  Two-sample hypothesis testing for random graphs arises naturally in
neuroscience, social networks, and machine learning. In this paper, we consider
a semiparametric problem of two-sample hypothesis testing for a class of latent
position random graphs. We formulate a notion of consistency in this context
and propose a valid test for the hypothesis that two finite-dimensional random
dot product graphs on a common vertex set have the same generating latent
positions or have generating latent positions that are scaled or diagonal
transformations of one another. Our test statistic is a function of a spectral
decomposition of the adjacency matrix for each graph and our test procedure is
consistent across a broad range of alternatives. We apply our test procedure to
real biological data: in a test-retest data set of neural connectome graphs, we
are able to distinguish between scans from different subjects; and in the {\em
C.elegans} connectome, we are able to distinguish between chemical and
electrical networks. The latter example is a concrete demonstration that our
test can have power even for small sample sizes. We conclude by discussing the
relationship between our test procedure and generalized likelihood ratio tests.
