  Comparing two population means of network data is of paramount importance in
a wide range of scientific applications. Many existing network inference
solutions focus on global testing of entire networks, without comparing
individual network links. Besides, the observed data often take the form of
vectors or matrices, and the problem is formulated as comparing two covariance
or precision matrices under a normal or matrix normal distribution. Moreover,
many tests suffer from a limited power under a small sample size. In this
article, we tackle the problem of network comparison, both global and
simultaneous inferences, when the data come in a different format, i.e., in the
form of a collection of symmetric matrices, each of which encodes the network
structure of an individual subject. Such data format commonly arises in
applications such as brain connectivity analysis and clinical genomics. We no
longer require the underlying data to follow a normal distribution, but instead
impose some moment conditions that are easily satisfied for numerous types of
network data. Furthermore, we propose a power enhancement procedure, and show
that it can control the false discovery, while it has the potential to
substantially enhance the power of the test. We investigate the efficacy of our
testing procedure through both an asymptotic analysis and a simulation study
under a finite sample size. We further illustrate our method with an example of
brain structural connectivity analysis.
