NL centrality is a semi-local measure that extends the DIL centrality [2] by accounting for the contributions of second-degree neighbors and the structural importance of edges [3]. The NL centrality of node \(i\) is defined as
\begin{equation*}
c_{NL}(i) = \sum_{j \in \mathcal{N}(i)} \left[ φ(j) +
\left( \frac{(d_i - \Delta_{ij} - 1)(d_j - \Delta_{ij} - 1)}{\Delta_{ij}/2 + 1} \right)
\left( \frac{d_i - 1}{d_i + d_j - 2} \right) \right],
\end{equation*}
where \(\mathcal{N}(i)\) is the set of neighbors of node \(i\), \(d_i\) is the degree of node \(i\), \(\Delta_{ij}\) denotes the number of triangles containing the edge \((i,j)\), and \(φ(j)=|\mathcal{N}^{(\leq 2)}(j)|\) counts the number of nodes within two steps of node \(j\). The NL centrality considers both edge-level clustering and the connectivity of second-degree neighbors, capturing information beyond immediate neighbors.

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, J., Xiong, Q., Shi, W., Shi, X., & Wang, K. (2016). Evaluating the importance of nodes in complex networks. Physica A: Statistical Mechanics and its Applications, 452, 209-219. doi: 10.1016/j.physa.2016.02.049.
[3] Shao, Z., Liu, S., Zhao, Y., & Liu, Y. (2019). Identifying influential nodes in complex networks based on Neighbours and edges. Peer-to-Peer Networking and Applications, 12, 1528-1537. doi: 10.1007/s12083-018-0681-x.