The random graph process is globally synchronizing

TL;DR

This paper proves that a random graph process becomes globally synchronizing with high probability once it is connected, confirming a conjecture and establishing connectivity as a threshold for synchronization.

Contribution

It confirms a conjecture that the random graph process is globally synchronizing once connected, establishing connectivity as a necessary and sufficient condition for synchronization.

Findings

Random graph process becomes globally synchronizing once connected

Connectivity is a threshold for global synchronization

Confirms a conjecture by Abdalla et al.

Abstract

The homogeneous Kuramoto model on a graph $G = (V,E)$ is a network of $|V|$ identical oscillators, one at each vertex, where every oscillator is coupled bidirectionally (with unit strength) to its neighbors in the graph. A graph $G$ is said to be globally synchronizing if, for almost every initial condition, the homogeneous Kuramoto model converges to the all-in-phase synchronous state. Confirming a conjecture of Abdalla, Bandeira, Kassabov, Souza, Strogatz, and Townsend, we show that with high probability, the random graph process becomes globally synchronizing as soon as it is connected. This is best possible, since connectivity is a necessary condition for global synchronization.