标题： 一类最佳响应意见动态的收敛性和共识性分析 显示英文标题

标题： Convergence and consensus analysis of a class of best-response opinion dynamics

摘要： 意见动态旨在理解个体的意见如何通过局部互动而演变。最近，意见动态被建模为网络博弈，在这些博弈中，个体更新他们的意见以减少由于与他人意见不合而产生的社会压力。本文中，我们研究了Mei等人引入的一类最佳响应意见动态模型，其中参数$\alpha > 0$控制意见差异的边际成本，从而连接了著名的机制，如DeGroot模型（$\alpha = 2$）和加权中位数模型（$\alpha = 1$）。我们对不同$\alpha$值如何影响系统的收敛性和共识行为进行了理论分析。对于$\alpha > 1$的情况，对应于增加的边际成本，我们建立了动力学的收敛性，并得出了形成共识的图论条件，这被证明与DeGroot模型中的条件相似。当$\alpha < 1$时，我们通过一个反例表明收敛并不总是得到保证，并提供了收敛和共识的充分条件。此外，在小世界网络上的数值模拟揭示了网络结构和$\alpha$如何共同影响意见多样性。 显示英文摘要

摘要： Opinion dynamics aims to understand how individuals' opinions evolve through local interactions. Recently, opinion dynamics have been modeled as network games, where individuals update their opinions in order to minimize the social pressure caused by disagreeing with others. In this paper, we study a class of best response opinion dynamics introduced by Mei et al., where a parameter $\alpha > 0$ controls the marginal cost of opinion differences, bridging well-known mechanisms such as the DeGroot model ($\alpha = 2$) and the weighted-median model ($\alpha = 1$). We conduct theoretical analysis on how different values of $\alpha$ affect the system's convergence and consensus behavior. For the case when $\alpha > 1$, corresponding to increasing marginal costs, we establish the convergence of the dynamics and derive graph-theoretic conditions for consensus formation, which is proved to be similar to those in the DeGroot model. When $\alpha < 1$, we show via a counterexample that convergence is not always guaranteed, and we provide sufficient conditions for convergence and consensus. Additionally, numerical simulations on small-world networks reveal how network structure and $\alpha$ together affect opinion diversity.