Guarantees for Spontaneous Synchronization on Random Geometric Graphs

TL;DR

This paper proves that the Kuramoto model on random geometric graphs on the sphere almost surely synchronizes as the number of nodes grows large, filling a gap in understanding the role of geometry in network synchronization.

Contribution

It provides the first rigorous proof of synchronization for the Kuramoto model on random geometric graphs, incorporating geometric structure into the analysis.

Findings

Synchronization occurs with probability tending to one as nodes increase.

First rigorous result for Kuramoto model on geometric random graphs.

Utilizes tools from random matrix theory and statistics.

Abstract

The Kuramoto model is a classical mathematical model in the field of non-linear dynamical systems that describes the evolution of coupled oscillators in a network that may reach a synchronous state. The relationship between the network's topology and whether the oscillators synchronize is a central question in the field of synchronization, and random graphs are often employed as a proxy for complex networks. On the other hand, the random graphs on which the Kuramoto model is rigorously analyzed in the literature are homogeneous models and fail to capture the underlying geometric structure that appears in several examples. In this work, we leverage tools from random matrix theory, random graphs, and mathematical statistics to prove that the Kuramoto model on a random geometric graph on the sphere synchronizes with probability tending to one as the number of nodes tends to infinity. To the best of our knowledge, this is the first rigorous result for the Kuramoto model on random geometric graphs.