The distance-weighted fragmentation (DF) centrality quantifies the effect of a node on the overall connectivity of a network by considering the average reciprocal distance among nodes after its removal [2, 3]. For a node \(i\), the DF centrality, denoted \(c_{DF}(i)\), is defined as
\begin{equation*}
c_{DF}(i) = 1 - \frac{\sum_{j \neq i} \sum_{k \neq i} d_{jk}^{-1}(G_i)}{(N-1)(N-2)},
\end{equation*}
where \(G_i\) is the subgraph obtained by removing node \(i\) from \(G\), and \(d_{jk}(G_i)\) is the shortest-path distance between nodes \(j\) and \(k\) in \(G_i\).
The DF centrality ranges from 0, when the network remains fully connected (as in a complete graph), to 1, when all nodes are isolated. Intermediate values indicate the extent to which the removal of a node increases distances in the network, thus reflecting its importance in maintaining overall network connectivity.

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] Borgatti, S. P. (2003). The key player problem. Washington, D.C.: National Academy of Sciences Press. pp. 241-252. doi: 10.2139/ssrn.1149843.
[3] Borgatti, S. P., & Everett, M. G. (2006). A graph-theoretic perspective on centrality. Social networks, 28(4), 466-484. doi: 10.1016/j.socnet.2005.11.005.