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 , we show that the distance to which the models can be coupled undergoes a phase transition from to 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 edges which allows for samples from a limit object to converge back to the limit itself must identify these models.

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中文翻译：

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稀疏块模型之间的最佳耦合

摘要：我们研究了将带有种植二等分的随机块模型耦合到具有相同平均度的均匀随机图的问题。着眼于平均度数相对于顶点数恒定的状态，我们表明，随着块模型中种植的二等分的变化，模型可以耦合的距离经历从到的相变。这解决了Bollobás和Riordan猜想的一半，并且对稀疏图极限理论有一些影响。特别是对于某些参数范围，块模型和相应的统一模型会产生必须收敛到同一极限点的样本。这意味着具有 允许来自极限对象的样本收敛到极限本身的边缘必须识别这些模型。

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