Dynamical and spectral properties of complex networks

TL;DR

This paper investigates how the spectral properties of the Laplacian matrix influence the dynamical behavior of complex networks, especially focusing on synchronization times and their dependence on eigenvalues.

Contribution

It demonstrates that the synchronization time primarily depends on the smallest nonzero Laplacian eigenvalue, highlighting its importance over adjacency spectrum.

Findings

Synchronization time correlates with the smallest nonzero Laplacian eigenvalue.

Laplacian spectral properties are more relevant for network dynamics than adjacency spectrum.

Different network types exhibit distinct synchronization behaviors based on Laplacian eigenvalues.

Abstract

Dynamical properties of complex networks are related to the spectral properties of the Laplacian matrix that describes the pattern of connectivity of the network. In particular we compute the synchronization time for different types of networks and different dynamics. We show that the main dependence of the synchronization time is on the smallest nonzero eigenvalue of the Laplacian matrix, in contrast to other proposals in terms of the spectrum of the adjacency matrix. Then, this topological property becomes the most relevant for the dynamics.