Optimal couplings between sparse block models

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 $O(\sqrt {n})$ to $\Omega (n)$ as the planted bisection in the block model varies. This settles half of a conjecture of Bollobás 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 $\Theta (n)$ edges which allows for samples from a limit object to converge back to the limit itself must identify these models.