How can one determine whether the realized differences between two stochastic networks are statistically significant? This paper considers a two-sample goodness-of-fit testing problem for network data in which the null hypothesis is that the networks are drawn from the same inhomogeneous random graph model. It first outlines a randomization test for the null hypothesis that controls size in finite samples based on any network statistic. It then focuses on two particular network statistics that produce tests powerful against a large class of alternative hypotheses. The statistics are based on the magnitude of the difference between the networks' adjacency matrices as measured by the 2\to2 and \infty\to1 operator norms. The power properties of the tests are examined analytically, in simulation, and through two real-world applications. A key finding is that while the test based on the \infty\to1 norm requires relatively new tools from the semidefinite programming literature to implement, it can be substantially more powerful than that based on the 2\to2 norm for the kinds of sparse and degree-heterogeneous networks common in economics.
 This paper considers a two-sample goodness-of-fit testing problem for network data in which the null hypothesis is that the networks are drawn from the same inhomogeneous random graph model. It first outlines a randomization test for the null hypothesis that controls size in finite samples based on any test statistic. It then focuses on two particular statistics that produce tests powerful against a large class of alternative hypotheses. The statistics are based on the magnitude of the difference between the networks' adjacency matrices as measured by the 2\to2 and \infty\to1 operator norms. The power properties of the tests are examined analytically, in simulation, and through two real-world applications. A key finding is that the test based on the \infty\to1 norm can be substantially more powerful than that based on the 2\to2 norm for the kinds of sparse and degree-heterogeneous networks common in economics.
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How can one determine whether the realized differences between two stochastic networks are statistically significant?
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requires relatively new tools from the semidefinite programming literature to implement, it