  Two-sample network hypothesis testing is an important inference task with
applications across diverse fields such as medicine, neuroscience, and
sociology. Many of these testing methodologies operate under the implicit
assumption that the vertex correspondence across networks is a priori known.
This assumption is often untrue, and the power of the subsequent test can
degrade when there are misaligned/label-shuffled vertices across networks. This
power loss due to shuffling is theoretically explored in the context of random
dot product and stochastic block model networks for a pair of hypothesis tests
based on Frobenius norm differences between estimated edge probability matrices
or between adjacency matrices. The loss in testing power is further reinforced
by numerous simulations and experiments, both in the stochastic block model and
in the random dot product graph model, where the power loss across multiple
recently proposed tests in the literature is considered. Lastly, the impact
that shuffling can have in real-data testing is demonstrated in a pair of
examples from neuroscience and from social network analysis.
