Computational Lower Bounds for Correlated Random Graphs via Algorithmic Contiguity
Zhangsong Li
TL;DR
The paper establishes computational lower bounds for two core problems in correlated random graphs under the low-degree conjecture: (i) partial recovery in sparse correlated ER graphs with $q=n^{-1+o(1)}$ and $\rho<\sqrt{\alpha}$, and (ii) detection between correlated SBMs and independent SBMs when $\varepsilon^2\lambda s<1$ and $s<\sqrt{\alpha}$. The authors introduce an algorithmic contiguity framework, tying bounded low-degree advantage to the impossibility of efficient one-sided tests and enabling reductions across inference tasks without strengthening the conjecture. They then apply this framework to prove hardness of partial recovery for both correlated ER graphs and SBMs, and to establish detection hardness for correlated SBMs against independent SBMs below the KS and Otter thresholds. The results elucidate sharp computational barriers in high-dimensional inference problems and provide a unified mechanism for transferring hardness across related tasks. The approach relies on careful low-degree analysis, hidden-sample constructions, and graph-structure decompositions to bound advantageous statistics and derive contiguity-based intractability results.
Abstract
In this paper, assuming the low-degree conjecture, we provide evidence of computational hardness for two problems: (1) the (partial) matching recovery problem in the sparse correlated Erdős-Rényi graphs $\mathcal G(n,q;ρ)$ when the edge-density $q=n^{-1+o(1)}$ and the correlation $ρ<\sqrtα$ lies below the Otter's threshold, this resolves a remaining problem in \cite{DDL23+}; (2) the detection problem between a pair of correlated sparse stochastic block models $\mathcal S(n,\tfracλ{n};k,ε;s)$ and a pair of independent stochastic block models $\mathcal S(n,\tfrac{λs}{n};k,ε)$ when $ε^2 λs<1$ lies below the Kesten-Stigum (KS) threshold and $s<\sqrtα$ lies below the Otter's threshold, this resolves a remaining problem in \cite{CDGL24+}. One of the main ingredient in our proof is to derive certain forms of \emph{algorithmic contiguity} between two probability measures based on bounds on their low-degree advantage. To be more precise, consider the high-dimensional hypothesis testing problem between two probability measures $\mathbb{P}$ and $\mathbb{Q}$ based on the sample $\mathsf Y$. We show that if the low-degree advantage $\mathsf{Adv}_{\leq D} \big( \frac{\mathrm{d}\mathbb{P}}{\mathrm{d}\mathbb{Q}} \big)=O(1)$, then (assuming the low-degree conjecture) there is no efficient algorithm $\mathcal A$ such that $\mathbb{Q}(\mathcal A(\mathsf Y)=0)=1-o(1)$ and $\mathbb{P}(\mathcal A(\mathsf Y)=1)=Ω(1)$. This framework provides a useful tool for performing reductions between different inference tasks, without requiring a strengthened version of the low-degree conjecture as in \cite{MW23+, DHSS25+}.