Random Geometric Graphs on Euclidean Balls

TL;DR

This paper studies a latent space model for random graphs on Euclidean balls, proposing estimators for latent norms and Gram matrices, analyzing their theoretical guarantees, and demonstrating the model's ability to produce power-law degree distributions.

Contribution

The paper introduces new estimators for latent norms and Gram matrices in a geometric graph model, with theoretical error bounds and insights into degree distribution properties.

Findings

Estimator for latent norms based on node degrees.

Frobenius error bounds for Gram matrix estimation.

Model can generate graphs with power-law degree distributions.

Abstract

We consider a latent space model for random graphs where a node $i$ is associated to a random latent point $X_i$ on the Euclidean unit ball. The probability that an edge exists between two nodes is determined by a ``link'' function, which corresponds to a dot product kernel. For a given class $\F$ of spherically symmetric distributions for $X_i$, we consider two estimation problems: latent norm recovery and latent Gram matrix estimation. We construct an estimator for the latent norms based on the degree of the nodes of an observed graph in the case of the model where the edge probability is given by $f(\langle X_i,X_j\rangle)=\mathbbm{1}_{\langle X_i,X_j\rangle\geq \tau}$, where $0<\tau<1$. We introduce an estimator for the Gram matrix based on the eigenvectors of observed graph and we establish Frobenius type guarantee for the error, provided that the link function is sufficiently regular in the Sobolev sense and that a spectral-gap-type condition holds. We prove that for certain link functions, the model considered here generates graphs with degree distribution that have tails with a power-law-type distribution, which can be seen as an advantage of the model presented here with respect to the classic Random Geometric Graph model on the Euclidean sphere. We illustrate our results with numerical experiments.