Contiguity and remote contiguity of some random graphs

Asymptotic properties of random graph sequences, like occurrence of a giant component or full connectivity in Erdos-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 (Pₙ) and (Qₙ) for two sequences of graph distributions, asymptotic equivalence (convergence in total-variation) and contiguity (Pₙ (Aₙ) =o (1) Qₙ (Aₙ) =o (1) ) have been considered by (Janson, 2010) and others; here we use so-called remote contiguity (for some fixed aₙ 0, Pₙ (Aₙ) =o (aₙ) Qₙ (Aₙ) =o (1) ) to show that connectivity properties are preserved in more heavily perturbed Erdos-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.