Locality via Global Ties: Stability of the 2-Core Against Misspecification

For many random graph models, the analysis of a related birth process suggests local sampling algorithms for the size of, e.g., the giant connected component, the $k$-core, the size and probability of an epidemic outbreak, etc. We study the question of when these algorithms are robust against misspecification of the graph model, for the special case of the 2-core. We show that, for locally converging graphs with bounded average degrees, under a weak notion of expansion, a local sampling algorithm provides robust estimates for the size of both the 2-core and its largest component. Our weak notion of expansion generalizes the classical definition of expansion, while holding for many well-studied random graph models. Our method involves a two-step sprinkling argument.