Phase Transitions for Detecting Latent Geometry in Random Graphs

TL;DR

This paper investigates the conditions under which random graphs with latent geometric structures become indistinguishable from Erdős-Rényi graphs, providing new theoretical thresholds for different models.

Contribution

It establishes precise phase transition thresholds for random intersection and geometric graphs converging to Erdős-Rényi models, resolving open problems and conjectures.

Findings

Random intersection graphs converge to Erdős-Rényi when d = (n^3)

Conditions for convergence of intersection size matrices to Poisson matrices

Geometric graphs on  spheres converge when d = (min{pn^3, p^2 n^{7/2}})

Abstract

Random graphs with latent geometric structure are popular models of social and biological networks, with applications ranging from network user profiling to circuit design. These graphs are also of purely theoretical interest within computer science, probability and statistics. A fundamental initial question regarding these models is: when are these random graphs affected by their latent geometry and when are they indistinguishable from simpler models without latent structure, such as the Erd\H{o}s-R\'{e}nyi graph $\mathcal{G}(n, p)$? We address this question for two of the most well-studied models of random graphs with latent geometry -- the random intersection and random geometric graph. Our results are as follows: (1) we prove that the random intersection graph converges in total variation to $\mathcal{G}(n, p)$ when $d = \tilde{\omega}(n^3)$, and does not if $d = o(n^3)$, resolving an open problem in Fill et al. (2000), Rybarczyk (2011) and Kim et al. (2018); (2) we provide conditions under which the matrix of intersection sizes of random family of sets converges in total variation to a symmetric matrix with independent Poisson entries, yielding the first total variation convergence result for $\tau$-random intersection graphs to $\mathcal{G}(n, p)$; and (3) we show that the random geometric graph on $\mathbb{S}^{d - 1}$ with edge density $p$ converges in total variation to $\mathcal{G}(n, p)$ when $d = \tilde{\omega}\left(\min\{ pn^3, p^2 n^{7/2} \} \right)$, yielding the first progress towards a conjecture of Bubeck et al. (2016). The first of these three results was obtained simultaneously and independently by Bubeck, Racz and Richey.