We study a system of nonlocal aggregation cross-diffusion PDEs that describe the evolution of opinion densities on a network. The PDEs are coupled with a system of ODEs that describe the time evolution of the agents on the network. Firstly, we apply the Deterministic Particle Approximation (DPA) method to the aforementioned system in order to prove the existence of solutions under suitable assumptions on the interactions between agents. Later on, we present an explicit model for opinion formation on an evolving network. The opinions evolve based on both the distance between the agents on the network and the 'attitude areas,' which depend on the distance between the agents' opinions. The position of the agents on the network evolves based on the distance between the agents' opinions. The goal is to study radicalization, polarization, and fragmentation of the population while changing its open-mindedness and the radius of interaction.

翻译：我们研究了一个描述网络上意见密度演化的非局部聚集交叉扩散偏微分方程组。该方程组与描述网络上主体时间演化的常微分方程组相耦合。首先，我们将确定性粒子近似（DPA）方法应用于上述系统，以证明在主体间相互作用满足适当假设的条件下解的存在性。随后，我们提出一个在演化网络上进行意见形成的显式模型。意见的演化同时依赖于网络上主体间的距离以及取决于主体意见距离的'态度区域'。主体在网络上的位置则基于主体间意见的距离而演化。本研究的目的是在改变群体的开放程度和交互半径的情况下，探讨其激进化、两极分化和碎片化现象。