The Kuramoto model on dynamic random graphs

TL;DR

This paper introduces a Kuramoto model on time-varying graphs driven by Markov processes, demonstrating synchronization properties and averaging principles for oscillators on dynamic random graphs.

Contribution

It develops a novel Kuramoto model on dynamic graphs with Markovian switching, proving synchronization and averaging results for fast-changing random environments.

Findings

Synchronization occurs regardless of random walk speed

Averaging principle holds for oscillators on dynamic graphs

High-probability global synchronization for fast processes

Abstract

We propose a Kuramoto model of coupled oscillators on a time-varying graph, whose dynamics is dictated by a Markov process in the space of graphs. The simplest representative is considering a base graph and then the subgraph determined by $N$ independent random walks on the underlying graph. We prove a synchronization result for solutions starting from a phase-cohesive set independent of the speed of the random walkers, an averaging principle and a global synchronization result with high probability for sufficiently fast processes. We also consider Kuramoto oscillators in a dynamical version of the Random Conductance Model.