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 an 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.


  1. Kuramoto model
  2. synchronization
  3. directed graphs
  4. oriented graphs

MSC codes

  1. 34D06
  2. 37N35

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Information & Authors


Published In

cover image SIAM Journal on Applied Dynamical Systems
SIAM Journal on Applied Dynamical Systems
Pages: 458 - 480
ISSN (online): 1536-0040


Submitted: 31 July 2018
Accepted: 16 January 2019
Published online: 28 February 2019


  1. Kuramoto model
  2. synchronization
  3. directed graphs
  4. oriented graphs

MSC codes

  1. 34D06
  2. 37N35



Funding Information

Eidgenössische Technische Hochschule Zürich https://doi.org/10.13039/501100003006 : PYAPP2 154275

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