The aim of the paper is to address the behavior in large population of diffusions interacting on a random, possibly diluted and inhomogeneous graph. This is the natural continuation of a previous work, where the homogeneous Erd\H os-Renyi case was considered. The class of graphs we consider includes disordered $W$-random graphs, with possibly unbounded graphons. The main result concerns a quenched convergence (that is true for almost every realization of the random graph) of the empirical measure of the system towards the solution of a nonlinear Fokker-Planck PDE with spatial extension, also appearing in different contexts, especially in neuroscience. The convergence of the spatial profile associated to the diffusions is also considered, and one proves that the limit is described in terms of a nonlinear integro-differential equation which matches the neural field equation in certain particular cases.

中文翻译：

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非齐次随机图上相互作用扩散的淬火渐近

本文的目的是解决大量扩散在随机、可能稀释和不均匀的图上相互作用的行为。这是之前工作的自然延续，其中考虑了同质的 Erd\H os-Renyi 案例。我们考虑的图类包括无序的 $W$-随机图，可能具有无限的图元。主要结果涉及系统的经验度量对具有空间扩展的非线性 Fokker-Planck PDE 解的淬灭收敛（对于随机图的几乎所有实现都是如此），也出现在不同的上下文中，尤其是在神经科学中. 还考虑了与扩散相关的空间剖面的收敛性，