Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Guarantees for Spontaneous Synchronization on Random Geometric Graphs

Pedro Abdalla, Afonso S. Bandeira, Clara Invernizzi

TL;DR

This work uses 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.

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.

Guarantees for Spontaneous Synchronization on Random Geometric Graphs

TL;DR

This work uses 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.

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.
Paper Structure (5 sections, 10 theorems, 33 equations)

This paper contains 5 sections, 10 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Kuramoto Model on random graphs
  3. Preliminaries on the Kuramoto Model
  4. The Random Geometric Graph on the Sphere
  5. Improvement when $p$ is sublinear.

Key Result

Theorem 1

(Simplified Statement of the Main Result Theorem ) Let $G$ be a random geometric graph with $n$ nodes on the unit sphere $\mathbb{S}^{d-1}$ such that each edge is connected with marginal probability $p$ (see the rigorous Definition def:RGG_sphere). If $p$,$n$ and $d$ satisfy at least one of the two Then the graph $G$ synchronizes with probability tending to one as $n$ goes to infinity.

Theorems & Definitions (17)

  • Theorem 1
  • Definition 1
  • Theorem 2
  • Definition 2: Erdős-Rényi Random Graph erdhos
  • Definition 3: Random Geometric Graph on the Sphere lugosi2017lecturespenrose2003random
  • Theorem 3
  • Lemma 1
  • proof
  • Proposition 1
  • Lemma 2
  • ...and 7 more