Optimal couplings between sparse block models

TL;DR

This paper investigates the coupling of sparse stochastic block models with planted bisections to uniform random graphs, revealing a phase transition in coupling distance and implications for graph limit theory.

Contribution

It establishes a phase transition in the coupling distance between block models and uniform graphs in the sparse regime, settling part of a conjecture and informing graph convergence concepts.

Findings

Coupling distance transitions from O(√n) to Ω(n) as parameters vary.

For certain parameters, models produce samples converging to the same limit.

Implications for graph limit theory and convergence notions.

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.