A nonparametric two-sample hypothesis testing problem for random dot product graphs

TL;DR

This paper introduces a kernel-based test for comparing two random dot product graphs' latent positions, providing a statistically rigorous method to determine if they are generated from the same or related distributions.

Contribution

The paper proposes a novel nonparametric test statistic based on spectral embeddings for two-sample testing of random dot product graphs, with proven consistency and limiting distribution.

Findings

Test statistic follows a known limiting distribution under the null hypothesis.

The test is consistent across various alternative hypotheses.

The method effectively distinguishes between identical and related latent distributions.

Abstract

We consider the problem of testing whether two finite-dimensional random dot product graphs have generating latent positions that are independently drawn from the same distribution, or distributions that are related via scaling or projection. We propose a test statistic that is a kernel-based function of the adjacency spectral embedding for each graph. We obtain a limiting distribution for our test statistic under the null and we show that our test procedure is consistent across a broad range of alternatives.