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  "path": "/abs/2603.24545v1",
  "publishedAt": "2026-03-26T00:00:00.000Z",
  "site": "https://arxiv.org",
  "tags": [
    "Jinho Bok",
    "Shuangping Li",
    "Sophie H. Yu"
  ],
  "textContent": "**Authors:** Jinho Bok, Shuangping Li, Sophie H. Yu\n\nWe 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 $\\ell \\geq 4$.",
  "title": "Detection of local geometry in random graphs: information-theoretic and computational limits"
}