A random geometric graph (RGG) with kernel $K$ is constructed by first sampling latent points $x_1,\ldots,x_n$ independently and uniformly from the $d$-dimensional unit sphere, then connecting each pair $(i,j)$ with probability $K(\langle x_i,x_j\rangle)$. We study the sharp detection threshold, namely the highest dimension at which an RGG can be distinguished from its Erdős--Rényi counterpart with the same edge density. For dense graphs, we show that for smooth kernels the critical scaling is $d = n^{3/4}$, substantially lower than the threshold $d = n^3$ known for the hard RGG with step-function kernels \cite{bubeck2016testing}. We further extend our results to kernels whose signal-to-noise ratio scales with $n$, and formulate a unifying conjecture that the critical dimension is determined by $n^3 \mathop{\rm tr}^2(κ^3) = 1$, where $κ$ is the standardized kernel operator on the sphere. Departing from the prevailing approach of bounding the Kullback-Leibler divergence by successively exposing latent points, which breaks down in the sublinear regime of $d=o(n)$, our key technical contribution is a careful analysis of the posterior distribution of the latent points given the observed graph, in particular, the overlap between two independent posterior samples. As a by-product, we establish that $d=\sqrt{n}$ is the critical dimension for non-trivial estimation of the latent vectors up to a global rotation.

翻译：具有核函数$K$的随机几何图（RGG）的构造方式如下：首先从$d$维单位球面上独立均匀地采样潜在点$x_1,\ldots,x_n$，随后以概率$K(\langle x_i,x_j\rangle)$连接每对节点$(i,j)$。我们研究尖锐检测阈值，即RGG能够与其具有相同边密度的Erdős–Rényi图相区分的最高维度。对于稠密图，我们证明对于光滑核函数，临界尺度为$d = n^{3/4}$，显著低于已知的具有阶跃函数核的硬RGG的阈值$d = n^3$ \cite{bubeck2016testing}。我们进一步将结果推广至信噪比随$n$变化的核函数，并提出一个统一猜想：临界维度由$n^3 \mathop{\rm tr}^2(κ^3) = 1$决定，其中$κ$是球面上的标准化核算子。不同于主流通过逐次暴露潜在点来界定Kullback-Leibler散度的方法（该方法在$d=o(n)$的次线性区域失效），我们的核心技术贡献是对给定观测图后潜在点后验分布的精细分析，特别是两个独立后验样本之间的重叠度。作为副产品，我们确立了$d=\sqrt{n}$是在全局旋转意义下对潜在向量进行非平凡估计的临界维度。