Latent Distance Estimation for Random Geometric Graphs

TL;DR

This paper introduces a spectral estimator for latent distances in random geometric graphs, demonstrating its convergence rate, efficiency, and ability to estimate the latent space dimension without prior knowledge.

Contribution

It proposes a novel spectral method for estimating latent distances and the dimension of the latent space in random geometric graphs, with proven convergence properties.

Findings

Estimator has convergence rate comparable to nonparametric function estimation.

Algorithm efficiently computes distances without knowing the link function.

Method can consistently estimate the latent space dimension.

Abstract

Random geometric graphs are a popular choice for a latent points generative model for networks. Their definition is based on a sample of $n$ points $X_1,X_2,\cdots,X_n$ on the Euclidean sphere~$\mathbb{S}^{d-1}$ which represents the latent positions of nodes of the network. The connection probabilities between the nodes are determined by an unknown function (referred to as the "link" function) evaluated at the distance between the latent points. We introduce a spectral estimator of the pairwise distance between latent points and we prove that its rate of convergence is the same as the nonparametric estimation of a function on $\mathbb{S}^{d-1}$, up to a logarithmic factor. In addition, we provide an efficient spectral algorithm to compute this estimator without any knowledge on the nonparametric link function. As a byproduct, our method can also consistently estimate the dimension $d$ of the latent space.