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Testing whether two graphs come from the same distribution is of interest inmany real world scenarios, including brain network analysis. Under the randomdot product graph model, the nonparametric hypothesis testing frame-workconsists of embedding the graphs using the adjacency spectral embedding (ASE),followed by aligning the embeddings using the median flip heuristic, andfinally applying the nonparametric maximum mean discrepancy(MMD) test to obtaina p-value. Using synthetic data generated from Drosophila brain networks, weshow that the median flip heuristic results in an invalid test, and demonstratethat optimal transport Procrustes (OTP) for alignment resolves the invalidity.We further demonstrate that substituting the MMD test with multiscale graphcorrelation(MGC) test leads to a more powerful test both in synthetic and insimulated data. Lastly, we apply this powerful test to the right and lefthemispheres of the larval Drosophila mushroom body brain networks, and concludethat there is not sufficient evidence to reject the null hypothesis that thetwo hemispheres are equally distributed.

Valid Two-Sample Graph Testing via Optimal Transport Procrustes and Multiscale Graph Correlation with Applications in Connectomics