Network-complement transitions, symmetries, and cluster synchronization

TL;DR

This paper investigates how specific network structures, called UCM networks, optimize synchronization thresholds and exhibit unique symmetry properties that influence cluster formation and coexistence with global synchronization.

Contribution

It introduces UCM networks as optimal for synchronization thresholds and explores their symmetry and cluster synchronization behaviors.

Findings

UCM networks have the lowest synchronization thresholds among comparable networks.

Symmetry properties of UCM networks lead to cluster synchronization.

Cluster states can coexist with global synchronization in these networks.

Abstract

Synchronization in networks of coupled oscillators is known to be largely determined by the spectral and symmetry properties of the interaction network. Here we leverage this relation to study a class of networks for which the threshold coupling strength for global synchronization is the lowest among all networks with the same number of nodes and links. These networks, defined as being uniform, complete, and multi-partite (UCM), appear at each of an infinite sequence of network-complement transitions in a larger class of networks characterized by having near-optimal thresholds for global synchronization. We show that the distinct symmetry structure of the UCM networks, which by design are optimized for global synchronizability, often leads to formation of clusters of synchronous oscillators, and that such states can coexist with the state of global synchronization.