The Kuramoto model on oriented and signed graphs

TL;DR

This paper extends the Kuramoto model to directed, weighted, and signed networks, providing complete synchronization characterizations for acyclic and cyclic structures, revealing conditions for stability and uniqueness of synchronous states.

Contribution

It introduces a generalized Kuramoto model for complex directed and signed networks and offers comprehensive conditions for synchronization, including stability and uniqueness results.

Findings

Complete characterization of synchronization in acyclic networks

Global synchronization in oriented cycles with identical frequencies

Finite number of stable states in cyclic networks

Abstract

Many real-world systems of coupled agents exhibit directed interactions, meaning that the influence of an agent on another is not reciprocal. Furthermore, interactions usually do not have identical amplitude and/or sign. To describe synchronization phenomena in such systems, we use a generalized Kuramoto model with oriented, weighted and signed interactions. Taking a bottom-up approach, we investigate the simplest possible oriented networks, namely acyclic oriented networks and oriented cycles. These two types of networks are fundamental building blocks from which many general oriented networks can be constructed. For acyclic, weighted and signed networks, we are able to completely characterize synchronization properties through necessary and sufficient conditions, which we show are optimal. Additionally, we prove that if it exists, a stable synchronous state is unique. In oriented, weighted and signed cycles with identical natural frequencies, we show that the system globally synchronizes and that the number of stable synchronous states is finite.