Recently the theory of noise-induced synchronization of Hegselmann-Krause (HK) dynamics has been well developed. As a typical opinion dynamics of bounded confidence, the HK model obeys a synchronous updating rule, i.e., \emph{all} agents check and update their opinions at each time point. However, whether asynchronous bounded confidence models, including the famous Deffuant-Weisbuch (DW) model, can be synchronized by noise have not been theoretically proved. In this paper, we propose a generalized bounded confidence model which possesses a stochastic asynchronous rule. The model takes the DW model and the HK model as special cases and can significantly generalize the bounded confidence models to practical application. We discover that the asynchronous model possesses a different noise-based synchronization behavior compared to the synchronous HK model. Generally, the HK dynamics can achieve quasi-synchronization \emph{almost surely} under the drive of noise. For the asynchronous dynamics, we prove that the model can achieve quasi-synchronization \emph{in mean}, which is a new type of quasi-synchronization weaker than the "almost surely" sense. The results unify the theory of noise-induced synchronization of bounded confidence opinion dynamics and hence proves the noise-induced synchronization of DW model theoretically for the first time. Moreover, the results provide a theoretical foundation for developing noise-based control strategy of more complex social opinion systems with stochastic asynchronous rules.

中文翻译：

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具有随机异步规则的有界置信意见动态的准同步

最近，Hegselmann-Krause (HK) 动力学的噪声诱导同步理论得到了很好的发展。作为典型的有界置信度的意见动态，HK 模型遵循同步更新规则，即 \emph{all} 代理在每个时间点检查和更新他们的意见。但是，包括著名的 Deffuant-Weisbuch (DW) 模型在内的异步有界置信模型是否可以通过噪声同步，尚未在理论上得到证实。在本文中，我们提出了一个具有随机异步规则的广义有界置信模型。该模型将 DW 模型和 HK 模型作为特例，可以将有界置信模型显着推广到实际应用中。我们发现与同步 HK 模型相比，异步模型具有不同的基于噪声的同步行为。通常，HK 动力学可以在噪声的驱动下实现准同步\emph{几乎肯定}。对于异步动力学，我们证明该模型可以实现准同步\emph{in mean}，这是一种比“几乎肯定”意义更弱的新型准同步。结果统一了有界置信意见动态的噪声诱导同步理论，从而首次从理论上证明了DW模型的噪声诱导同步。此外，该结果为开发具有随机异步规则的更复杂的社会舆论系统的基于噪声的控制策略提供了理论基础。HK动力学可以在噪声的驱动下实现准同步\emph{几乎肯定}。对于异步动力学，我们证明该模型可以实现准同步\emph{in mean}，这是一种比“几乎肯定”意义更弱的新型准同步。结果统一了有界置信意见动态的噪声诱导同步理论，从而首次从理论上证明了DW模型的噪声诱导同步。此外，该结果为开发具有随机异步规则的更复杂的社会舆论系统的基于噪声的控制策略提供了理论基础。HK动力学可以在噪声的驱动下实现准同步\emph{几乎肯定}。对于异步动力学，我们证明该模型可以实现准同步\emph{in mean}，这是一种比“几乎肯定”意义更弱的新型准同步。结果统一了有界置信意见动态的噪声诱导同步理论，从而首次从理论上证明了DW模型的噪声诱导同步。此外，该结果为开发具有随机异步规则的更复杂的社会舆论系统的基于噪声的控制策略提供了理论基础。这是一种比“几乎肯定”意义更弱的新型准同步。结果统一了有界置信意见动态的噪声诱导同步理论，从而首次从理论上证明了DW模型的噪声诱导同步。此外，该结果为开发具有随机异步规则的更复杂的社会舆论系统的基于噪声的控制策略提供了理论基础。这是一种比“几乎肯定”意义更弱的新型准同步。结果统一了有界置信意见动态的噪声诱导同步理论，从而首次从理论上证明了DW模型的噪声诱导同步。此外，该结果为开发具有随机异步规则的更复杂的社会舆论系统的基于噪声的控制策略提供了理论基础。