Laplacian centrality

The concept of Laplacian centrality quantifies the importance of a node based on the network's response to its removal [2]. Formally, the Laplacian centrality \(c_{\text{Laplacian}}(i)\) of node \(i\) is defined as the relative drop in the Laplacian energy of the graph \(G\) upon deletion of \(i\):\begin{equation*}c_{\text{Laplacian}}(i) = \frac{E_L(G) - E_L(G_i)}{E_L(G)},\end{equation*}where \(E_L(G)\) denotes the Laplacian energy of \(G\), given by\begin{equation*}E_L(G) = \sum_{k=1}^{N} λ_k^2 = \sum_{i=1}^{N} \left(\sum_{j=1}^{N} w_{ij} \right)^2 + 2 \sum_{i et al. [2] also showed that the Laplacian centrality of a node is closely related to the number of 2-walks in which it participates.

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] Qi, X., Fuller, E., Wu, Q., Wu, Y., & Zhang, C. Q. (2012). Laplacian centrality: A new centrality measure for weighted networks. Information Sciences, 194, 240-253. doi: 10.1016/j.ins.2011.12.027.