Node Regression on Latent Position Random Graphs via Local Averaging

Node regression consists in predicting the value of a graph label at a node, given observations at the other nodes. We perform a theoretical study where the graph is generated by a Latent Position Model: each node has a latent position and the probability of connection depends on the distance between latent positions. We begin by studying the simplest estimator: averaging 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, with the same rate of convergence. One issue with this estimator is that it averages over all neighbors of a node, which may be too large or too small a region depending on the graph model. An alternative consists in first estimating the "true" distances between latent positions, then injecting these 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 even when the graph neighborhood is too large or too small.