On connectivity in random graph models with limited dependencies

Abstract

AbstractWe consider random graph models in which the events describing the inclusion of potential edges have to be independent of each other if the corresponding edges are non‐adjacent and ask: what is the minimum probability , such that for any distribution (in this model) on graphs with vertices in which each potential edge has a marginal probability of being present at least , a graph drawn from is connected with non‐zero probability? The answer to this question is sensitive to the formalization of the independence condition. We introduce a strict hierarchy of five conditions, which give rise to at least three different functions . For each condition, we provide upper and lower bounds for . For the strongest condition, the coloring model, we show that for . In contrast, for the weakest condition, pairwise independence, we show that lies within of the threshold for completely arbitrary distributions.