Harmonic centrality
Harmonic centrality
(also known as Latora closeness centrality [2], nodal efficiency [3], reciprocal closeness [4] or efficiency centrality [5]) was introduced in [6, 7] and discussed in [8]. It is an extension of closeness centrality, in which the centrality of node \(i\) is computed as the sum of the inverse distances to all other nodes, i.e.,
\begin{equation*}
c_{harmonic}(i) = \sum_{j \neq i}{\frac{1}{d_{ij}}}.
\end{equation*}
where \(d_{ij}\) is the length of the shortest path from node \(i\) to node \(j\). Intuitively, harmonic centrality quantifies a node’s closeness to all others by summing the reciprocals of shortest-path distances, remaining well-defined even in disconnected networks: if no path exists between a pair of nodes, the shortest-path distance is considered infinite, and consequently, their contribution to the sum is taken as zero.