Convergence rate of opinion dynamics with complex interaction types

Lingling Yao, Aming Li
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Abstract

The convergence rate is a crucial issue in opinion dynamics, which characterizes how quickly opinions reach a consensus and tells when the collective behavior can be formed. However, the key factors that determine the convergence rate of opinions are elusive, especially when individuals interact with complex interaction types such as friend/foe, ally/adversary, or trust/mistrust. In this paper, using random matrix theory and low-rank perturbation theory, we present a new body of theory to comprehensively study the convergence rate of opinion dynamics. First, we divide the complex interaction types into five typical scenarios: mutual trust $(+/+)$, mutual mistrust $(-/-)$, trust$/$mistrust $(+/-)$, unilateral trust $(+/0)$, and unilateral mistrust $(-/0)$. For diverse interaction types, we derive the mathematical expression of the convergence rate, and further establish the direct connection between the convergence rate and population size, the density of interactions (network connectivity), and individuals' self-confidence level. Second, taking advantage of these connections, we prove that for the $(+/+)$, $(+/-)$, $(+/0)$, and random mixture of different interaction types, the convergence rate is proportional to the population size and network connectivity, while it is inversely proportional to the individuals' self-confidence level. However, for the $(-/-)$ and $(-/0)$ scenarios, we draw the exact opposite conclusions. Third, for the $(+/+,-/-)$ and $(-/-,-/0)$ scenarios, we derive the optimal proportion of different interaction types to ensure the fast convergence of opinions. Finally, simulation examples are provided to illustrate the effectiveness and robustness of our theoretical findings.
具有复杂互动类型的舆论动态收敛率
意见趋同率是意见动力学中的一个关键问题,它描述了意见达成共识的速度,并告诉人们何时可以形成集体行为。然而,决定意见收敛速度的关键因素却难以捉摸,尤其是当个体与复杂的互动类型(如朋友/敌人、盟友/对手或信任/不信任)发生互动时。本文利用随机矩阵理论和低秩扰动理论,提出了一套新的理论体系来全面研究意见动态的趋同率。首先,我们把复杂的互动类型分为五种典型情况:相互信任 $(+/+)$、相互不信任 $(-/-)$、信任 $/$ 不信任 $(+/-)$、单边信任 $(+/0)$、单边不信任 $(-/0)$。其次,利用这些联系,我们证明了对于$(+/+)$、$(+/-)$、$(+/0)$和不同互动类型的随机混合,收敛率与种群规模和网络连接成正比,而与个体的自信水平成反比。然而,对于$(-/-)$和$(-/0)$情景,我们得出了完全相反的结论。第三,对于$(+/+,-/-)$和$(-/-,-/0)$情景,我们推导出了不同互动类型的最佳比例,以确保意见的快速收敛。最后,我们提供了一些模拟实例,以说明我们理论发现的有效性和稳健性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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