AI 中文总结
研究完全连接社交网络中含多种意见的随机意见动态模型，证明快速共识等特性，考虑通信偏差参数时系统呈现相变，负参数下多数参与者有限时间内停止表达，否则无参与者停止。
AI 中文摘要
我们考虑一个完全连接的社交网络上的随机意见动态模型，其中\(N\)个参与者通过表达一组\(M\)种意见进行互动。在任何时间\(t\geq0\)，每个参与者都与一个\(M\)元组相关联，该元组表示对该参与者每种意见施加的社会压力。包含所有参与者对所有意见的社会压力的矩阵的演化是一个马尔可夫跳跃过程。每个参与者倾向于根据其社会压力向量表达意见，并且这种倾向由极化系数调节。当一个参与者表达意见\(o\)时，其对所有意见的社会压力重置为零，而对于其他参与者，对\(o\)的社会压力增加\(1\)，对其他意见的社会压力减少\(1/(M - 1)\)。在此设置下，我们证明了快速共识形成、唯一不变测度的存在以及高度极化网络中的亚稳定性。此外，通过考虑通信偏差参数，系统表现出如下相变。对于负通信偏差参数，除一个参与者外的所有参与者几乎肯定在有限时间内停止表达。否则，没有参与者停止表达意见。
英文摘要
We consider a stochastic opinion dynamics model on a fully connected social network with $N$ actors interacting by expressing opinions from a set of $M$ opinions. At any time $t\geq 0$, each actor is associated to an $M$-tuple representing the social pressure exerted on this actor for each opinion. The evolution of the matrix containing the social pressure of all actors for all opinions is a Markov jump process. Each actor tends to express opinions according to their social pressure vector and this tendency is modulated by a polarization coefficient. When an actor expresses an opinion $o$, its social pressure for all opinions is reset to zero, while for other actors the social pressure for $o$ increases by 1 and the social pressure for other opinions decreases by $1/(M-1)$. In this setting, we prove fast consensus formation, existence of a unique invariant measure and metastability in a highly polarized network. Moreover, by considering a communication bias parameter, the system exhibits a phase transition described as follows. With a negative communication bias parameter, all actors except one stop expressing in a finite time almost surely. Otherwise, no actor stops expressing opinions.
