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 curves on the space of measure-valued graphons. Measure-valued graphons, introduced by Lov\'{a}sz and Szegedy in \cite{lovasz2010decorated}, 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 suitable assumptions and a specified 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 converges to a deterministic gradient flow curve on the space of graphons introduced in\cite{Oh2023}. A novel feature of this approach is that it provides a precise exponential convergence rate for the Metropolis chain in a certain limiting regime. The connection between a natural Metropolis chain commonly used in exponential random graph models and gradient flows on graphons, to the best of our knowledge, is new in the literature as well.

翻译：我们考虑两类在有限无标记图上的自然随机过程：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一种改进版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取值于测度值图空间的确定性曲线。测度值图由Lovász和Szegedy在文献\cite{lovasz2010decorated}中引入，是对图概念的精细化，能够区分产生相同图极限的两个无穷可交换数组。我们在此空间上引入新度量，为极限定理提供自然的收敛概念，该概念等价于无穷可交换数组的收敛性。在适当假设和特定时间尺度下，当顶点数趋于无穷时，Metropolis链具有扩散极限。随后我们证明，在恰当设定的零噪声极限下，该扩散过程的邻接矩阵随机过程收敛到文献\cite{Oh2023}中引入的图空间上的确定性梯度流曲线。该方法的一个新颖之处在于，它能在特定极限下为Metropolis链提供精确的指数收敛速率。据我们所知，指数随机图模型中常用的自然Metropolis链与图上的梯度流之间的联系，在文献中也是首次提出。