Expander graphs are globally synchronizing

TL;DR

This paper proves that graphs with strong expansion properties ensure global synchronization in the Kuramoto model, including Erdős-Rényi and Ramanujan graphs, advancing understanding of network synchronization.

Contribution

It establishes that sufficiently expanding graphs guarantee global synchronization in the Kuramoto model, confirming a conjecture and extending results to various random and regular graphs.

Findings

Erdős-Rényi graphs with p ≥ (1+ε)(log n)/n are almost surely globally synchronizing.

Ramanujan graphs exhibit global synchronization in the Kuramoto model.

Large degree regular graphs tend to be globally synchronizing.

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 $d$.