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.

References

[1] Shvydun, S. (2025). Zoo of Centralities: Encyclopedia of Node Metrics in Complex Networks. arXiv: 2511.05122 https://doi.org/10.48550/arXiv.2511.05122
[2] Latora, V., & Marchiori, M. (2001). Efficient behavior of small-world networks. Physical review letters, 87(19), 198701. doi: 10.1103/PhysRevLett.87.198701.
[3] Achard, S., & Bullmore, E. (2007). Efficiency and cost of economical brain functional networks. PLoS computational biology, 3(2), e17. doi: 10.1371/journal.pcbi.0030017.
[4] Agneessens, F., Borgatti, S. P., & Everett, M. G. (2017). Geodesic based centrality: Unifying the local and the global. Social Networks, 49, 12-26. doi: 10.1016/j.socnet.2016.09.005.
[5] Zhou, X., Zhang, F. M., Li, K. W., Hui, X. B., & Wu, H. S. (2012). Finding vital node by node importance evaluation matrix in complex networks. Acta Phys. Sin., 61(5): 050201. doi: 10.7498/aps.61.050201.
[6] Harris, C. D. (1954). The, Market as a Factor in the Localization of Industry in the United States. Annals of the association of American geographers, 44(4), 315-348. doi: 10.1080/00045605409352140.
[7] Beauchamp, M. A. (1965). An improved index of centrality. Behavioral science, 10(2), 161-163. doi: 10.1002/bs.3830100205.
[8] Rochat, Y. (2009). Closeness centrality extended to unconnected graphs: The harmonic centrality index (Tech. Rep.).