A computational transition for detecting correlated stochastic block models by low-degree polynomials

Detection of correlation in a pair of random graphs is a fundamental statistical and computational problem that has been extensively studied in recent years. In this work, we consider a pair of correlated (sparse) stochastic block models $\mathcal{S}(n,\tfracλ{n};k,ε;s)$ that are subsampled from a common parent stochastic block model $\mathcal S(n,\tfracλ{n};k,ε)$ with $k=O(1)$ symmetric communities, average degree $λ=O(1)$, divergence parameter $ε$, and subsampling probability $s$. For the detection problem of distinguishing this model from a pair of independent Erdős-Rényi graphs with the same edge density $\mathcal{G}(n,\tfrac{λs}{n})$, we focus on tests based on \emph{low-degree polynomials} of the entries of the adjacency matrices, and we determine the threshold that separates the easy and hard regimes. More precisely, we show that this class of tests can distinguish these two models if and only if $s> \min { \sqrtα, \frac{1}{λε^2} }$, where $α\approx 0.338$ is the Otter’s constant and $\frac{1}{λε^2}$ is the Kesten-Stigum threshold. Combining a reduction argument in \cite{Li25+}, our hardness result also implies low-degree hardness for partial recovery and detection (to independent block models) when $s< \min { \sqrtα, \frac{1}{λε^2} }$. Finally, our proof of low-degree hardness is based on a conditional variant of the low-degree likelihood calculation.

💡 Research Summary

The paper studies the problem of detecting correlation between two sparse stochastic block models (SBMs) that are generated by subsampling a common parent SBM. Formally, a parent SBM 𝒮(n,λ/n;k,ε) with a constant number k of symmetric communities, constant average degree λ, and divergence parameter ε is sampled. Each edge of the parent graph is retained independently with probability s, producing two observed graphs A and B. The detection task is to distinguish the joint distribution Pₙ of (A,B) from the null distribution Qₙ, which consists of two independent Erdős–Rényi graphs 𝒢(n,λs/n) with the same edge density.

The authors focus on tests that are low-degree polynomials in the entries of the adjacency matrices. A degree‑D polynomial P(A,B) is said to separate the two hypotheses if there exists a threshold τ such that the probability under Pₙ that P(A,B)≥τ is close to 1 while under Qₙ the same event has probability close to 0. The “low‑degree” regime considered is D=ω(1) but D=o(log n·log log n), which captures many known polynomial‑time algorithms (spectral methods, approximate message passing, small‑subgraph counting, etc.).

The main result (Theorem 1.3) identifies a sharp computational transition for low‑degree tests. There are two thresholds:

1. The Kesten–Stigum (KS) threshold 1/(λ ε²). When the subsampling probability s exceeds this value, low‑degree tests succeed.
2. The Otter constant threshold √α, where α≈0.338 is Otter’s constant (the growth rate of rooted regular trees). When s exceeds √α, low‑degree tests also succeed.

If either condition holds (i.e., s > min{1/(λ ε²), √α}), the authors construct an explicit algorithm based on counting common rooted trees in the two graphs. This statistic can be expressed as a low‑degree polynomial, runs in time n^{2+o(1)}, and achieves vanishing error probability.

Conversely, when s < min{1/(λ ε²), √α}, the paper proves that any low‑degree polynomial test fails. The difficulty stems from the divergence of the second moment of the low‑degree likelihood ratio, caused by rare “bad” events under the planted model. To overcome this, the authors condition on a “good” event (admissible graphs) that excludes pathological configurations. Within this conditioned space they compute the expectations of orthogonal polynomial bases and bound the conditional second moment. They then show that the contribution of the excluded bad events is negligible, leading to a rigorous low‑degree hardness result.

The hardness proof is then leveraged via a reduction from Li et al.