Synchronization of Kuramoto Oscillators in Dense Networks

TL;DR

This paper investigates the synchronization of Kuramoto oscillators on dense networks, focusing on the energy landscape's properties, and improves bounds on the network density needed for all local maxima to be global.

Contribution

The paper provides a refined bound on the minimum degree proportion for which all local maxima of the energy landscape are global, improving previous results.

Findings

Improved the critical density bound to 0.7889 from 0.7929.

Confirmed the conjecture that the critical value might be 0.75.

Analyzed the energy landscape of Kuramoto oscillators on dense graphs.

Abstract

We study synchronization properties of systems of Kuramoto oscillators. The problem can also be understood as a question about the properties of an energy landscape created by a graph. More formally, let $G=(V,E)$ be a connected graph and $(a_{ij})_{i,j=1}^{n}$ denotes its adjacency matrix. Let the function $f:\mathbb{T}^n \rightarrow \mathbb{R}$ be given by $$ f(\theta_1, \dots, \theta_n) = \sum_{i,j=1}^{n}{ a_{ij} \cos{(\theta_i - \theta_j)}}.$$ This function has a global maximum when $\theta_i = \theta$ for all $1\leq i \leq n$. It is known that if every vertex is connected to at least $\mu(n-1)$ other vertices for $\mu$ sufficiently large, then every local maximum is global. Taylor proved this for $\mu \geq 0.9395$ and Ling, Xu \& Bandeira improved this to $\mu \geq 0.7929$. We give a slight improvement to $\mu \geq 0.7889$. Townsend, Stillman \& Strogatz suggested that the critical value might be $\mu_c = 0.75$.