The planted partition model (also known as the stochastic blockmodel) is a classical cluster-exhibiting random graph model that has been extensively studied in statistics, physics, and computer science. In its simplest form, the planted partition model is a model for random graphs on $n$ nodes with two equal-sized clusters, with an between-class edge probability of $q$ and a within-class edge probability of $p$. Although most of the literature on this model has focused on the case of increasing degrees (ie.\ $pn, qn \to \infty$ as $n \to \infty$), the sparse case $p, q = O(1/n)$ is interesting both from a mathematical and an applied point of view.
A striking conjecture of Decelle, Krzkala, Moore and Zdeborová based on deep, non-rigorous ideas from statistical physics gave a precise prediction for the algorithmic threshold of clustering in the sparse planted partition model. In particular, if $p = a/n$ and $q = b/n$, then Decelle et al.\ conjectured that it is possible to cluster in a way correlated with the true partition if $(a - b)^2 > 2(a + b)$, and impossible if $(a - b)^2 < 2(a + b)$. By comparison, the best-known rigorous result is that of Coja-Oghlan, who showed that clustering is possible if $(a - b)^2 > C (a + b)$ for some sufficiently large $C$.
We prove half of their prediction, showing that it is indeed impossible to cluster if $(a - b)^2 < 2(a + b)$. Furthermore we show that it is impossible even to estimate the model parameters from the graph when $(a - b)^2 < 2(a + b)$; on the other hand, we provide a simple and efficient algorithm for estimating $a$ and $b$ when $(a - b)^2 > 2(a + b)$. Following Decelle et al, our work establishes a rigorous connection between the clustering problem, spin-glass models on the Bethe lattice and the so called reconstruction problem. This connection points to fascinating applications and open problems.
Community detection in the sparse hypergraph stochastic block model
University of Washington（华盛顿大学）；University of California, San Diego（加州大学圣地亚哥分校）
本文研究了稀疏随机超图中的社团检测问题。Angelini et al.（2015）证明了在由超图随机块模型生成的稀疏超图中，社区检测的模型参数存在一个尖锐的阈值。我们解决了两个块的情况下的猜想的积极的一部分：在阈值以上，有一个谱算法，渐近几乎肯定构建一个分区的超图与真正的分区。我们的方法是Massoulié（2014）为稀疏随机图开发的方法的随机超图的推广。
Non-Reconstructability in the Stochastic Block Model
我们考虑随机块模型中的聚类（或重建）问题，在该政权的平均度是常数。对于大小相等的两个簇的情况下，最近的结果由Mossel，Neeman和Sly，和Massoulie，表明重构性经历了一个相变的Kesten-Stigum界的$λ_2^2 d = 1$，其中$λ_2$是第二大特征值的相关随机阵和$d$是平均度。在本文中，我们解决了一般情况下，两个以上的集群和/或不平衡的集群大小。我们的主要结果是一个充分条件聚类是不可能的，这符合现有的结果，两个集群的大小相等。我们的结果中的一个关键因素是块模型的不可重构性和不可重构性之间的一个新的连接，从一个Erdens-Rényi模型具有相同的平均度。我们还表明，它是有时可能的重建，即使当$λ_2^2 d < 1$。我们的研究结果提供了证据支持Decelle，Krzkala，Moore和Zdeborová关于随机块模型的可重构性和可扩展性（但不解决它们）的一系列假设。
Squared Wasserstein-2 Distance for Efficient Reconstruction of Stochastic Differential Equations
Courant Institute of Mathematical Sciences（库朗数学科学研究所）；New York University（纽约大学）；University of California Los Angeles（加州大学洛杉矶分校）；University of Oxford（牛津大学）
