On The Fourier Coefficients of High-Dimensional Random Geometric Graphs
Kiril Bangachev, Guy Bresler
TL;DR
This work analyzes the low-degree Fourier coefficients of two high-dimensional latent-geometry graph models, spherical RGGe(n, S^{d-1}, p) and Gaussian RGGe(n, N(0, d^{-1}I_d), p). The authors introduce a novel two-step approach: localize edge dependencies to a small fragile set of edges and apply a noise operator to the remaining edges, facilitated by an edge-independent latent-basis from Gram-Schmidt. They derive sharp bounds on Fourier coefficients via the ordered/strong edge independence numbers (OEI/SOEI) and apply these bounds to distinguish RGGe from Erdős–Rényi under full and masked observations, exhibit a statistical-computational gap for RGGe vs planted coloring, and reprove bounds on the second eigenvalue of RGGe. The results illuminate when low-degree polynomial tests are optimal or incomplete and provide a versatile framework for analyzing latent-space distributions beyond RGGe. The methods have potential implications for testing, spectral analysis, and understanding computational-statistical gaps in high-dimensional latent-geometry models.
Abstract
The random geometric graph $\mathsf{RGG}(n,\mathbb{S}^{d-1}, p)$ is formed by sampling $n$ i.i.d. vectors $\{V_i\}_{i = 1}^n$ uniformly on $\mathbb{S}^{d-1}$ and placing an edge between pairs of vertices $i$ and $j$ for which $\langle V_i,V_j\rangle \ge τ^p_d,$ where $τ^p_d$ is such that the expected density is $p.$ We study the low-degree Fourier coefficients of the distribution $\mathsf{RGG}(n,\mathbb{S}^{d-1}, p)$ and its Gaussian analogue. Our main conceptual contribution is a novel two-step strategy for bounding Fourier coefficients which we believe is more widely applicable to studying latent space distributions. First, we localize the dependence among edges to few fragile edges. Second, we partition the space of latent vector configurations $(\mathsf{RGG}(n,\mathbb{S}^{d-1}, p))^{\otimes n}$ based on the set of fragile edges and on each subset of configurations, we define a noise operator acting independently on edges not incident (in an appropriate sense) to fragile edges. We apply the resulting bounds to: 1) Settle the low-degree polynomial complexity of distinguishing spherical and Gaussian random geometric graphs from Erdos-Renyi both in the case of observing a complete set of edges and in the non-adaptively chosen mask $\mathcal{M}$ model recently introduced by [MVW24]; 2) Exhibit a statistical-computational gap for distinguishing $\mathsf{RGG}$ and the planted coloring model [KVWX23] in a regime when $\mathsf{RGG}$ is distinguishable from Erdos-Renyi; 3) Reprove known bounds on the second eigenvalue of random geometric graphs.