Density centrality is a semi-local measure inspired by Newton’s gravity formula [2]. The method draws an analogy to area density in physics, measuring how much “mass” (node degree) is distributed within a fixed spatial region.
The centrality \(c_{\textsc{Density}}(i)\) of node \(i\) is defined as
\begin{equation*}
c_{\textsc{Density}}(i) = \sum_{j \in \mathcal{N}^{(\leq l)}(i)} \frac{d_i}{π\, d_{ij}^2},
\end{equation*}
where \(\mathcal{N}^{(\leq l)}(i)\) denote the set of nodes within \(l\)-hop neighborhood of node \(i\), \(d_{ij}\) is the shortest-path distance between nodes \(i\) and \(j\), and \(d_i\) is the degree of node \(i\). Ibnoulouafi and El Haziti [2] set \(l = 3\) as the truncated radius.

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] Ibnoulouafi, A., El Haziti, M., & Cherifi, H. (2018). M-centrality: identifying key nodes based on global position and local degree variation. Journal of Statistical Mechanics: Theory and Experiment, 2018(7), 073407. doi: 10.1088/1742-5468/aace08.