Phase transition in noisy high-dimensional random geometric graphs

Abstract

We study the problem of detecting latent geometric structure in random graphs. To this end, we consider the soft high-dimensional random geometric graph $G(n,p,d,q)$, where each of the n vertices corresponds to an independent random point distributed uniformly on the sphere ${\mathbb{S}}^{d-1}$, and the probability that two vertices are connected by an edge is a decreasing function of the Euclidean distance between the points. The probability of connection is parametrized by $q\in [0,1]$, with smaller q corresponding to weaker dependence on the geometry; this can also be interpreted as the level of noise in the geometric graph. In particular, the model smoothly interpolates between the hard spherical random geometric graph $G(n,p,d)$ (corresponding to $q=1$) and the Erdős–Rényi model $G(n,p)$ (corresponding to $q=0$). We focus on the dense regime (i.e., p is a constant).

We show that if $nq\to 0$ or $d»n^3{q}^2$, then geometry is lost: $G(n,p,d,q)$ is asymptotically indistinguishable from $G(n,p)$. On the other hand, if $d«n^3{q}^6$, then the signed triangle statistic provides an asymptotically powerful test for detecting geometry. These results generalize those of Bubeck, Ding, Eldan, and Rácz (2016) for $G(n,p,d)$, and give quantitative bounds on how the noise level affects the dimension threshold for losing geometry. We also prove analogous results under a related but different distributional assumption which corresponds to the random dot product graph.