%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 }