Decay centrality is a centrality measure that accounts for path lengths by assigning a weight to each path that decreases exponentially with its length [2]. Specifically, the contribution of a path is given by a decay parameter \(δ \in (0,1)\) raised to the power of the path length, summarizing the diminishing influence of distant nodes, i.e.,
\begin{equation*}
c_{decay}(i) = \sum_{j \neq i}{δ^{d_{ij}}}.
\end{equation*}
The decay centrality can be interpreted as the expected number of nodes that can reach \(i\) via shortest paths, where the probability of a successful move is defined by \(δ\). The value of \(δ\) depends on the network, but it is commonly assumed to be \(δ = 0.5\).

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] Jackson, M. O. (2008). Social and economic networks (Vol. 3, p. 519). Princeton: Princeton university press. doi: 10.2307/j.ctvcm4gh1.