Laplacian Spectra as a Diagnostic Tool for Network Structure and Dynamics

TL;DR

This paper investigates how Laplacian spectra reflect network topology and influence synchronization dynamics, revealing that spectral features can diagnose and predict complex synchronization behaviors.

Contribution

It introduces a spectral diagnostic approach linking Laplacian eigenvalues and eigenvectors to network structure and synchronization states, advancing understanding of network dynamics.

Findings

Laplacian spectra encode topological features like clustering and degree distribution.

Synchronization transitions can be visualized through eigenmode analysis.

Partially synchronized states involve specific eigenmodes remaining unlocked.

Abstract

We examine numerically the three-way relationships among structure, Laplacian spectra and frequency synchronization dynamics on complex networks. We study the effects of clustering, degree distribution and a particular type of coupling asymmetry (input normalization), all of which are known to have effects on the synchronizability of oscillator networks. We find that these topological factors produce marked signatures in the Laplacian eigenvalue distribution and in the localization properties of individual eigenvectors. Using a set of coordinates based on the Laplacian eigenvectors as a diagnostic tool for synchronization dynamics, we find that the process of frequency synchronization can be visualized as a series of quasi-independent transitions involving different normal modes. Particular features of the partially synchronized state can be understood in terms of the behavior of particular modes or groups of modes. For example, there are important partially synchronized states in which a set of low-lying modes remain unlocked while those in the main spectral peak are locked. We find therefore that spectra influence dynamics in ways that go beyond results relating a single threshold to a single extremal eigenvalue.