Contiguity and remote contiguity of some random graphs

Abstract

Asymptotic properties of random graph sequences, like occurrence of a giant component or full connectivity in Erd\H{o}s-R\'enyi graphs, are usually derived with very specific choices for defining parameters. The question arises to which extent those parameters choices may be perturbed, without losing the asymptotic property. Writing $(P_n)$ and $(Q_n)$ for two sequences of graph distributions, asymptotic equivalence (convergence in total-variation) and contiguity ($P_n(A_n)=o(1) \implies Q_n(A_n)=o(1)$) have been considered by (Janson, 2010) and others; here we use so-called remote contiguity (for some fixed $a_n\downarrow 0$, $P_n(A_n)=o(a_n) \implies Q_n(A_n)=o(1)$) to show that connectivity properties are preserved in more heavily perturbed Erd\H{o}s-R\'enyi graphs. The techniques we demonstrate with random graphs here, extend to general asymptotic properties, e.g. in more complex large-graph limits, scaling limits, large-sample limits, etc.