Detection of local geometry in random graphs: information-theoretic and computational limits

arXiv:2603.24545v1 Announce Type: cross
Abstract: 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 ErdH{o}s–R’enyi 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 $d = widetilde{Theta}(k^2 vee k^6/n^3)$ 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 $ell geq 4$.