Nonparametric two-sample hypothesis testing for low-rank random graphs of differing sizes

TL;DR

This paper introduces a nonparametric two-sample test for low-rank random graphs of different sizes, leveraging graph embeddings and optimal transport to determine if two networks are from the same distribution.

Contribution

It proposes a novel test statistic based on maximum mean discrepancy and optimal transport, applicable to low-rank random graphs with differing sizes.

Findings

Test statistic is consistent for sufficiently dense graphs.

Convergence studied under various sparsity regimes.

Numerical simulations demonstrate effectiveness.

Abstract

Given two networks of differing sizes, it is of interest to test whether the two networks belong to the same distribution. We formalize the notion of "equality of distribution" under the framework of the generalized random dot product graph, which considers as special cases a number of popular network models with low-rank expectations. We then propose a nonparametric two-sample test statistic to conduct this test, assuming only that the networks have independent edges generated from low-rank probability matrices. Our proposed test statistic involves using the maximum mean discrepancy applied to suitably rotated rows of a graph embedding, where the rotation is estimated using optimal transport. We show that our test statistic, appropriately scaled, is consistent for sufficiently dense graphs, and we study its convergence under different sparsity regimes, and our results are demonstrated in numerical simulations.