Phase Synchronization in Random Geometric Graphs on the 2D Sphere

TL;DR

This work analyzes phase synchronization of the Kuramoto model on a random geometric graph built from $n$ i.i.d. points on $\mathbb{S}^2$ with local interactions of range $\sqrt{\epsilon}$. A central contribution is a scaling limit: solutions of the discrete Kuramoto dynamics converge to the heat equation $\partial_t u = \kappa \Delta_{\mathbb{S}^2} u$ on $\mathbb{S}^2$ (with $\mathbb{S}^1$-valued states interpreted via a real-valued lift) on finite time intervals, provided the initial data converge to a smooth function $u_0$; this links discrete network dynamics to a well-understood PDE. Leveraging the heat equation’s global synchronization and a basin-of-attraction result for any connected graph, the authors deduce that the Kuramoto system synchronizes with high probability as $n\to\infty$ when the initial data are smooth and the regime $\epsilon\to0$ with $\epsilon^2 n/\log n\to\infty$ holds. They develop an integral-equation framework, prove well-posedness and convergence of intermediate dynamics to the heat flow, and establish key Lipschitz and operator-convergence estimates to justify the continuum limit. The study highlights how sphere topology and simple-connectivity influence synchronization and suggests extensions to more general simply-connected manifolds, while noting potential failure of global synchronization on non-simply-connected spaces.

Abstract

The Kuramoto model is a classical nonlinear ODE system designed to study synchronization phenomena. Each equation represents the phase of an oscillator and the coupling between them is determined by a graph. There is an increasing interest in understanding the relation between the graph topology and the spontaneous synchronization of the oscillators. Abdalla, Bandeira and Invernizzi considered random geometric graphs on the $d$-dimensional sphere and proved that the system synchronizes with high probability as long as the mean number of neighbors and the dimension $d$ go to infinity. They posed the question about the behavior when $d$ is small. In this paper, we prove that synchronization holds for random geometric graphs on the two-dimensional sphere, with high probability as the number of nodes goes to infinity, as long as the initial conditions converge to a smooth function. We conjecture a similar behavior for more general simply-connected closed Riemannian manifolds but we expect global synchronization to fail if the manifold is not simply-connected, as was shown in [11] and suggested in [9].

Figures (2)

• Figure 1: A connected random geometric graph on $\mathbb{S}^2$, where each node supports a Kuramoto oscillator whose phase on $\mathbb{S}^1$ is colored.
• Figure 2: Solar time. Different colors represent different values of $u$. Since the function is interpreted as taking values in $\mathbb S^1$, the function is continuous at the "International Day Line", but not at the poles. Our theorem does not discard the possibility of a stable state close to it, although numerical computations suggest it is not the case.

Theorems & Definitions (19)

• Theorem 2.1
• Remark 2.1
• proof
• Proposition 2.2
• Proposition 2.3: cirelli2024scaling
• Theorem 2.4
• Proposition 3.1
• proof
• Proposition 3.2
• proof