The k -shell centrality (also called node coreness, coreness centrality, or k -coreness centrality [2]) is based on the k -shell (or k -core) decomposition of a network. The k -core of a graph \(G\) is defined as the maximal subgraph in which every node has degree at least \(k\) [3, 4].Nodes in the \(k\)-core may also belong to higher-order \((k{+}1)\)-cores. The k -shell is then defined as the set of nodes that belong to the \(k\)-core but not to any higher-order \((k{+}1)\)-core. The identification of the \(k\)-shell can be performed iteratively as follows:


  1. Remove all nodes with degree less than 1 and their incident links. Repeat until no such nodes remain. The removed nodes constitute the \(k_s = 1\) shell.

  2. Remove all nodes with degree less than 2 in the remaining network and their links. Repeat until no such nodes remain. These nodes form the \(k_s = 2\) shell.

  3. Continue this iterative process for increasing \(k\) until all nodes are assigned a shell index \(k_s\).


The k -shell centrality of a node \(i\) is the index \(k_s\) of the highest-order core containing \(i\) [5]. Intuitively, nodes in higher-order shells are considered more central because they are embedded in denser, more interconnected regions of the network. Kitsak et al. [6] showed that nodes with high k -shell indices often play a more influential role in spreading processes than nodes with high degree. However, a limitation of k -shell centrality is its relatively low resolution: since the k -shell index takes only a small number of discrete values, it may not always distinguish effectively between nodes in large or heterogeneous networks.

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] Van Mieghem P. \emph{Performance Analysis of Complex Networks and Systems}. Cambridge University Press; 2014. doi: 10.1017/CBO9781107415874.
[3] Seidman, S. B. (1983). Network structure and minimum degree. Social networks, 5(3), 269-287. doi: 10.1016/0378-8733(83)90028-X.
[4] Bollobás, B. (1984). Graph Theory and Combinatorics: Proceedings of the Cambridge Combinatorial Conference in Honor of P. Erdös Vol. 35 (Academic, 1984).
[5] Batagelj V., Zaveršnik M. (2002) Generalized Cores, arXiv:0202039. doi: 10.48550/arXiv.cs/0202039.
[6] Kitsak, M., Gallos, L. K., Havlin, S., Liljeros, F., Muchnik, L., Stanley, H. E., & Makse, H. A. (2010). Identification of influential spreaders in complex networks. Nature physics, 6(11), 888-893. doi: 10.1038/nphys1746.