Network synchronization: Optimal and Pessimal Scale-Free Topologies

TL;DR

This paper presents a method to design and analyze scale-free networks for optimal and minimal synchronization, revealing how degree correlations influence network stability and introducing the concept of 'pessimal' networks.

Contribution

It introduces an optimization algorithm for designing networks with specific synchronizability properties and analyzes the effects of degree correlations on these properties.

Findings

Optimally synchronizable networks tend to be disassortative.

Pessimal networks are highly assortative and string-like.

A lower bound for the Laplacian eigenvalue ratio is derived.

Abstract

By employing a recently introduced optimization algorithm we explicitely design optimally synchronizable (unweighted) networks for any given scale-free degree distribution. We explore how the optimization process affects degree-degree correlations and observe a generic tendency towards disassortativity. Still, we show that there is not a one-to-one correspondence between synchronizability and disassortativity. On the other hand, we study the nature of optimally un-synchronizable networks, that is, networks whose topology minimizes the range of stability of the synchronous state. The resulting ``pessimal networks'' turn out to have a highly assortative string-like structure. We also derive a rigorous lower bound for the Laplacian eigenvalue ratio controlling synchronizability, which helps understanding the impact of degree correlations on network synchronizability.