Analytical foundation for adversarial synchronization control in oscillator networks

TL;DR

This paper develops an analytical framework for adversarial control of synchronization in oscillator networks, revealing how small perturbations can significantly influence collective behavior.

Contribution

It derives an exact formula for the effect of perturbations on synchronization and extends the analysis to complex networks, providing insights into control strategies.

Findings

A finite, coupling-independent increase in synchronization from each perturbation.

Asymmetry between enhancement and suppression mechanisms.

Theoretical model accurately predicts behavior in various network types.

Abstract

This study provides an analytical foundation for adversarial synchronization control in Kuramoto oscillator networks, where small gradient-based perturbations applied repeatedly to oscillator phases can dramatically enhance or suppress collective synchronization. Using the Ott--Antonsen reduction, we derive an exact closed-form expression for the effect of a single adversarial perturbation (kick) on the order parameter. A key finding is that each kick produces a finite, coupling-independent increment in the order parameter even when synchronization is arbitrarily weak, which combined with slow relaxation near the critical coupling and mean-field feedback explains the disproportionate amplification previously observed in numerical simulations. Fixed-point analysis further reveals a fundamental asymmetry between enhancement and suppression, with the latter governed by noise-induced escape in finite systems. Extending the framework to networks via the annealed network approximation, we show that the theory captures the synchronization behavior of representative model networks and identify a decoupling between kick sensitivity and mean-field dominance in scale-free networks. These results offer a tractable theoretical basis for understanding and designing kick-based synchronization control in oscillator networks.