A probabilistic view of latent space graphs and phase transitions

TL;DR

This paper investigates phase transitions in random graphs with latent geometric structure, identifying when such geometry can be detected based on graph parameters and proposing efficient statistical tests.

Contribution

It introduces a unified probabilistic framework for latent space graphs, analyzing phase transitions and providing computationally efficient methods for geometry detection.

Findings

Phase transitions depend on dimension and variance parameters.

A signed triangle statistic can distinguish geometric from Erdős–Rényi graphs.

High dimension or variance leads to graphs indistinguishable from Erdős–Rényi.

Abstract

We study random graphs with latent geometric structure, where the probability of each edge depends on the underlying random positions corresponding to the two endpoints. We focus on the setting where this conditional probability is a general monotone increasing function of the inner product of two vectors; such a function can naturally be viewed as the cumulative distribution function of some independent random variable. We consider a one-parameter family of random graphs, characterized by the variance of this random variable, that smoothly interpolates between a random dot product graph and an Erd\H{o}s--R\'enyi random graph. We prove phase transitions of detecting geometry in these graphs, in terms of the dimension of the underlying geometric space and the variance parameter of the conditional probability. When the dimension is high or the variance is large, the graph is similar to an Erd\H{o}s--R\'enyi graph with the same edge density that does not possess geometry; in other parameter regimes, there is a computationally efficient signed triangle statistic that distinguishes them. The proofs make use of information-theoretic inequalities and concentration of measure phenomena.