The \(k\)-shell iteration factor (KS-IF) ranks nodes by combining their degree and \(k\)-shell decomposition information [2]. The centrality of node \(i\) is defined as\begin{equation*}c_{KS-IF}(i) = δ(i) d_i + \sum_{j \in \mathcal{N}(i)} δ(j) d_j,\end{equation*}where \(d_i\) is the degree of node \(i\) and \(δ(i)\) is the \(k\)-shell iteration factor given by\begin{equation*}δ(i) = k_s(i) \left( 1 + \frac{n(i)}{m(i)} \right).\end{equation*}Here, \(k_s(i)\) is the \(k\)-shell index of node \(i\), \(n(i)\) is the iteration at which node \(i\) is removed, and \(m(i)\) is the total number of iterations in that step. KS-IF combines node degree and \(k\)-shell position to better identify influential nodes.

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] Wang, Z., Zhao, Y., Xi, J., & Du, C. (2016). Fast ranking influential nodes in complex networks using a k-shell iteration factor. Physica A: Statistical Mechanics and its Applications, 461, 171-181. doi: 10.1016/j.physa.2016.05.048.