Network Global Testing and Estimating Number of Network Communities

Given a symmetrical social network, the network global testing is where we use the adjacency matrix of the network to test whether it has only one community or multiple

communities. It's also naturally connected to the problem of estimating the number of network communities, which is arguably one of the most important problem in network

analysis area. Despite many interesting works in recent years, it remains unclear how to find test statistics and estimators that are (a) applicable to networks with severe degree heterogeneity and mixed-memberships with varying sparsity, and is (b) optimal. This thesis aims to design statistics to solve the above two problems, under a more realistic network model. To assess optimality, we use the phase transition framework, which includes the standard minimax argument, but is more informative.