We consider two classes of natural stochastic processes on finite unlabeled graphs. These are Euclidean stochastic optimization algorithms on the adjacency matrix of weighted graphs and a modified version of the Metropolis MCMC algorithm on stochastic block models over unweighted graphs. In both cases we show that, as the size of the graph goes to infinity, the random trajectories of the stochastic processes converge to deterministic limits. These deterministic limits are curves on the space of measure-valued graphons. Measure-valued graphons, introduced by Lov\'{a}sz and Szegedy, are a refinement of the concept of graphons that can distinguish between two infinite exchangeable arrays that give rise to the same graphon limit. We introduce new metrics on this space which provide us with a natural notion of convergence for our limit theorems. This notion is equivalent to the convergence of infinite-exchangeable arrays. Under a suitable time-scaling, the Metropolis chain admits a diffusion limit as the number of vertices go to infinity. We then demonstrate that, in an appropriately formulated zero-noise limit, the stochastic process of adjacency matrices of this diffusion converge to a deterministic gradient flow curve on the space of graphons introduced in arXiv:2111.09459 [math.PR]. Under suitable assumptions, this allows us to estimate an exponential convergence rate for the Metropolis chain in a certain limiting regime. To the best of our knowledge, both the actual rate and the connection between a natural Metropolis chain commonly used in exponential random graph models and gradient flows on graphons are new in the literature.

翻译：我们考虑有限无标号图上的两类自然随机过程。这些是加权图邻接矩阵上的欧几里得随机优化算法，以及无加权图上随机块模型的修正版Metropolis MCMC算法。在这两种情况下，我们证明当图的大小趋于无穷时，随机过程的随机轨迹收敛到确定性的极限。这些确定性极限是测度值图极限空间上的曲线。由Lovász和Szegedy引入的测度值图极限是对图极限概念的细化，能够区分产生相同图极限的两个无穷可交换阵列。我们在该空间上引入新的度量，为极限定理提供了自然的收敛概念。该概念等价于无穷可交换阵列的收敛性。在适当的时间缩放下，Metropolis链在顶点数趋于无穷时具有扩散极限。接着我们证明，在适当制定的零噪声极限下，该扩散过程的邻接矩阵随机过程收敛到arXiv:2111.09459 [math.PR]中引入的图极限空间上的确定性梯度流曲线。在适当假设下，这使我们能够在特定极限状态下估计Metropolis链的指数收敛速率。据我们所知，无论是实际速率，还是指数随机图模型中常用的自然Metropolis链与图极限上梯度流之间的关联，均为文献中的新发现。