Path convergence of Markov chains on large graphs

TL;DR

This paper studies how stochastic processes on large graphs, including optimization algorithms and MCMC methods, converge to deterministic curves on measure-valued graphons, providing new insights into their asymptotic behavior and convergence rates.

Contribution

It introduces a framework for analyzing the convergence of stochastic processes on large graphs to measure-valued graphons, including new metrics and a diffusion limit for the Metropolis chain.

Findings

Stochastic processes converge to deterministic measure-valued graphon curves as graph size increases.

A diffusion limit for the Metropolis chain is established under certain conditions.

Exponential convergence rates for the Metropolis chain are derived in the large graph limit.

Abstract

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.