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 $> 0$ and $p (1 + \varepsilon) (n) / n$, the Kuramoto model on the Erd\Hos$x2013$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.
%0 Journal Article
%1 abdalla2022expander
%A Abdalla, Pedro
%A Bandeira, Afonso S.
%A Kassabov, Martin
%A Souza, Victor
%A Strogatz, Steven H.
%A Townsend, Alex
%D 2022
%K combinatorics complexity mathematics nonlinear_dynamics synchronization
%T Expander graphs are globally synchronising
%U http://arxiv.org/abs/2210.12788
%X 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 $> 0$ and $p (1 + \varepsilon) (n) / n$, the Kuramoto model on the Erd\Hos$x2013$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.
@article{abdalla2022expander,
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 $> 0$ and $p (1 + \varepsilon) (n) / n$, the Kuramoto model on the Erd\Hos$x2013$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.},
added-at = {2023-09-03T12:05:23.000+0200},
author = {Abdalla, Pedro and Bandeira, Afonso S. and Kassabov, Martin and Souza, Victor and Strogatz, Steven H. and Townsend, Alex},
biburl = {https://www.bibsonomy.org/bibtex/2261c2eaaa1f960b0841e1b332f674b7d/tabularii},
interhash = {4e846a5f44d6f7745c53762453130082},
intrahash = {261c2eaaa1f960b0841e1b332f674b7d},
keywords = {combinatorics complexity mathematics nonlinear_dynamics synchronization},
timestamp = {2023-09-03T12:28:17.000+0200},
title = {Expander graphs are globally synchronising},
type = {Publication},
url = {http://arxiv.org/abs/2210.12788},
year = 2022
}