Testing Thresholds and Spectral Properties of High-Dimensional Random Toroidal Graphs via Edgeworth-Style Expansions

Abstract: We study high-dimensional random geometric graphs (RGGs) of edge-density $p$ with vertices uniformly distributed on the $d$-dimensional torus and edges inserted between sufficiently close vertices with respect to an $L_q$-norm. We focus on distinguishing an RGG from an Erd\H{o}s--R\'enyi (ER) graph if both models have edge probability $p$. So far, most results considered either spherical RGGs with $L_2$-distance or toroidal RGGs under $L_\infty$-distance. However, for general $L_q$-distances, many questions remain open, especially if $p$ is allowed to depend on $n$. The main reason for this is that RGGs under $L_q$-distances can not easily be represented as the logical AND of their 1-dimensional counterparts, as for $L_\infty$ geometries. To overcome this, we devise a novel technique for quantifying the dependence between edges based on modified Edgeworth expansions. Our technique yields the first tight algorithmic upper bounds for distinguishing toroidal RGGs under general $L_q$ norms from ER-graphs for fixed $p$ and $q$. We achieve this by showing that signed triangles can distinguish the two models when $d\ll n^3p^3$ for the whole regime of $c/n<p<1$. Additionally, our technique yields an improved information-theoretic lower bound for this task, showing that the two distributions converge whenever $d=\tilde{\Omega}(n^3p^2)$, which is just as strong as the currently best known lower bound for spherical RGGs in case of general $p$ from Liu et al. [STOC'22]. Finally, our expansions allow us to tightly characterize the spectral properties of toroidal RGGs both under $L_q$-distances for fixed $1\le q<\infty$, and $L_\infty$-distance. Our results partially resolve a conjecture of Bangachev and Bresler [COLT'24] and prove that the distance metric, rather than the underlying space, is responsible for the observed differences in the behavior of spherical and toroidal RGGs.