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

Abstract: 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. In this paper, 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. In the first step, we use sprinkling to establish the existence of a non-empty $2$-core inside the giant, while in the second, we use this non-empty $2$-core as seed for a second sprinkling argument to establish that the giant contains a linear sized $2$-core. The second step is based on a novel coloring scheme for the vertices in the tree-part. Our algorithmic results follow from the structural properties for the $2$-core established in the course of our sprinkling arguments. The run-time of our local algorithm is constant independent of the graph size, with the value of the constant depending on the desired asymptotic accuracy $\epsilon$. But given the existential nature of local limits, our arguments do not give any bound on the functional dependence of this constant on $\epsilon$, nor do they give a bound on how large the graph has to be for the asymptotic additive error bound $\epsilon$ to hold.