Combinatorial-Probabilistic Trade-Off: P-Values of Community Property Test in the Stochastic Block Models

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 and provide p-values to assess the statistical uncertainty. 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 present a general inferential framework that can be applied to all symmetric community properties. To ease the challenges caused by the combinatorial nature of community 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 property test. Our test is honest as long as the product of the combinatorial distance between two community classes and the probabilistic distance between two assignment probabilities is sufficiently large. On the other hand, we show that such a 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-Protein Interaction networks to show the validity of our method.