Expander graphs are globally synchronizing

Abstract

The Kuramoto model is fundamental to the study of synchronization. It consists of a collection of oscillators with interactions given by a network, which we identify respectively with vertices and edges of a graph. In this paper, we show that a graph with sufficient expansion must be globally synchronizing, meaning that a homogeneous Kuramoto model of identical oscillators on such a graph will converge to the fully synchronized state with all the oscillators having the same phase, for every initial state up to a set of measure zero. In particular, we show that for any ε>0 and p⩾(1+ε)(log⁡n)/n, the homogeneous Kuramoto model on the Erdős–Rényi random graph G(n,p) is globally synchronizing with probability tending to one as n goes to infinity. This improves on a previous result of Kassabov, Strogatz, and Townsend and solves a conjecture of Ling, Xu, and Bandeira. We also show that the Kuramoto model is globally synchronizing on any d -regular Ramanujan graph, and on typical d -regular graphs, for d⩾600.