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:Nrd\H{o}s-R\ enyi 随机图中的一个主要问题是了解它包含给定子图的概率。本研究起源于Erd\H{o}s和R\'enyi(1960年)的古典著作。最近的工作研究,在建立尖锐与粗粗转型的一般理论(Friedgut和Bourgain,1999年;Hadami,2012年)以及转型地点的结果(Kahn和Kalai,2007年;Talagrand,2010年;Frankston、Kahn、Narayanan、Park,2019年;Park和Pham,2022年。推断问题,常常研究推断性判断力的最佳准确性作为噪音的函数。在各种恢复问题中,Arqal-kai(Ao)现象出现:随着噪音的数量逐渐增加,在临界值值中,我们从接近断层恢复到接近零准确度的数据(Gairnik和Zadik,在201717年的直径理论中,Refereals) 研究中,我们的直径直径直径直径直系的直判研究显示。在A。在2021中,直系的研究显示直系的直系的直系的直系的直系的直系的直系,直系的直系的直系的直系的直系的直系的直系的直系的直系的直系的直系直系直径径向向向向向向向向,直系的直径向,直系的直径向向向,直向,直向,直系的直系的直系的直系的直系的直向,直系的直系的直系的直向,直系的直系的直系的直系的直系的直系的直系的直系的直系的直系,直系的直系的直系的直系的直系的直系的直系的直系的直系的直系的直系的直系的直系的直系的直。</s>