Node Regression on Latent Position Random Graphs via Local Averaging

TL;DR

This paper studies node regression on latent position random graphs, showing that simple averaging estimators converge to a Nadaraya Watson estimator in the latent space and proposing an improved method using estimated distances for better convergence rates.

Contribution

It introduces a theoretical analysis of node regression estimators on latent position models and proposes a novel distance estimation approach to improve convergence rates.

Findings

Averaging over neighbors converges to a Nadaraya Watson estimator in latent space.

Estimating latent distances allows for adaptive averaging regions.

The proposed method achieves standard nonparametric rates under certain conditions.

Abstract

Node regression consists in predicting the value of a graph label at a node, given observations at the other nodes. To gain some insight into the performance of various estimators for this task, we perform a theoretical study in a context where the graph is random. Specifically, we assume that the graph is generated by a Latent Position Model, where each node of the graph has a latent position, and the probability that two nodes are connected depend on the distance between the latent positions of the two nodes. In this context, we begin by studying the simplest possible estimator for graph regression, which consists in averaging the value of the label at all neighboring nodes. We show that in Latent Position Models this estimator tends to a Nadaraya Watson estimator in the latent space, and that its rate of convergence is in fact the same. One issue with this standard estimator is that it averages over a region consisting of all neighbors of a node, and that depending on the graph model this may be too much or too little. An alternative consists in first estimating the true distances between the latent positions, then injecting these estimated distances into a classical Nadaraya Watson estimator. This enables averaging in regions either smaller or larger than the typical graph neighborhood. We show that this method can achieve standard nonparametric rates in certain instances even when the graph neighborhood is too large or too small.