The k -shell Physarum centrality is a hybrid measure that combines Physarum centrality and \(k\)-shell centrality to identify influential nodes in weighted networks [2]. The centrality of node \(i\) is defined as
\[
c_{kp}(i) = c_{\mathrm{Physarum}}(i) \cdot \frac{k_s(i)}{\sum_{j=1}^N k_s(j)},
\]
where \(c_{\mathrm{Physarum}}(i)\) is the Physarum centrality of node \(i\) and \(k_s(i)\) is its \(k\)-shell index. This formulation weights the Physarum centrality by the relative position of the node within the \(k\)-shell decomposition, emphasizing nodes that are both structurally central and carry significant flux.

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] Gao, C., Lan, X., Zhang, X., & Deng, Y. (2013). A bio-inspired methodology of identifying influential nodes in complex networks. PloS one, 8(6), e66732. doi: 10.1371/journal.pone.0066732.