A Bootstrap-based Method for Testing Network Similarity

This paper studies the matched network inference problem, where the goal is to determine if two networks, defined on a common set of nodes, exhibit a specific form of stochastic similarity. Two notions of similarity are considered: (i) equality, i.e., testing whether the networks arise from the same random graph model, and (ii) scaling, i.e., testing whether their probability matrices are proportional for some unknown scaling constant. We develop a testing framework based on a parametric bootstrap approach and a Frobenius norm-based test statistic. The proposed approach is highly versatile as it covers both the equality and scaling problems, and ensures adaptability under various model settings, including stochastic blockmodels, Chung-Lu models, and random dot product graph models. We establish theoretical consistency of the proposed tests and demonstrate their empirical performance through extensive simulations under a wide range of model classes. Our results establish the flexibility and computational efficiency of the proposed method compared to existing approaches. We also report a real-world application involving the Aarhus network dataset, which reveals meaningful sociological patterns across different communication layers.