  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
Bollob\'{a}s 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.
