Optimal synchronization of Kuramoto oscillators: a dimensional reduction approach

TL;DR

This paper introduces a dimensional reduction method to analytically optimize network topologies for synchronization in Kuramoto oscillators, providing a simpler way to achieve desired synchronization properties.

Contribution

It presents a novel analytical condition for optimal synchronization based on maximizing a quadratic form involving natural frequencies and the network Laplacian.

Findings

Analytical maximization condition for synchronization.

Efficient hill climb rewiring algorithm for network optimization.

Applicability to Kuramoto models with attractive and repulsive interactions.

Abstract

A recently proposed dimensional reduction approach for studying synchronization in the Kuramoto model is employed to build optimal network topologies to favor or to suppress synchronization. The approach is based in the introduction of a collective coordinate for the time evolution of the phase locked oscillators, in the spirit of the Ott-Antonsen ansatz. We show that the optimal synchronization of a Kuramoto network demands the maximization of the quadratic function $\omega^T L \omega$, where $\omega$ stands for the vector of the natural frequencies of the oscillators, and $L$ for the network Laplacian matrix. Many recently obtained numerical results can be re-obtained analytically and in a simpler way from our maximization condition. A computationally efficient {hill climb} rewiring algorithm is proposed to generate networks with optimal synchronization properties. Our approach can be easily adapted to the case of the Kuramoto models with both attractive and repulsive interactions, and again many recent numerical results can be rederived in a simpler and clearer analytical manner.