We study the problem of detecting local geometry in random graphs. We introduce a model $\mathcal{G}(n, p, d, k)$, where a hidden community of average size $k$ has edges drawn as a random geometric graph on $\mathbb{S}^{d-1}$, while all remaining edges follow the Erdős--Rényi model $\mathcal{G}(n, p)$. The random geometric graph is generated by thresholding inner products of latent vectors on $\mathbb{S}^{d-1}$, with each edge having marginal probability equal to $p$. This implies that $\mathcal{G}(n, p, d, k)$ and $\mathcal{G}(n, p)$ are indistinguishable at the level of the marginals, and the signal lies entirely in the edge dependencies induced by the local geometry. We investigate both the information-theoretic and computational limits of detection. On the information-theoretic side, our upper bounds follow from three tests based on signed triangle counts: a global test, a scan test, and a constrained scan test; our lower bounds follow from two complementary methods: truncated second moment via Wishart--GOE comparison, and tensorization of KL divergence. These results together settle the detection threshold at $d = \widetildeΘ(k^2 \vee k^6/n^3)$ for fixed $p$, and extend the state-of-the-art bounds from the full model (i.e., $k = n$) for vanishing $p$. On the computational side, we identify a computational--statistical gap and provide evidence via the low-degree polynomial framework, as well as the suboptimality of signed cycle counts of length $\ell \geq 4$.

翻译：研究随机图中局部几何结构的检测问题。我们引入模型 $\mathcal{G}(n, p, d, k)$，其中隐藏社区的平均规模为 $k$，其边按照 $\mathbb{S}^{d-1}$ 上的随机几何图生成，而所有其余边遵循 Erdős--Rényi 模型 $\mathcal{G}(n, p)$。随机几何图通过对 $\mathbb{S}^{d-1}$ 上潜向量的内积设置阈值生成，每条边的边际概率等于 $p$。这意味着 $\mathcal{G}(n, p, d, k)$ 与 $\mathcal{G}(n, p)$ 在边际层面上不可区分，信号完全来源于由局部几何结构引起的边依赖关系。我们从信息论和计算两个角度研究检测极限。在信息论方面，上界基于符号三角形计数的三种检验：全局检验、扫描检验和约束扫描检验；下界则来自两种互补方法：通过 Wishart--GOE 比较的截断二阶矩，以及 KL 散度的张量化。这些结果共同确定了固定 $p$ 下检测阈值为 $d = \widetildeΘ(k^2 \vee k^6/n^3)$，并将全模型（即 $k = n$）在 $p$ 趋近于零时的最新界加以推广。在计算方面，我们识别出计算-统计差距，并通过低次多项式框架以及长度 $\ell \geq 4$ 的有符号环计数非最优性提供证据。