论文标题
密集的随机图上弱相互作用的振荡器
Weakly interacting oscillators on dense random graphs
论文作者
论文摘要
我们考虑一类平均场类型的弱相互作用的粒子系统。粒子之间的相互作用是用图序列编码的，即，当且仅当它们连接在基础图中时，两个粒子才会相互作用。我们为系统的经验度量建立了大量的定律，该定律每当图形序列在图形限制理论（即与图形）的意义上收敛时所保持的系统。该极限显示为通过（可能是随机）Graphon限制加权的非线性fokker-Planck方程的解决方案。与现有文献相反，我们的分析侧重于未标记的图形：在图表上没有规律性假设，我们能够包括一般图形序列，例如可交换的随机图。最后，我们确定了随机和确定性的图的序列，相关的经验度量将其收敛到平均场限制，即，即经典的McKean-Vlasov方程的解决方案。
We consider a class of weakly interacting particle systems of mean-field type. The interactions between the particles are encoded in a graph sequence, i.e., two particles are interacting if and only if they are connected in the underlying graph. We establish a Law of Large Numbers for the empirical measure of the system that holds whenever the graph sequence is convergent in the sense of graph limits theory, i.e., to a graphon. The limit is shown to be the solution of a non-linear Fokker-Planck equation weighted by the (possibly random) graphon limit. In contrast with the existing literature, our analysis focuses on unlabeled graphons: no regularity assumptions are made on the graph limit and we are able to include general graph sequences such as exchangeable random graphs. Finally, we identify the sequences of graphs, both random and deterministic, for which the associated empirical measure converges to the mean-field limit, i.e., to the solution of a classical McKean-Vlasov equation.
