Determining the number of communities K is crucial in many network models. In the first part of this talk, we present a universal approach for inference via residual sub-sampling to test and estimate K in various models. Our method constructs a test statistic by sampling residual matrix entries after extracting spiked components. The test statistic converges to the standard normal under the null hypothesis and diverges to infinity with probability one under the alternative hypothesis. In the second part, we propose an approach for testing and estimating K in the global network model when individuals have limited partial information available. Our procedure constructs a test statistic based on singular values and eigenvalues of partitioned matrices derived from a centered and rescaled partial adjacency matrix. We establish the asymptotic null distribution for testing and demonstrate consistency in estimating K using random matrix theory results. Our proposed methods are demonstrated to be effective and useful through simulation and real data examples. |
