We consider the problem of estimating the number of communities in a weighted balanced Stochastic Block Model. We construct hypothesis tests based on semidefinite programming and with a statistic coming from a GOE matrix to distinguish between any two candidate numbers of communities. This is possible due to a universality result for a semidefinite programming-based function that we also prove. The tests are then used to form a sequential test to estimate the number of communities. Furthermore, we also construct estimators of the communities themselves.

翻译：本文研究加权平衡随机块模型中社区数量的估计问题。我们基于半定规划构建假设检验，并采用源自高斯正交系综矩阵的统计量来区分任意两个候选社区数量。这一方法的可行性源于我们证明的基于半定规划函数的普适性结果。随后利用这些检验构建序列检验来估计社区数量。此外，我们还构建了社区本身的估计量。