The random graph process is globally synchronizing
Vishesh Jain, Clayton Mizgerd, Mehtaab Sawhney
TL;DR
The paper proves that the random graph process becomes globally synchronizing as soon as it is connected, resolving a strong form of a conjecture linking connectivity to Kuramoto synchronization on random graphs. It introduces the defective expander framework, showing that a core-expander structure with a small defect set suffices to rule out spurious stable states in the homogeneous Kuramoto model. The authors develop an amplification machinery that propagates phase concentration from small to large sets of vertices, overcoming the challenges posed by low-degree defects. They then extend the result from the static core to the dynamic random graph process G(n,m), using a careful coupling and partitioning argument that yields global synchronization whp for all m \\ge \\tau, thereby matching the connectivity threshold up to constants. The work advances the understanding of synchronization on sparse random networks and confirms the sharp transition from connectivity to global synchrony.
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.