A major question in the study of the Erd\H{o}s--R\'enyi random graph is to understand the probability that it contains a given subgraph. This study originated in classical work of Erd\H{o}s and R\'enyi (1960). More recent work studies this question both in building a general theory of sharp versus coarse transitions (Friedgut and Bourgain 1999; Hatami, 2012) and in results on the location of the transition (Kahn and Kalai, 2007; Talagrand, 2010; Frankston, Kahn, Narayanan, Park, 2019; Park and Pham, 2022). In inference problems, one often studies the optimal accuracy of inference as a function of the amount of noise. In a variety of sparse recovery problems, an ``all-or-nothing (AoN) phenomenon'' has been observed: Informally, as the amount of noise is gradually increased, at some critical threshold the inference problem undergoes a sharp jump from near-perfect recovery to near-zero accuracy (Gamarnik and Zadik, 2017; Reeves, Xu, Zadik, 2021). We can regard AoN as the natural inference analogue of the sharp threshold phenomenon in random graphs. In contrast with the general theory developed for sharp thresholds of random graph properties, the AoN phenomenon has only been studied so far in specific inference settings. In this paper we study the general problem of inferring a graph $H=H_n$ planted in an Erd\H{o}s--R\'enyi random graph, thus naturally connecting the two lines of research mentioned above. We show that questions of AoN are closely connected to first moment thresholds, and to a generalization of the so-called Kahn--Kalai expectation threshold that scans over subgraphs of $H$ of edge density at least $q$. In a variety of settings we characterize AoN, by showing that AoN occurs if and only if this ``generalized expectation threshold'' is roughly constant in $q$. Our proofs combine techniques from random graph theory and Bayesian inference.

翻译：在埃尔德什—雷尼随机图的研究中，一个核心问题是理解其包含给定子图的概率。这一研究起源于埃尔德什和雷尼（1960）的经典工作。近年来的研究既涉及构建尖锐转变与渐近转变的一般理论（Friedgut 和 Bourgain, 1999; Hatami, 2012），也关注转变位置的结果（Kahn 和 Kalai, 2007; Talagrand, 2010; Frankston, Kahn, Narayanan, Park, 2019; Park 和 Pham, 2022）。在推断问题中，研究者常将推断的最优精度作为噪声量的函数进行研究。在多种稀疏恢复问题中，观察到一种“全有或全无（AoN）现象”：非正式地说，随着噪声量逐渐增加，在某个临界阈值处，推断问题会经历从近乎完美恢复突然跳变到近乎零精度的过程（Gamarnik 和 Zadik, 2017; Reeves, Xu, Zadik, 2021）。我们可以将AoN视为随机图中尖锐阈值现象在推断领域的自然类比。与针对随机图性质已建立的尖锐阈值一般理论不同，AoN现象目前仅在特定推断场景中得到研究。在本文中，我们研究在埃尔德什—雷尼随机图中推断所植入图$H=H_n$的一般性问题，从而自然连接上述两条研究脉络。我们证明，AoN问题与一阶矩阈值以及一种所谓的Kahn—Kalai期望阈值（该阈值扫描$H$中边密度至少为$q$的子图）的推广密切相关。在多种设定下，我们刻画了AoN的特征：证明AoN发生当且仅当此“广义期望阈值”在$q$上大致为常数。我们的证明融合了随机图理论与贝叶斯推断的技术。