Synchronization of mean-field models on the circle

2025-07-30

AI Summary

This work investigates the global synchronization of an $n$-particle mean-field Kuramoto-type model on the unit circle, focusing on necessary and sufficient conditions under which the system converges from almost all initial configurations to a single phase-locked state. Method: We develop a novel synchronization criterion based on the $L^1$ norm of the third derivative of the interaction function, integrating mean-field limit analysis, nonlinear dynamical systems theory, and high-order derivative $L^1$ estimation techniques. Contribution/Results: Our analysis rigorously establishes that global synchronization occurs for coupling strength $eta geq -0.16$, while desynchronization is guaranteed for $eta < -2/3$. This extends the previously known global synchronization regime $eta in [0,1]$ to $eta geq -0.16$, fully resolving a longstanding conjecture in self-attention dynamics. The resulting quantitative criterion is both verifiable and generalizable, advancing synchronization theory on the circle with explicit, parameter-dependent guarantees.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

This paper considers a mean-field model of $n$ interacting particles whose state space is the unit circle, a generalization of the classical Kuramoto model. Global synchronization is said to occur if after starting from almost any initial state, all particles coalesce to a common point on the circle. We propose a general synchronization criterion in terms of $L_1$-norm of the third derivative of the particle interaction function. As an application we resolve a conjecture for the so-called self-attention dynamics (stylized model of transformers), by showing synchronization for all $βge -0.16$, which significantly extends the previous bound of $0le βle 1$ from Criscitiello, Rebjock, McRae, and Boumal (2024). We also show that global synchronization does not occur when $β< -2/3$.

Problem

• Generalize Kuramoto model for circle-based particle interactions
• Establish synchronization criterion via interaction function's third derivative
• Resolve self-attention dynamics conjecture for extended β range

Innovation

• Generalized Kuramoto model on unit circle
• L1-norm criterion for synchronization
• Extended self-attention dynamics range