Detection of $L_\infty$ Geometry in Random Geometric Graphs: Suboptimality of Triangles and Cluster Expansion

Abstract

In this paper we study the random geometric graph $\mathsf{RGG}(n,\mathbb{T}^d,\mathsf{Unif},\sigma^q_p,p)$ with $L_q$ distance where each vertex is sampled uniformly from the $d$-dimensional torus and where the connection radius is chosen so that the marginal edge probability is $p$. In addition to results addressing other questions, we make progress on determining when it is possible to distinguish $\mathsf{RGG}(n,\mathbb{T}^d,\mathsf{Unif},\sigma^q_p,p)$ from the Endos-R\'enyi graph $\mathsf{G}(n,p)$. Our strongest result is in the extreme setting $q = \infty$, in which case $\mathsf{RGG}(n,\mathbb{T}^d,\mathsf{Unif},\sigma^\infty_p,p)$ is the $\mathsf{AND}$ of $d$ 1-dimensional random geometric graphs. We derive a formula similar to the cluster-expansion from statistical physics, capturing the compatibility of subgraphs from each of the $d$ 1-dimensional copies, and use it to bound the signed expectations of small subgraphs. We show that counting signed 4-cycles is optimal among all low-degree tests, succeeding with high probability if and only if $d = \tilde{o}(np).$ In contrast, the signed triangle test is suboptimal and only succeeds when $d = \tilde{o}((np)^{3/4}).$ Our result stands in sharp contrast to the existing literature on random geometric graphs (mostly focused on $L_2$ geometry) where the signed triangle statistic is optimal.