Graphop mean-field limits and synchronization for the stochastic Kuramoto model.

Area of Science:

• Complex Systems
• Network Science
• Theoretical Physics

Background:

• Coupled oscillator networks are crucial for modeling synchronization in diverse systems.
• The standard Kuramoto model assumes uniform coupling, limiting its applicability to real-world heterogeneous networks.
• Heterogeneous connectivity and coupling strengths significantly impact the transition to collective synchronization.

Purpose of the Study:

• To develop a general mean-field theory for stochastic Kuramoto-type oscillator models with heterogeneous coupling.
• To derive an exact formula for the critical synchronization threshold in such networks.
• To validate the theoretical predictions against numerical simulations.

Main Methods:

• Formulation of a general mean-field theory using the Vlasov-Fokker-Planck equation.
• Application of graphop theory in the mean-field limit for heterogeneous networks.
• Mathematical derivation of the critical threshold for symmetric odd-valued coupling functions.
• Numerical testing using large finite-size network models.

Main Results:

• An exact formula for the critical synchronization threshold was mathematically proven.
• Numerical simulations showed excellent agreement with the theoretical predictions for many graph models.
• Prediction accuracy decreased for sufficiently sparse network structures.

Conclusions:

• The developed mean-field theory accurately predicts synchronization thresholds in heterogeneous coupled oscillator networks.
• The findings extend beyond the limitations of the traditional Kuramoto model.
• Further research is needed to refine mean-field theories for sparse heterogeneous systems.