Synchronization on Effective Networks

TL;DR

This paper investigates effective networks, revealing that their spectral properties determine synchronizability, while synchronization time depends mainly on network depth and link distribution, providing new insights into network synchronization dynamics.

Contribution

It introduces the concept of effective networks, proves their spectral uniformity, and analyzes how network depth and link types influence synchronization time.

Findings

All effective networks of the same size share the same spectra.

Synchronization time increases with network depth.

Horizontal links slow down synchronization, vertical links speed it up.

Abstract

The study of network synchronization has attracted increasing attention recently. In this paper, we strictly define a class of networks, namely effective networks, which are synchronizable and orientable networks. We can prove that all the effective networks with the same size have the same spectra, and are of the best synchronizability according to the master stability analysis. However, it is found that the synchronization time for different effective networks can be quite different. Further analysis show that the key ingredient affecting the synchronization time is the maximal depth of an effective network: the larger depth results in a longer synchronization time. The secondary factor is the number of links. The more links connecting the nodes in the same layer (horizontal links) will lead to longer synchronization time, while the increasing number of links connecting nodes in neighboring layers (vertical links) will accelerate the synchronization. Our findings provide insights into the roles of horizontal and vertical links in synchronizing process, and suggest that the spectral analysis is helpful yet insufficient for the understanding of network synchronization.