Expander graphs are globally synchronizing

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 varepsilon > 0 and p ≥ (1 + varepsilon) (log n) / n, the homogeneous Kuramoto model on the ErdHos-R'enyi 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 model is globally synchronizing on any d-regular Ramanujan graph, and on typical d-regular graphs, for large enough degree d.