In this paper, we propose a new spectral-based approach to hypothesis testing for populations of networks. The primary goal is to develop a test to determine whether two given samples of networks come from the same random model or distribution. Our test statistic is based on the trace of the third order for a centered and scaled adjacency matrix, which we prove converges to the standard normal distribution as the number of nodes tends to infinity. The asymptotic power guarantee of the test is also provided. The proper interplay between the number of networks and the number of nodes for each network is explored in characterizing the theoretical properties of the proposed testing statistics. Our tests are applicable to both binary and weighted networks, operate under a very general framework where the networks are allowed to be large and sparse, and can be extended to multiple-sample testing. We provide an extensive simulation study to demonstrate the superior performance of our test over existing methods and apply our test to three real datasets.
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