网络上意见形成的动力学模型的渐近与稳定性分析：一种 Allen-Cahn 方法
Asymptotic and stability analysis of kinetic models for opinion formation on networks: an Allen-Cahn approach
AI总结：
本文从 Boltzmann 型动力学模型出发，将网络上二元对立意见的多数决演化过程约化为 Allen-Cahn 型方程，并利用线性稳定性分析、积分算子及广义相对熵工具研究其稳态与稳定性。
摘要：
我们针对社会网络上存在两种对立意见、并按照类似多数决规则演化的意见形成过程，给出了其稳态平衡及其稳定性的分析。出发点是基于微观相互作用规则导出的动力学 Boltzmann 型模型，该规则描述具有一定连接度的个体之间的意见交换。关键思想是从动力学模型导出一个关于持有两种意见之一的个体比例的 Allen-Cahn 型方程。后者可以通过线性稳定性分析以及利用积分算子分析来研究。虽然这对于三元相互作用成立，但对于二元相互作用，所导出的感兴趣方程是一个线性散射方程，可以通过广义相对熵工具和积分算子来研究。
英文摘要：
We present the analysis of the stationary equilibria and their stability in case of an opinion formation process in presence of binary opposite opinions evolving according to majority-like rules on social networks. The starting point is a kinetic Boltzmann-type model derived from microscopic interactions rules for the opinion exchange among individuals holding a certain degree of connectivity. The key idea is to derive from the kinetic model an Allen-Cahn type equation for the fraction of individuals holding one of the two opinions. The latter can be studied by means of a linear stability analysis and by exploiting integral operator analysis. While this is true for ternary interactions, for binary interactions the derived equation of interest is a linear scattering equation, that can be studied by means of General Relative Entropy tools and integral operators.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
