  It has become an increasingly common practice for scientists in modern
science and engineering to collect samples of multiple network data in which a
network serves as a basic data object. The increasing prevalence of multiple
network data calls for developments of models and theory that can deal with
inference problems for populations of networks. In this work, we propose a
general procedure for hypothesis testing of networks and in particular, for
differentiating distributions of two samples of networks. We consider a very
general framework which allows us to perform tests on large and sparse
networks. Our contribution is two-fold: (1) We propose a test statistics based
on the singular value of a generalized Wigner matrix. The asymptotic null
distribution of the statistics is shown to follow the Tracy--Widom distribution
as the number of nodes tends to infinity. The test also yields asymptotic power
guarantee with the power tending to one under the alternative; (2) The test
procedure is adapted for change-point detection in dynamic networks which is
proven to be consistent in detecting the change-points. In addition to
theoretical guarantees, another appealing feature of this adapted procedure is
that it provides a principled and simple method for selecting the threshold
that is also allowed to vary with time. Extensive simulation studies and real
data analyses demonstrate the superior performance of our procedure with
competitors.
