Mean field Kuramoto models on graphs

TL;DR

This paper introduces first and second order Kuramoto models on graphs using discrete optimal transport, analyzes their synchronization behaviors, and provides analytical formulas and numerical examples for these models.

Contribution

It develops a novel framework combining Kuramoto models with discrete optimal transport on graphs, including analytical solutions and numerical demonstrations.

Findings

Synchronization behaviors depend on graph structure.

Analytical formulas derived for two-point graphs.

Numerical examples illustrate model dynamics.

Abstract

One of a classical synchronization model is the Kuramoto model. We propose both first and second order Kuramoto dynamical models on graphs using discrete optimal transport dynamics. We analyze the synchronization behaviors for some examples of Kuramoto models on graphs. We also provide a generalized Hopf-Cole transformation for discrete optimal transport systems. Focus on the two points graph, we derive analytical formulas of the Kuramoto dynamics with various potential induced from entropy functionals. Several numerical examples for the Kuramoto model on general graphs are presented.