We study the problem of detecting a faint geometric signal hidden in an otherwise random graph. Formally, we consider a hypothesis testing problem in which, under the null, the observed graph is an Erdős--Rényi random graph $\mathcal{G}(n,q)$, while under the alternative a random geometric graph $\mathcal{G}(k,q,d)$ is planted on $k\le n$ vertices. The planted subgraph is generated from independent random points on the unit sphere $\mathbb{S}^{d-1}$, with edges determined by latent geometric proximity and calibrated to have edge density $q$. Our goal is to characterize the statistical and computational limits of detecting this hidden geometry. We derive sharp information-theoretic lower bounds that identify regimes where detection is impossible and provide algorithms that achieve these limits whenever detection is feasible. We further investigate the computational complexity of the problem and determine when efficient polynomial-time tests exist. The model exhibits an \emph{easy--hard--impossible} phase transition: some regimes allow efficient detection, others permit detection only with computationally intractable procedures, and still others render detection impossible even with unlimited computational power. As evidence for the computational barrier, we prove that all low-degree polynomial algorithms fail throughout the conjecturally hard regime, demonstrating a sharp gap between statistical and computational feasibility.

翻译：我们研究了在随机图中检测隐藏微弱几何信号的问题。形式上，我们考虑一个假设检验问题：在原假设下，观测图是埃尔迪斯-雷尼随机图 $\mathcal{G}(n,q)$；而在备择假设下，一个随机几何图 $\mathcal{G}(k,q,d)$ 被植入到 $k\le n$ 个顶点中。该植入子图由单位球面 $\mathbb{S}^{d-1}$ 上独立随机点生成，边由潜在几何邻近性决定，并校准至边密度为 $q$。我们的目标是刻画检测这一隐藏几何结构的统计与计算极限。我们推导出尖锐的信息论下界，识别检测不可能存在的机制，并提供在检测可行时达到这些极限的算法。我们进一步研究该问题的计算复杂性，并确定高效多项式时间检验存在的条件。该模型呈现出“简单-困难-不可能”的相变：某些机制允许高效检测，其他机制仅能通过计算不可行的程序实现检测，而剩余机制即使拥有无限计算能力也无法进行检测。作为计算壁垒的证据，我们证明所有低阶多项式算法在推测的困难机制中均失效，揭示了统计可行性与计算可行性之间的尖锐差距。