Synchronization in random networks of identical phase oscillators: A graphon approach

Networks of coupled nonlinear oscillators have been used to model circadian rhythms, flashing fireflies, Josephson junction arrays, high-voltage electric grids, and many other kinds of self-organizing systems. Recently, several authors have sought to understand how coupled oscillators behave when they interact according to a random graph. Here we consider interaction networks generated by a graphon model known as a \(W\)-random network, and examine the dynamics of an infinite number of identical phase oscillators, following an approach pioneered by Medvedev. We show that with sufficient regularity on \(W\), the solution to the dynamical system over a \(W\)-random network of size \(n\) converges in the \(L^{\infty}\) norm to the solution of the continuous graphon system, with high probability as \(n\rightarrow\infty\). This result suggests a framework for studying synchronization properties in large but finite random networks. In this paper, we leverage our convergence result in the \(L^{\infty}\) norm to prove synchronization results for two classes of identical phase oscillators on Erdős-Rényi random graphs. First, we show that the Kuramoto model on the Erdős-Rényi graph \(G(n, \alpha_n)\) achieves phase synchronization with high probability as \(n\) goes to infinity, if the edge probability \(\alpha_n\) exceeds \((\log n)/n\), the connectivity threshold of an Erdős-Rényi random graph. Then we show that the Sakaguchi-Kuramoto model on the Erdős-Rényi graph \(G(n, p)\) achieves frequency synchronization with high probability as \(n\) goes to infinity, assuming a fixed edge probability \(p\in(0,1]\) and a certain regime for the model's phase shift parameter. A notable feature of the latter result is that it holds for an oscillator model whose dynamics are not simply given by a gradient flow.