In this paper we continue to rigorously establish the predictions in ground breaking work in statistical physics by Decelle, Krzakala, Moore, Zdeborov\'a (2011) regarding the block model, in particular in the case of $q=3$ and $q=4$ communities. We prove that for $q=3$ and $q=4$ there is no computational-statistical gap if the average degree is above some constant by showing it is information theoretically impossible to detect below the Kesten-Stigum bound. The proof is based on showing that for the broadcast process on Galton-Watson trees, reconstruction is impossible for $q=3$ and $q=4$ if the average degree is sufficiently large. This improves on the result of Sly (2009), who proved similar results for regular trees for $q=3$. Our analysis of the critical case $q=4$ provides a detailed picture showing that the tightness of the Kesten-Stigum bound in the antiferromagnetic case depends on the average degree of the tree. We also prove that for $q\geq 5$, the Kestin-Stigum bound is not sharp. Our results prove conjectures of Decelle, Krzakala, Moore, Zdeborov\'a (2011), Moore (2017), Abbe and Sandon (2018) and Ricci-Tersenghi, Semerjian, and Zdeborov{\'a} (2019). Our proofs are based on a new general coupling of the tree and graph processes and on a refined analysis of the broadcast process on the tree.
翻译:在本文中,我们继续严格地预测Decelle、Krzakala、Moore、Zdeborov\'a(2011)在统计物理领域对地破碎工作的预测,特别是在3美元=3美元和4美元=4美元社区的情况下。我们证明,如果平均水平高于某些常数,则不存在计算-统计差距,显示在理论上无法检测到Kesten-Stigum约束下方的Kesten-Stigum。证据是显示Galton-Watson树的广播进程,如果平均水平足够大,则无法重建3美元和4美元=4美元。我们证明,Sly(2009年)的结果显示,正常树的类似结果为3美元=3美元和4美元。我们对关键案例的分析显示,在防火塔-Stigum约束下的Kesten-Stigum案例的紧凑密性,取决于树的平均水平。我们还证明,在Sqqq=3美元和Sqreguealal 5ralal 上,我们的Sqreqeqeal-ralalalal 和Keqral-revalalalalalalalal 。