Testing thresholds for high-dimensional sparse random geometric graphs

Abstract

The random geometric graph model Geo d (n, p) is a distribution over graphs in which the edges capture a latent geometry. To sample G ∼ Geo d (n, p), we identify each of our n vertices with an independently and uniformly sampled vector from the d-dimensional unit sphere S d−1 , and we connect pairs of vertices whose vectors are "sufficiently close," such that the marginal probability of an edge is p. Because of the underlying geometry, this model is natural for applications in data science and beyond.We investigate the problem of testing for this latent geometry, or in other words, distinguishing

Cliques in High-Dimensional Geometric Inhomogeneous Random Graphs

“…for which values of d) our model converges in total variation distance to its non-geometric counterpart. A large body of work has already been devoted to this question for RGGs on the sphere [17,10,9,33,32] and recently also for random intersection graphs [9]. While the question when these models lose their geometry in the dense case is already largely answered, it remains open for the sparse case (where the marginal connection probability is proportional to 1/n) and progress has only been made recently [9,32].…”

“…A large body of work has already been devoted to this question for RGGs on the sphere [17,10,9,33,32] and recently also for random intersection graphs [9]. While the question when these models lose their geometry in the dense case is already largely answered, it remains open for the sparse case (where the marginal connection probability is proportional to 1/n) and progress has only been made recently [9,32]. It would be interesting to tightly characterize when our model loses its geometry both for the case of constant and for the case of inhomogeneous weights.…”

Cliques in High-Dimensional Geometric Inhomogeneous Random Graphs

“…for which values of d) our model converges in total variation distance to its non-geometric counterpart. A large body of work has already been devoted to this question for RGGs on the sphere [17,10,9,32,31] and recently also for random intersection graphs [9]. While the question when these models lose their geometry in the dense case is already largely answered, it remains open for the sparse case (where the marginal connection probability is proportional to 1/n) and progress has only been made recently [9,31].…”

“…A large body of work has already been devoted to this question for RGGs on the sphere [17,10,9,32,31] and recently also for random intersection graphs [9]. While the question when these models lose their geometry in the dense case is already largely answered, it remains open for the sparse case (where the marginal connection probability is proportional to 1/n) and progress has only been made recently [9,31]. It would be interesting to tightly characterize when our model loses its geometry both for the case of constant and for the case of inhomogeneous weights.…”