  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.
