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The Kuramoto model is a prototypical model used for rigorous mathematical analysis in the field of synchronisation and nonlinear dynamics. A realisation of this model consists of a collection of identical 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 synchronising, meaning that the Kuramoto model on such a graph will converge to the fully synchronised state with all the oscillators with 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)(logn)/n, the Kuramoto model on the Erdős–Rényi graph G(n,p) is globally synchronising 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 synchronising on any d-regular Ramanujan graph with d≥600 and that, for the same range of degrees, a d-regular random graph is typically globally synchronising.