Quasi-Laplacian centrality (QC)
The
quasi-Laplacian centrality
(QC) is an eigenvalue-based method for identifying influential nodes in complex networks, derived from the concept of graph energy [2]. The main idea behind QC is that the importance (centrality) of a node \(i\) is reflected by the variation in the quasi-Laplacian energy resulting from the removal of node \(i\) from the network.
The quasi-Laplacian centrality \(c_{QC}(i)\) of node \(i\) is defined as
\[
c_{QC}(i) = E_Q(G) - E_Q(G_i),
\]
where \(G_i\) denotes the subgraph of \(G\) obtained by removing node \(i\), and \(E_Q(G)\) is the quasi-Laplacian energy of the network \(G\), given by
\[
E_Q(G) = \sum_{j=1}^{N} μ_j^2,
\]
where \(μ_1, \dots, μ_N\) are the eigenvalues of the
quasi-Laplacian
matrix \(Q = D + A\), with \(D\) being the diagonal degree matrix and \(A\) the adjacency matrix.
The quasi-Laplacian centrality \(c_{QC}(i)\) can be further simplified as
\[
c_{QC}(i) = d_i^2 + d_i + \sum_{j \in \mathcal{N}(i)} d_j,
\]
where \(d_i\) denotes the degree of node \(i\) and \(\mathcal{N}(i)\) is the set of its neighboring nodes.
The quasi-Laplacian centrality has been tested on Zachary’s karate club network and several terrorist networks, and its effectiveness has been validated using the susceptible-infected-recovered (SIR) epidemic model.