Shuyang Gong

Guanyi Chen, Jian Ding, Shuyang Gong et al.

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.

Shuyang Gong, Zhangsong Li

Motivated by the problem of matching two correlated random geometric graphs, we study the problem of matching two Gaussian geometric models correlated through a latent node permutation. Specifically, given an unknown permutation $π^*$ on $\{1,\ldots,n\}$ and given $n$ i.i.d. pairs of correlated Gaussian vectors $\{X_{π^*(i)},Y_i\}$ in $\mathbb{R}^d$ with noise parameter $σ$, we consider two types of (correlated) weighted complete graphs with edge weights given by $A_{i,j}=\langle X_i,X_j \rangle$, $B_{i,j}=\langle Y_i,Y_j \rangle$. The goal is to recover the hidden vertex correspondence $π^*$ based on the observed matrices $A$ and $B$. For the low-dimensional regime where $d=O(\log n)$, Wang, Wu, Xu, and Yolou [WWXY22+] established the information thresholds for exact and almost exact recovery in matching correlated Gaussian geometric models. They also conducted numerical experiments for the classical Umeyama algorithm. In our work, we prove that this algorithm achieves exact recovery of $π^*$ when the noise parameter $σ=o(d^{-3}n^{-2/d})$, and almost exact recovery when $σ=o(d^{-3}n^{-1/d})$. Our results approach the information thresholds up to a $\operatorname{poly}(d)$ factor in the low-dimensional regime.