Low-degree conjecture for distinguishing symmetric stochastic block models from Erdős–Rényi graphs

Prove the low-degree conjecture for the symmetric k-stochastic block model versus Erdős–Rényi graphs: Let P be the distribution SSBM(n, d/n, ε, k) and Q be the Erdős–Rényi distribution G(n, d/n). For functions f: {0,1}^{n×n} → R, define the normalized advantage R_{P,Q}(f) = (E_{Y∼P}[f(Y)] − E_{Y∼Q}[f(Y)]) / sqrt(Var_{Y∼Q}(f(Y))). Establish that if, for a fixed constant δ ∈ [0,1], every polynomial f of degree at most n^δ satisfies R_{P,Q}(f) = O(1), then every [0,1]-valued function f computable in time exp(n^δ) also satisfies R_{P,Q}(f) = O(1).

Background

The work studies computational hardness in the symmetric stochastic block model (SSBM) using the low-degree method. The conjecture asserts that if all low-degree polynomials have bounded distinguishing power between SSBM and Erdős–Rényi (ER) graphs, then no subexponential-time algorithm can achieve nontrivial distinguishing power either.

This conjecture, formalized in prior work, underpins the paper’s conditional lower bounds: assuming it holds for SSBM vs ER, the authors translate known low-degree lower bounds into algorithmic lower bounds for weak recovery and parameter learning below the Kesten–Stigum threshold.

References

In this paper, we focus on the implications of the following conjecture of [moitra2023precise] in the context of the SBM. Let P be a distribution from the k-stochastic block model and Q be a distribution of o-R random graphs. For functions f : {n \times n} \to \mathbb{R}, consider the parameter P,Q(f)\coloneqq \frac{\E_{Y\sim P} f(Y)-\E_{Y\sim Q} f(Y)}{\sqrt{\text{Var}_{Y\sim Q}(f(Y))}. Suppose that for fixed constant \delta\in [0,1], and every polynomial f(\cdot) of degree at most n\delta, we have P,Q(f) = O(1). Then, for any function f(\cdot) computable in time \exp(n{\delta}) taking values in [0,1], we have P,Q(f) = O(1).

— Low degree conjecture implies sharp computational thresholds in stochastic block model (2502.15024 - Ding et al., 20 Feb 2025) in Section 1, The low-degree method