We derive two types of stochastic Hegselmann–Krause opinion formation models with leadership, study their different asymptotic behaviors and the noise effect to the leader’s control. Firstly, we propose a stochastic model with a multiplicative noise, which mimics the randomness from communication uncertainty. Using the Lyapunov functional approach, we provide a sufficient condition leading to the almost surely consensus behavior, which partially explains the mechanism that the group with large size and strong leadership can tolerate strong noise produced by communication uncertainty. The second noise comes from environmental uncertainty, fluctuates the system and makes opinions diverse and inclusive. We derive a stochastic model with additive noise, show that the noise from the environment destroys the leadership. However, the relative fluctuations of the followers’ opinions around the leader’s opinion have a uniformly bounded variance, which means they are still in a same group with the leader’s control. Finally, numerical simulations are performed to confirm theoretical results and explore more findings.

中文翻译：

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具有两种噪声类型的非齐次 Hegselmann-Krause 模型


我们推导出了两种类型的随机 Hegselmann-Krause 领导力意见形成模型，研究了它们不同的渐近行为和对领导者控制的噪声效应。首先，我们提出了一个带有乘法噪声的随机模型，它模拟了通信不确定性的随机性。使用 Lyapunov 函数方法，我们提供了一个导致几乎肯定的共识行为的充分条件，这部分解释了具有大规模和强大领导能力的群体可以容忍通信不确定性产生的强烈噪音的机制。第二个噪音来自环境不确定性，使系统波动，使观点多样化和包容性。我们推导出了一个带有加性噪声的随机模型，表明来自环境的噪声会破坏领导层。然而，追随者围绕领导者观点的相对波动具有均匀的有界方差，这意味着他们仍然与领导者的控制权处于同一群体中。最后，进行数值模拟以确认理论结果并探索更多发现。