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.

本文研究弱相互作用粒子系统，探讨其在图序列中的表现，建立了一个连续的图极限类上的经验测度的大数定律，并且证明了这个极限是一个非线性协方差方程的解。