Combinatorial-Probabilistic Trade-Off: Community Properties Test in the Stochastic Block Models
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In this paper, we propose an inferential framework testing the general community combinatorial properties of the stochastic block model. Instead of estimating the community assignments, we aim to test the hypothesis on whether a certain community property is satisfied. For instance, we propose to test whether a given set of nodes belong to the same community or whether different network communities have the same size. We propose a general inference framework that can be applied to all symmetric community properties. To ease the challenges caused by the combinatorial nature of communities properties, we develop a novel shadowing bootstrap testing method. By utilizing the symmetry, our method can find a shadowing representative of the true assignment and the number of assignments to be tested in the alternative can be largely reduced. In theory, we introduce a combinatorial distance between two community classes and show a combinatorial-probabilistic trade-off phenomenon in the community properties test. Our test is honest as long as the product of combinatorial distance between two communities and the probabilistic distance between two assignment probabilities is sufficiently large. On the other hand, we shows that such trade-off also exists in the information-theoretic lower bound of the community property test. We also implement numerical experiments on both the synthetic data and the protein interaction application to show the validity of our method.
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