Teacher Knows It Best: Spontaneous Symmetry Breaking and Tipping Points in Networked Langevin Dynamics AI Sycophancy

2026-07-27

AI Summary

This study addresses the problem of algorithmic sycophancy induced by large language models, which can trap populations in spirals of erroneous collective beliefs. To counteract this cognitive bias, the authors construct a networked stochastic dynamical system incorporating a minority of topologically central “teacher” nodes designed to correct group-level misperceptions. By employing degree-weighted mean-field approximation, they reduce the high-dimensional Langevin equations to a macroscopic drift equation, thereby offering the first analytical framework that integrates statistical physics with social network theory to elucidate AI-induced sycophancy. Key contributions include an analytical solution for the critical tipping time based on saddle-node bifurcation, a proof that centralized rapid intervention outperforms distributed slow strategies, and demonstration of universal data collapse and theoretical bounds across diverse network topologies. Under strict budget constraints, the work further derives an optimal intervention policy.

Abstract

We formulate a statistical physics framework to model a networked stochastic dynamical system exhibiting bistability, driven by additive noise and social conformity. We apply this model to understand and mitigate AI-induced delusional spiraling-a phenomenon where algorithmic sycophancy from Large Language Models continuously reinforces inaccurate beliefs within a socially interacting society. By partitioning the network into a majority of regular agents and a minority of "aware" nodes (Teachers) placed at topological hubs, we use a degree-weighted mean-field approximation to reduce high-dimensional coupled Langevin equations into a single macroscopic drift equation. We provide a closed-form analytical derivation for the deterministic critical tipping time through a saddle-node bifurcation. We validate this analytical boundary using finite-size scaling and demonstrate a universal data collapse across diverse network topologies. Finally, we optimize an intervention strategy under a strict budget constraint that balances the topological footprint against driving velocity. We prove mathematically that under certain conditions, a highly concentrated, rapid intervention targeting massive hubs strictly outperforms a distributed, slow approach to rescue the network.

Problem

• AI sycophancy
• delusional spiraling
• spontaneous symmetry breaking
• tipping points
• bistability

Innovation

• spontaneous symmetry breaking
• networked Langevin dynamics
• tipping points
• mean-field approximation
• optimal intervention