锚定二元意见动力学的均匀时间玻尔兹曼平均场极限
Uniform-in-Time Boltzmann Mean-Field Limits for Anchored Binary Opinion Dynamics
- Georgia Institute of Technology（佐治亚理工学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该论文针对锚定二元意见动力学，在稳定性假设下建立了均匀时间的混沌传播，证明有限系统与非线性玻尔兹曼极限指数收敛至平稳分布，并给出两个应用及模拟验证。
AI中文摘要：
我们研究了一个连续时间意见模型，其中智能体成对交互，每个智能体都有一个固定的锚点。在每个交互时刻，两个意见根据一个可能非线性的规则并带有随机输入进行更新。在适当的稳定性和矩假设下，我们建立了均匀时间的混沌传播：随着群体规模的增长，任意固定数量智能体的联合分布随时间一致地收敛到求解非线性玻尔兹曼方程的解的相应乘积分布。我们首先在有限时间区间上获得全变差距离下的图形估计，仅假设交互规则是可测的。我们通过证明有限系统及其非线性极限都指数快速地收敛到它们各自的平稳分布，将该近似推广到均匀时间。我们还证明了在均方收缩条件下，二阶Wasserstein距离的均匀时间收敛性。为此结果，我们假设状态空间有界或存在一个漂移条件，使得大于二阶的某些阶矩一致有界。我们考察了两个应用。对于具有随机系数的锚定Friedkin-Johnsen模型，我们推导了意见与其锚点之间平稳偏差的仿射随机递推关系。我们利用该递推关系给出了次高斯或幂律尾部的条件，并刻画了幂律情形下的指数。我们还研究了一个有偏同化模型，其中智能体偏好与其当前意见一致的信息。当锚点和传入证据保持在中性值时，我们确定了交互何时会使意见偏离中性值更远，即使它从中性值附近开始。模拟验证了尾部预测，并展示了意见分布中的单个峰如何分裂为两个峰。
英文摘要：
We study a continuous-time opinion model in which agents interact in pairs and each agent has a fixed anchor. At each interaction, both opinions are updated according to a possibly nonlinear rule with random inputs. Under suitable stability and moment assumptions, we establish uniform-in-time propagation of chaos: as the population grows, any fixed number of agents become asymptotically independent, with a common law solving a nonlinear Boltzmann equation. The proof combines a graphical approximation on finite time intervals with exponential convergence of the finite system and its nonlinear limit to their respective stationary laws. We examine two applications. For an anchored variant of the Friedkin-Johnsen model with random coefficients, we give conditions for sub-Gaussian or power-law tails of stationary deviations from the anchors. Occasional overreaction can produce power-law tails with Gaussian anchors and noise, even as the nonlinear law converges exponentially fast to stationarity. We also study an anchored model of biased assimilation, determining when a nearby opinion moves toward or away from neutrality with the anchor and incoming evidence held at neutral values. Simulations illustrate the tail predictions and show how a single peak in the smoothed empirical opinion profile can split into two.
