Coupling sparse models and dense extremal problems

Abstract

We study the problem of coupling a stochastic block model with a planted bisection to a uniform random graph having the same average degree. Focusing on the regime where the average degree is a constant relative to the number of vertices n, we show that the distance to which the models can be coupled undergoes a phase transition from O [square root of n to [omega]n) as the planted bisection in the block model varies. This settles half of a conjecture of Bollobas and Riordan and has some implications for sparse graph limit theory. In particular, for certain ranges of parameters, a block model and the corresponding uniform model produce samples which must converge to the same limit point. This implies that any notion of convergence for sequences of graphs with [theta] edges which allows for samples from a limit object to converge back to the limit itself must identify these models. On the other hand, we demonstrate that the existing theory of dense graph limits is a powerful tool for dealing with extremal problems on graphs with [theta](n2) edges. The language of graphons along with the flag algebra method allow us to obtain many results which would otherwise be out of reach or at least difficult to manage. We study graph profiles which capture correlations between different graphs in a larger network. Further, we give proofs in the flag algebra of some inducibility-like problems which have gained some particular interest recently.