Asymptotic equivalence and contiguity of some random graphs

TL;DR

This paper demonstrates that two random graph models with similar edge probabilities are asymptotically equivalent under weaker conditions than previously thought, simplifying proofs of related graph distance results.

Contribution

It establishes a strong form of asymptotic equivalence and contiguity between certain random graph models under relaxed assumptions.

Findings

Asymptotic equivalence holds under weaker conditions.

Simplified proof of graph distance equivalence.

Applicable to models with slight differences in edge probabilities.

Abstract

We show that asymptotic equivalence, in a strong form, holds between two random graph models with slightly differing edge probabilities under substantially weaker conditions than what might naively be expected. One application is a simple proof of a recent result by van den Esker, van der Hofstad and Hooghiemstra on the equivalence between graph distances for some random graph models.