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arXiv · 2210.12788

Expander graphs are globally synchronizing

Pedro Abdalla·Afonso S. Bandeira·Martin Kassabov·Victor Souza·Steven H. Strogatz·Alex Townsend
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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 $\varepsilon > 0$ and $p \geq (1 + \varepsilon) (\log n) / n$, the homogeneous Kuramoto model on the Erd\H{o}s-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 $d$.

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Pedro Abdalla, Afonso S. Bandeira, Martin Kassabov, Victor Souza, Steven H. Strogatz, Alex Townsend. 2022-10-23. Expander graphs are globally synchronizing. https://doi.org/10.1016/j.aim.2025.110773

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