The Node local centrality (NLC) is a centrality measure for identifying influential spreaders by combining network embedding (NE) with local network information [2].
For a node \(i\), the NLC centrality \(c_{\textsc{NLC}}(i)\) is defined as
\[
c_{\textsc{NLC}}(i) = \sum_{j \in \mathcal{N}(i)} k_s(i) \, e^{-\|x_i - x_j\|^2},
\]
where \(k_s(i)\) is the \(k\)-shell index of node \(i\) and \(x_i \in \mathbb{R}^{r \times 1}\) is the vector embedding of node \(i\) obtained via the DeepWalk network representation method [3].
Yang et al. [2] set the embedding dimension to \(r = N/2\). The measure captures both the hierarchical position of a node in the network (via \(k\)-shell) and its proximity in the embedded low-dimensional space, reflecting local structural similarity.

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] Yang, X. H., Xiong, Z., Ma, F., Chen, X., Ruan, Z., Jiang, P., & Xu, X. (2021). Identifying influential spreaders in complex networks based on network embedding and node local centrality. Physica A: Statistical Mechanics and its Applications, 573, 125971. doi: 10.1016/j.physa.2021.125971.
[3] Perozzi, B., Al-Rfou, R., & Skiena, S. (2014). Deepwalk: Online learning of social representations. In Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining (pp. 701-710). doi: 10.1145/2623330.2623732.