Threshold for detecting high dimensional geometry in anisotropic random geometric graphs

Abstract

In the anisotropic random geometric graph model, vertices correspond to points drawn from a high‐dimensional Gaussian distribution and two vertices are connected if their distance is smaller than a specified threshold. We study when it is possible to hypothesis test between such a graph and an Erdős‐Rényi graph with the same edge probability. If is the number of vertices and is the vector of eigenvalues, Eldan and Mikulincer, Geo. Aspects Func. Analysis: Israel seminar, 2017 shows that detection is possible when and impossible when . We show detection is impossible when , closing this gap and affirmatively resolving the conjecture of Eldan and Mikulincer, Geo. Aspects Func. Analysis: Israel seminar, 2017.