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We study the behavior of the eigenvectors associated with the smallesteigenvalues of the Laplacian matrix of temporal networks. We consider themultilayer representation of temporal networks, i.e. a set of networks linkedthrough ordinal interconnected layers. We analyze the Laplacian matrix, knownas supra-Laplacian, constructed through the supra-adjacency matrix associatedwith the multilayer formulation of temporal networks, using a constant blockJacobi model which has closed-form solution. To do this, we assume that theinter-layer weights are perturbations of the Kronecker sum of the separateadjacency matrices forming the temporal network. Thus we investigate theproperties of the eigenvectors associated with the smallest eigenvalues (closeto zero) of the supra-Laplacian matrix. Using arguments of perturbation theory,we show that these eigenvectors can be approximated by linear combinations ofthe zero eigenvectors of the individual time layers. This finding is crucial inreconsidering and generalizing the role of the Fielder vector insupra-Laplacian matrices.

Spectral properties of the Laplacian of temporal networks following a constant block Jacobi model