  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.
