Dense multigraphon-valued stochastic processes and edge-changing dynamics in the configuration model

TL;DR

This paper develops a rigorous mathematical framework for large, evolving multigraphs using multigraphon limits, extending previous models to include edge addition and removal, and establishing process-level convergence.

Contribution

It generalizes existing edge-flipping dynamics to include edge addition and removal, providing non-deterministic process-level limits for large multigraphs.

Findings

Established weak convergence of multigraph processes

Extended configuration model dynamics to include edge changes

Provided non-deterministic limit processes for large graphs

Abstract

Time-evolving random graph models have appeared and have been studied in various fields of research over the past decades. However, the rigorous mathematical treatment of large graphs and their limits at the process-level is still in its infancy. In this article, we adapt the approach of Athreya, den Hollander and R\"ollin (2021+) to the setting of multigraphs and multigraphons, introduced by Kolossv\'ary and R\'ath (2011). We then generalise the work of R\'ath (2012) and R\'ath and Szak\'acs (2012), who analysed edge-flipping dynamics on the configuration model -- in contrast to their work, we establish weak convergence at the process-level, and by allowing removal and addition of edges, these limits are non-deterministic.