Graphon-valued processes with vertex-level fluctuations

TL;DR

This paper studies a class of stochastic processes where vertex types fluctuate over time, influencing the evolution of the graph's structure, with a focus on the limiting behavior of the empirical graphon as the number of vertices grows large.

Contribution

It introduces a framework for analyzing graph-valued processes driven primarily by vertex-type fluctuations, including large deviation principles and convergence results.

Findings

Established large deviation principles for the processes.

Proved convergence of the empirical graphon in the large vertex limit.

Demonstrated the approach's flexibility with models depending on both vertex types and graph states.

Abstract

We consider a class of graph-valued stochastic processes in which each vertex has a type that fluctuates randomly over time. Collectively, the paths of the vertex types up to a given time determine the probabilities that the edges are active or inactive at that time. Our focus is on the evolution of the associated empirical graphon in the limit as the number of vertices tends to infinity, in the setting where fluctuations in the graph-valued process are more likely to be caused by fluctuations in the vertex types than by fluctuations in the states of the edges given these types. We derive both sample-path large deviation principles and convergence of stochastic processes. We demonstrate the flexibility of our approach by treating a class of stochastic processes where the edge probabilities depend not only on the fluctuations in the vertex types but also on the state of the graph itself.