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
with \(λ_k\) representing the eigenvalues of the Laplacian matrix of \(G\), and \(w_{ij}\) the weight of the edge between nodes \(i\) and \(j\). Here, \(G_i\) denotes the graph obtained by deleting node \(i\) from \(G\). Qi 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.