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Random graphs are statistical models that have many applications, rangingfrom neuroscience to social network analysis. Of particular interest in someapplications is the problem of testing two random graphs for equality ofgenerating distributions. Tang et al. (2017) propose a test for this setting.This test consists of embedding the graph into a low-dimensional space via theadjacency spectral embedding (ASE) and subsequently using a kernel two-sampletest based on the maximum mean discrepancy. However, if the two graphs beingcompared have an unequal number of vertices, the test of Tang et al. (2017) maynot be valid. We demonstrate the intuition behind this invalidity and propose acorrection that makes any subsequent kernel- or distance-based test valid. Ourmethod relies on sampling based on the asymptotic distribution for the ASE. Wecall these altered embeddings the corrected adjacency spectral embeddings(CASE). We also show that CASE remedies the exchangeability problem of theoriginal test and demonstrate the validity and consistency of the test thatuses CASE via a simulation study. Lastly, we apply our proposed test to theproblem of determining equivalence of generating distributions in humanconnectomes extracted from diffusion magnetic resonance imaging (dMRI) atdifferent scales.

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