We study the weak recovery problem on the $r$-uniform hypergraph stochastic block model ($r$-HSBM) with two balanced communities. In this model, $n$ vertices are randomly divided into two communities, and size-$r$ hyperedges are added randomly depending on whether all vertices in the hyperedge are in the same community. The goal of the weak recovery problem is to recover a non-trivial fraction of the communities given the hypergraph. Previously, Pal and Zhu (2021) established that weak recovery is always possible above a natural threshold called the Kesten-Stigum (KS) threshold. Gu and Polyanskiy (2023) proved that the KS threshold is tight if $r\le 4$ or the expected degree $d$ is small. It remained open whether the KS threshold is tight for $r\ge 5$ and large $d$. In this paper we determine the tightness of the KS threshold for any fixed $r$ and large $d$. We prove that for $r\le 6$ and $d$ large enough, the KS threshold is tight. This shows that there is no information-computation gap in this regime. This partially confirms a conjecture of Angelini et al. (2015). For $r\ge 7$, we prove that for $d$ large enough, the KS threshold is not tight, providing more evidence supporting the existence of an information-computation gap in this regime. Furthermore, we establish asymptotic bounds on the weak recovery threshold for fixed $r$ and large $d$. We also obtain a number of results regarding the closely-related broadcasting on hypertrees (BOHT) model, including the asymptotics of the reconstruction threshold for $r\ge 7$ and impossibility of robust reconstruction at criticality.

翻译：我们研究具有两个平衡社区的$r$一致超图随机块模型($r$-HSBM)中的弱恢复问题。在该模型中，$n$个顶点被随机划分为两个社区，且大小为$r$的超边根据超边中所有顶点是否属于同一社区而随机添加。弱恢复问题的目标是给定超图后恢复非平凡比例的社区信息。此前，Pal和Zhu(2021)证明，在称为Kesten-Stigum(KS)阈值的自然阈值之上，弱恢复总是可能的。Gu和Polyanskiy(2023)证明，当$r\le 4$或期望度$d$较小时，KS阈值是紧的。对于$r\ge 5$且$d$较大的情形，KS阈值是否紧仍为开放问题。本文确定了任意固定$r$且$d$较大时KS阈值的紧性。我们证明，当$r\le 6$且$d$足够大时，KS阈值是紧的，这表明该区域不存在信息-计算鸿沟。这部分证实了Angelini等人(2015)的猜想。对于$r\ge 7$，我们证明当$d$足够大时，KS阈值不是紧的，为该区域存在信息-计算鸿沟提供了更多证据。此外，我们建立了固定$r$且$d$较大时弱恢复阈值的渐近界。针对密切相关的超树广播(BOHT)模型，我们还获得了多项结果，包括$r\ge 7$时重构阈值的渐近性质以及临界状态下鲁棒重构的不可能性。