Model reduction for Kuramoto models with complex topologies

TL;DR

This paper introduces a novel model reduction framework for the Kuramoto model on complex networks, effectively capturing synchronization dynamics in finite and large networks with complex topologies.

Contribution

It generalizes a collective coordinates approach by incorporating the graph Laplacian, enabling low-dimensional descriptions of complex oscillator networks.

Findings

Accurately captures synchronization in finite-size networks.

Works well for both clustered and non-clustered networks.

Effective in the thermodynamic limit for ER networks.

Abstract

Synchronisation of coupled oscillators is a ubiquitous phenomenon, occurring in topics ranging from biology and physics, to social networks and technology. A fundamental and long-time goal in the study of synchronisation has been to find low-order descriptions of complex oscillator networks and their collective dynamics. However, for the Kuramoto model - the most widely used model of coupled oscillators - this goal has remained surprisingly challenging, in particular for finite-size networks. Here, we propose a model reduction framework that effectively captures synchronisation behaviour in complex network topologies. This framework generalises a collective coordinates approach for all-to-all networks [Gottwald (2015) Chaos 25, 053111] by incorporating the graph Laplacian matrix in the collective coordinates. We first derive low dimensional evolution equations for both clustered and non-clustered oscillator networks. We then demonstrate in numerical simulations for Erdos-Renyi (ER) networks that the collective coordinates capture the synchronisation behaviour in both finite-size networks as well as in the thermodynamic limit, even in the presence of interacting clusters.