The Kuramoto model in complex networks

TL;DR

This paper reviews the Kuramoto model's application to complex networks, highlighting how network structure influences synchronization, recent variations of the model, and future research directions across various scientific fields.

Contribution

It provides a comprehensive overview of synchronization in Kuramoto oscillator networks, including analytical methods, numerical results, and recent model variations with noise and inertia.

Findings

Network topology significantly affects synchronization patterns.

Recent model variations include noise and inertia effects.

Open problems and future research directions are identified.

Abstract

Synchronization of an ensemble of oscillators is an emergent phenomenon present in several complex systems, ranging from social and physical to biological and technological systems. The most successful approach to describe how coherent behavior emerges in these complex systems is given by the paradigmatic Kuramoto model. This model has been traditionally studied in complete graphs. However, besides being intrinsically dynamical, complex systems present very heterogeneous structure, which can be represented as complex networks. This report is dedicated to review main contributions in the field of synchronization in networks of Kuramoto oscillators. In particular, we provide an overview of the impact of network patterns on the local and global dynamics of coupled phase oscillators. We cover many relevant topics, which encompass a description of the most used analytical approaches and the analysis of several numerical results. Furthermore, we discuss recent developments on variations of the Kuramoto model in networks, including the presence of noise and inertia. The rich potential for applications is discussed for special fields in engineering, neuroscience, physics and Earth science. Finally, we conclude by discussing problems that remain open after the last decade of intensive research on the Kuramoto model and point out some promising directions for future research.