Abstract: 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 a centered and scaled adjacency matrix to the third power, which we prove converges to the standard normal distribution as the number of nodes tends to infinity. We also provide the asymptotic power guarantee of the test. We explore the relationship between the number of networks and the number of nodes in each network when characterizing the theoretical properties of the proposed test statistic. Our test can be applied to both binary and weighted networks, operates under a very general framework in which the networks are allowed to be large and sparse, and can be extended to multiple-sample testing. We present a simulation study that demonstrates the superior performance of our tests over that of existing methods, and apply our tests to three real data sets.

Key words and phrases: Hypothesis testing, populations of networks, random matrix theory.