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arXiv · 1910.05442

Optimal couplings between sparse block models

James Hirst
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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 $O(\sqrt{n})$ 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.

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James Hirst. 2019-10-11. Optimal couplings between sparse block models. https://arxiv.org/abs/1910.05442

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