We study the inference of communities in stochastic block models with a growing number of communities. For block models with $n$ vertices and a fixed number of communities $q$, it was predicted in Decelle et al. (2011) that there are computationally efficient algorithms for recovering the communities above the Kesten–Stigum (KS) bound and that efficient recovery is impossible below the KS bound. This conjecture has since stimulated a lot of interest, with the achievability side proven in a line of research that culminated in the work of Abbe and Sandon (2018). Conversely, recent work provides evidence for the hardness part using the low-degree paradigm. In this paper we investigate community recovery in the regime $q=q_n \to \infty$ as $n\to\infty$ where no such predictions exist. We show that efficient inference of communities remains possible above the KS bound. Furthermore, we show that recovery of block models is low-degree hard below the KS bound when the number of communities satisfies $q\ll \sqrt{n}$. Perhaps surprisingly, we find that when $q \gg \sqrt{n}$, there is an efficient algorithm based on non-backtracking walks for recovery even below the KS bound. We identify a new threshold and ask if it is the threshold for efficient recovery in this regime. Finally, we show that detection is easy and identify (up to a constant) the information-theoretic threshold for community recovery as the number of communities $q$ diverges. Our low-degree hardness results also naturally have consequences for graphon estimation, improving results of Luo and Gao (2024).

For a large class of random constraint satisfaction problems (CSP), deep but non-rigorous theory from statistical physics predict the location of the sharp satisfiability transition. The works of Ding, Sly, Sun (2014, 2016) and Coja-Oghlan, Panagiotou (2014) established the satisfiability threshold for random regular $k$-NAE-SAT, random $k$-SAT, and random regular $k$-SAT for large enough $k\geq k_0$ where $k_0$ is a large non-explicit constant. Establishing the same for small values of $k\geq 3$ remains an important open problem in the study of random CSPs. In this work, we study two closely related models of random CSPs, namely the $2$-coloring on random $d$-regular $k$-uniform hypergraphs and the random $d$-regular $k$-NAE-SAT model. For every $k\geq 3$, we prove that there is an explicit $d_{\ast}(k)$ which gives a satisfiability upper bound for both of the models. Our upper bound $d_{\ast}(k)$ for $k\geq 3$ matches the prediction from statistical physics for the hypergraph $2$-coloring by Dall'Asta, Ramezanpour, Zecchina (2008), thus conjectured to be sharp. Moreover, $d_{\ast}(k)$ coincides with the satisfiability threshold of random regular $k$-NAE-SAT for large enough $k\geq k_0$ by Ding, Sly, Sun (2014).

The local behavior of typical solutions of random constraint satisfaction problems (csp) describes many important phenomena including clustering thresholds, decay of correlations, and the behavior of message passing algorithms. When the constraint density is low, studying the planted model is a powerful technique for determining this local behavior which in many examples has a simple Markovian structure. Work of Coja-Oghlan, Kapetanopoulos, M'uller (2020) showed that for a wide class of models, this description applies up to the so-called condensation threshold. Understanding the local behavior after the condensation threshold is more complex due to long-range correlations. In this work, we revisit the random regular nae-sat model in the condensation regime and determine the local weak limit which describes a random solution around a typical variable. This limit exhibits a complicated non-Markovian structure arising from the space of solutions being dominated by a small number of large clusters. This is the first description of the local weak limit in the condensation regime for any sparse random csps in the one-step replica symmetry breaking (1rsb) class. Our result is non-asymptotic, and characterizes the tight fluctuation O(n−1/2) around the limit. Our proof is based on coupling the local neighborhoods of an infinite spin system, which encodes the structure of the clusters, to a broadcast model on trees whose channel is given by the 1rsb belief-propagation fixed point. We believe that our proof technique has broad applicability to random csps in the 1rsb class.

We connect the study of phase transitions in high-dimensional statistical inference to the study of threshold phenomena in random graphs. A major question in the study of the Erdős–Rényi random graph $G(n,p)$ is to understand the probability, as a function of $p$, that $G(n,p)$ contains a given subgraph $H=H_n$. This was studied for many specific examples of $H$, starting with classical work of Erdős and Rényi (1960). More recent work studies this question for general $H$, 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 theorydeveloped for sharp thresholds of random graph properties, the AoNphenomenon has only been studied so far in specific inference settings, anda general theory behind its appearance remains elusive.In this paper we study the general problem of inferring a graph $H=H_n$ planted in an Erdős–Rényi 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 \emph{if and only if} this “generalized expectation threshold” is roughly constant in $q$. Our proofs combine techniques from random graph theory and Bayesian inference.

In this paper, we rigorously establish the predictions in ground breaking work in statistical physics by Decelle, Krzakala, Moore, Zdeborová (2011) regarding the block model, in particular in the case of q=3 and q=4 communities.

In a broad class of sparse random constraint satisfaction problems (CSP), deep heuristics from statistical physics predict that there is a condensation phase transition before the satisfiability threshold, governed by one-step replica symmetry breaking (1RSB). In fact, in random regular k-NAE-SAT, which is one of such random CSPS, it was verified [1] that its free energy is well-defined and the explicit value follows the 1RSB prediction. However, for any model of sparse random CSP, it has been unknown whether the solution space indeed condensates on O(1) clusters according to the 1RSB prediction. In this paper, we give an affirmative answer to this question for the random regular k-NAE-SAT model. Namely, we prove that with probability close to one, most of the solutions lie inside a bounded number of solution clusters whose sizes are comparable to the scale of the free energy. Furthermore, we establish that the overlap between two independently drawn solutions concentrates precisely at two values. This is the defining property of the one-step replica symmetry breaking class which we establish for the first time in a sparse random CSP, Our proof is based on a detailed moment analysis of a spin system, which has an infinite spin space that encodes the structure of solution clusters. We develop new techniques to study the partition function as well as enhance previous approaches which were only applicable to spin systems with finitely many spins. We believe that our method is applicable to a broad range of random CSPS in the 1RSB universality class.