The improved neighbors’ k -core (INK) method , also known as neighborhood coreness or INK score, is an extension of the classical \(k\)-shell centrality [2, 3]. While nodes with the largest \(k\)-core values may have different spreading influence, INK accounts for the \(k\)-core values of neighboring nodes to better distinguish their influence.
The centrality of node \(i\) is defined as
\begin{equation*}
c_{\mathrm{INK}}(i) = \sum_{j \in \mathcal{N}(i)} k_s(j)^{α},
\end{equation*}
where \(k_s(j)\) is the \(k\)-core value of neighbor \(j\), \(\mathcal{N}(i)\) denotes the set of neighbors of node \(i\), and \(α\) is a tunable parameter controlling the contribution of neighbors’ influence.
Nodes connected to influential neighbors attain higher INK scores. The parameter \(α\) modulates this effect: for \(α < 1\), the influence of neighbors with large \(k\)-core values is weakened; when \(α = 1\), the INK score reduces to the sum of the neighbors’ \(k\)-core values.

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] Bae, J., & Kim, S. (2014). Identifying and ranking influential spreaders in complex networks by neighborhood coreness. Physica A: Statistical Mechanics and its Applications, 395, 549-559. doi: 10.1016/j.physa.2013.10.047.
[3] Lin, J. H., Guo, Q., Dong, W. Z., Tang, L. Y., & Liu, J. G. (2014). Identifying the node spreading influence with largest k-core values. Physics Letters A, 378(45), 3279-3284. doi: 10.1016/j.physleta.2014.09.054.