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.