Nieminen’s closeness centrality
Nieminen's closeness centrality , originally designed for weakly connected directed graphs, measures a node's centrality by combining the total distance from node \(i\) to all other reachable nodes with the ability to reach a large number of nodes [2]. Let \(RP(i)\) denote the set of nodes reachable from \(i\) in the network \(G\). By definition, \(i \in RP(i)\). Then, the Nieminen's closeness centrality of node \(i\) is defined as\begin{equation*}c_{Nieminen}(i) =\begin{cases}\sum_{j \in RP(i)} \left(|RP(i)| - d_{ij} \right), & \text{if } |RP(i)| \geq 2,\\0, & \text{otherwise}.\end{cases}\end{equation*}where \(d_{ij}\) denotes the shortest-path distance between nodes \(i\) and \(j\).For unweighted and strongly connected networks, the Nieminen's closeness centrality can be expressed as\[c_{Nieminen}(i) = N^2 - \sum_{j=1}^N d_{ij},\]which is directly related to the sum of shortest-path distances from node \(i\) to all other nodes. In this case, the ranking of nodes by Nieminen's closeness is identical to the ranking obtained from the closeness centrality.