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

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