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

TL;DR

This paper analyzes the spectral properties of the Laplacian matrix in temporal networks modeled as multilayer systems, revealing that eigenvectors associated with small eigenvalues can be approximated by combinations of layer-specific eigenvectors, thus generalizing the Fielder vector concept.

Contribution

It introduces a perturbation-based analysis of the supra-Laplacian in temporal networks, providing a closed-form approximation for eigenvectors related to small eigenvalues.

Findings

Eigenvectors near zero eigenvalues are approximated by layer eigenvectors.

The analysis generalizes the Fielder vector concept for multilayer temporal networks.

Perturbation theory is used to understand eigenvector behavior.

Abstract

We study the behavior of the eigenvectors associated with the smallest eigenvalues of the Laplacian matrix of temporal networks. We consider the multilayer representation of temporal networks, i.e. a set of networks linked through ordinal interconnected layers. We analyze the Laplacian matrix, known as supra-Laplacian, constructed through the supra-adjacency matrix associated with the multilayer formulation of temporal networks, using a constant block Jacobi model which has closed-form solution. To do this, we assume that the inter-layer weights are perturbations of the Kronecker sum of the separate adjacency matrices forming the temporal network. Thus we investigate the properties of the eigenvectors associated with the smallest eigenvalues (close to zero) of the supra-Laplacian matrix. Using arguments of perturbation theory, we show that these eigenvectors can be approximated by linear combinations of the zero eigenvectors of the individual time layers. This finding is crucial in reconsidering and generalizing the role of the Fielder vector in supra-Laplacian matrices.