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Expander graphs are globally synchronising

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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. 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 p ≥ (1 + eps)(log n)/n, the Kuramoto model on the Erdős—Rényi graph G(n,p) is globally synchronising with high probability, settling a conjecture of Ling, Xu and Bandeira. We also show the global synchrony of any d-regular Ramanujan graph with d ≥ 600.

Joint work with P. Abdalla, A. Bandeira, M. Kassabov, S. Strogatz and A. Townsend.

This talk is part of the Probability series.

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