Optimal synchronization of Kuramoto oscillators: A dimensional reduction approach

Area of Science:

• Complex systems
• Network science
• Statistical physics

Background:

• The Kuramoto model is a fundamental tool for studying synchronization phenomena in coupled oscillator systems.
• Designing network topologies to control synchronization remains a significant challenge in complex systems research.

Purpose of the Study:

• To develop an analytical framework for constructing optimal network topologies that either promote or inhibit synchronization in the Kuramoto model.
• To provide a computationally efficient method for generating such optimized networks.

Main Methods:

• Employing a dimensional reduction approach inspired by the Ott-Antonsen ansatz to introduce a collective coordinate.
• Deriving the condition for optimal synchronization by maximizing the quadratic function ω(T)Lω, involving natural frequencies (ω) and the network Laplacian (L).
• Utilizing a hill climb rewiring algorithm for efficient network generation.

Main Results:

• Analytical derivation of the condition for optimal synchronization, simplifying and unifying recent numerical findings.
• Demonstration that optimal synchronization requires maximizing a specific quadratic function of frequencies and the network Laplacian.
• Development of an efficient algorithm to generate networks with desired synchronization properties.

Conclusions:

• The proposed dimensional reduction approach offers a powerful analytical tool for understanding and designing synchronization in complex networks.
• The derived maximization condition provides a clear guideline for network topology optimization.
• The method is adaptable to various interaction types (attractive/repulsive) within Kuramoto models, offering broad applicability.