The improved k -shell decomposition (IKSD) is a variant of the standard \(k\)-shell decomposition [2], designed to better capture the hierarchical importance of nodes in a network. Unlike the standard \(k\)-shell, IKSD refines node ranking by iteratively removing nodes starting from those with the lowest degree and recalculating degrees after each removal. The process proceeds as follows:


  1. Initialize \(k = 1\) and consider the current graph \(G\).

  2. Identify all nodes in \(G\) with the minimum degree.

  3. Assign an IKSD value of \(k\) to these nodes.

  4. Remove the identified nodes from the graph and update the degrees of remaining nodes.

  5. Increment \(k\) and repeat steps 2-4 until all nodes have been assigned an IKSD value.


Nodes with higher IKSD values are located in the core of the network and are considered structurally more influential, whereas nodes with lower IKSD values occupy the periphery.

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] Liu, Z., Jiang, C., Wang, J., & Yu, H. (2015). The node importance in actual complex networks based on a multi-attribute ranking method. Knowledge-Based Systems, 84, 56-66. doi: 10.1016/j.knosys.2015.03.026.