Strong couplings for static locally tree-like random graphs

TL;DR

This paper establishes a general coupling method for exploration processes of various random graphs with their local weak limits, including models with complex vertex attributes and influence, valid at any fixed exploration depth.

Contribution

It introduces a universal coupling framework for exploration processes of diverse random graph models with local weak limits as marked Galton-Watson processes.

Findings

Coupling holds for undirected and directed graphs.

Applicable to models with vertex attributes in metric spaces.

Valid for any fixed exploration depth.

Abstract

The goal of this paper is to provide a general purpose result for the coupling of exploration processes of random graphs, both undirected and directed, with their local weak limits when this limit is a marked Galton-Watson process. This class includes in particular the configuration model and the family of inhomogeneous random graphs with rank-1 kernel. Vertices in the graph are allowed to have attributes on a general separable metric space and can potentially influence the construction of the graph itself. The coupling holds for any fixed depth of a breadth-first exploration process.