时变权集体动力学模型的平均场和图极限
Mean-field and graph limits for collective dynamics models with time-varying weights
摘要：
在本文中，我们研究了一个模型的意见动力学的影响力权重的时间演变通过一个方程，这是耦合的意见的演变。我们探索自然问题的大人口极限与两种方法：现在经典的平均场极限和最近的图形极限。在建立了我们将要考虑的模型的解的存在性和唯一性之后，我们为在一般情况下取图极限提供了严格的数学证明。然后，我们建立了考虑平均场极限的一个必要框架--不可约性的关键概念，证明了在此背景下平均场极限从属于图极限。这实际上为平均场极限提供了另一种（但较弱的）证明。最后，我们通过一些数值模拟来说明我们的结果。
英文摘要：
In this paper, we study a model for opinion dynamics where the influence weights of agents evolve in time via an equation which is coupled with the opinions' evolution. We explore the natural question of the large population limit with two approaches: the now classical mean-field limit and the more recent graph limit. After establishing the existence and uniqueness of solutions to the models that we will consider, we provide a rigorous mathematical justification for taking the graph limit in a general context. Then, establishing the key notion of indistinguishability, which is a necessary framework to consider the mean-field limit, we prove the subordination of the mean-field limit to the graph one in that context. This actually provides an alternative (but weaker) proof for the mean-field limit. We conclude by showing some numerical simulations to illustrate our results.
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