Correcting a Nonparametric Two-sample Graph Hypothesis Test for Graphs with Different Numbers of Vertices with Applications to Connectomics

TL;DR

This paper identifies a flaw in existing graph hypothesis tests for graphs with different sizes and proposes a correction using asymptotic sampling, improving validity and applicability to connectomics data.

Contribution

It introduces the corrected adjacency spectral embeddings (CASE) method, enabling valid two-sample tests for graphs with unequal numbers of vertices, with theoretical validation and practical application.

Findings

CASE makes the test valid for graphs with different sizes

The corrected test is consistent and reliable in simulations

Application to connectomes demonstrates practical utility

Abstract

Random graphs are statistical models that have many applications, ranging from neuroscience to social network analysis. Of particular interest in some applications is the problem of testing two random graphs for equality of generating distributions. Tang et al. (2017) propose a test for this setting. This test consists of embedding the graph into a low-dimensional space via the adjacency spectral embedding (ASE) and subsequently using a kernel two-sample test based on the maximum mean discrepancy. However, if the two graphs being compared have an unequal number of vertices, the test of Tang et al. (2017) may not be valid. We demonstrate the intuition behind this invalidity and propose a correction that makes any subsequent kernel- or distance-based test valid. Our method relies on sampling based on the asymptotic distribution for the ASE. We call these altered embeddings the corrected adjacency spectral embeddings (CASE). We also show that CASE remedies the exchangeability problem of the original test and demonstrate the validity and consistency of the test that uses CASE via a simulation study. Lastly, we apply our proposed test to the problem of determining equivalence of generating distributions in human connectomes extracted from diffusion magnetic resonance imaging (dMRI) at different scales.