Abstract: Comparing two population means of network data is of paramount importance in a wide range of scientific applications. Numerous existing network inference solutions focus on global tests of entire networks, without comparing individual net- work links. 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, these tests often suffer from limited power under a small sample size. In this study, we examine the problem of network comparisons, with both global and simultaneous inferences, when the data are in the form of a collection of symmetric matrices, each of which encodes the network structure of an individual subject. Such data are common in applications such as brain connectivity analyses and clinical genomics. Rather than requiring that the underlying data follow a normal distribution, we impose some moment conditions that are easily satisfied for numerous types of network data. Furthermore, we propose a power enhancement procedure that controls the false discovery, while substantially enhancing the power of the test. We investigate the efficacy of our testing procedure using an asymptotic analysis and a simulation study under a finite sample size. We further illustrate our method using a brain connectivity analysis.

Key words and phrases: Auxiliary information, false discovery rate, multiple testing, network data, power enhancement.