  Network data is a major object data type that has been widely collected or
derived from common sources such as brain imaging. Such data contains numeric,
topological, and geometrical information, and may be necessarily considered in
certain non-Euclidean space for appropriate statistical analysis. The
development of statistical methodologies for network data is challenging and
currently at its infancy; for instance, the non-Euclidean counterpart of basic
two-sample tests for network data is scarce in literature. In this study, a
novel framework is presented for two independent sample comparison of networks.
Specifically, an approximation distance metric to quotient Euclidean distance
is proposed, and then combined with network spectral distance to quantify the
local and global dissimilarity of networks simultaneously. A permutational
non-Euclidean analysis of variance is adapted to the proposed distance metric
for the comparison of two independent groups of networks. Comprehensive
simulation studies and real applications are conducted to demonstrate the
superior performance of our method over other alternatives. The asymptotic
properties of the proposed test are investigated and its high-dimensional
extension is discussed as well.
