A random algebraic graph is defined by a group $G$ with a uniform distribution over it and a connection $\sigma:G\longrightarrow[0,1]$ with expectation $p,$ satisfying $\sigma(g)=\sigma(g^{-1}).$ The random graph $\mathsf{RAG}(n,G,p,\sigma)$ with vertex set $[n]$ is formed as follows. First, $n$ independent vectors $x_1,\ldots,x_n$ are sampled uniformly from $G.$ Then, vertices $i,j$ are connected with probability $\sigma(x_ix_j^{-1}).$ This model captures random geometric graphs over the sphere and the hypercube, certain regimes of the stochastic block model, and random subgraphs of Cayley graphs. The main question of interest to the current paper is: when is a random algebraic graph statistically and/or computationally distinguishable from $\mathsf{G}(n,p)$? Our results fall into two categories. 1) Geometric. We focus on the case $G =\{\pm1\}^d$ and use Fourier-analytic tools. For hard threshold connections, we match [LMSY22b] for $p = \omega(1/n)$ and for $1/(r\sqrt{d})$-Lipschitz connections we extend the results of [LR21b] when $d = \Omega(n\log n)$ to the non-monotone setting. We study other connections such as indicators of interval unions and low-degree polynomials. 2) Algebraic. We provide evidence for an exponential statistical-computational gap. Consider any finite group $G$ and let $A\subseteq G$ be a set of elements formed by including each set of the form $\{g, g^{-1}\}$ independently with probability $1/2.$ Let $\Gamma_n(G,A)$ be the distribution of random graphs formed by taking a uniformly random induced subgraph of size $n$ of the Cayley graph $\Gamma(G,A).$ Then, $\Gamma_n(G,A)$ and $\mathsf{G}(n,1/2)$ are statistically indistinguishable with high probability over $A$ if and only if $\log|G|\gtrsim n.$ However, low-degree polynomial tests fail to distinguish $\Gamma_n(G,A)$ and $\mathsf{G}(n,1/2)$ with high probability over $A$ when $\log |G|=\log^{\Omega(1)}n.$

翻译：随机代数图由群$G$（其上服从均匀分布）和连接函数$\sigma:G\longrightarrow[0,1]$（期望为$p$且满足$\sigma(g)=\sigma(g^{-1})$）定义。以顶点集$[n]$构建的随机图$\mathsf{RAG}(n,G,p,\sigma)$生成方式如下：首先，从$G$中均匀独立采样$n$个向量$x_1,\ldots,x_n$；然后，顶点$i$与$j$以概率$\sigma(x_ix_j^{-1})$相连。该模型涵盖球面与超立方体上的随机几何图、随机块模型的特定状态以及Cayley图的随机子图。本文关注的核心问题是：随机代数图在统计和/或计算意义上何时可与$\mathsf{G}(n,p)$区分？我们的结果分为两类：1) 几何情形。针对$G =\{\pm1\}^d$情形，运用傅里叶分析工具。对于硬阈值连接，我们匹配了[LMSY22b]中$p = \omega(1/n)$的结论；对于$1/(r\sqrt{d})$-Lipschitz连接，我们将[LR21b]中$d = \Omega(n\log n)$的结果推广至非单调设定。此外，我们研究了区间并集指示函数及低次多项式等其他连接形式。2) 代数情形。我们提供了指数级统计-计算裂口的证据。考虑任意有限群$G$，设$A\subseteq G$为由每个形如$\{g, g^{-1}\}$的集合以概率$1/2$独立选取构成的元素集。令$\Gamma_n(G,A)$为从Cayley图$\Gamma(G,A)$中均匀随机抽取大小为$n$的诱导子图所得到的随机图分布。则当且仅当$\log|G|\gtrsim n$时，$\Gamma_n(G,A)$与$\mathsf{G}(n,1/2)$能在高概率下（对$A$而言）实现统计不可区分。然而，当$\log |G|=\log^{\Omega(1)}n$时，低次多项式检验无法在高概率下（对$A$而言）区分$\Gamma_n(G,A)$与$\mathsf{G}(n,1/2)$。