ \in \mathbb{R}\), reflecting the importance of geodesic distances in the network. The \(δ\)-closeness of a node \(i\) can be expressed as\begin{equation*}c_{δ-cl}(i) = \frac{\sum_{j \neq i}d_{ij}^{-δ}}{N-1},\end{equation*}where \(d_{ij}\) is the length of the shortest path from node \(i\) to node \(j\). Agneessens et al. [2] demonstrated that, by varying the parameter \(δ\), degree centrality and harmonic centrality can be viewed as specific instances of the generalized \(δ\)-closeness centrality: for \(δ \rightarrow \infty\), the index is proportional to degree centrality, whereas for \(δ = 1\), \(δ\)-closeness coincides with harmonic centrality.