Expander graphs are globally synchronizing

arXiv2210.12788MaRDI QIDQ6506861

Pedro Abdalla, Victor Souza, Martin Kassabov, Afonso S. Bandeira, Steven Strogatz, Alex Townsend

Abstract: 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 $varepsilon>0$ and $pgeq(1+varepsilon)(logn)/n$, the Kuramoto model on the ErdH{o}s$unicodex2013$R'enyi 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 $dgeq600$ and that, for the same range of degrees, a $d$-regular random graph is typically globally synchronising.

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