Network data arise in a wide range of disciplines, including social science, economics, biology, and computer science. Over the past several decades, many statistical models have been developed for network data. Among them, the block model family is one of the most widely used and includes four popular models: the stochastic block model (SBM), the degree-corrected block model (DCBM), the mixed-membership stochastic block model (MMSBM), and the degree-corrected mixed-membership model (DCMM). More recently, cycle count statistics have emerged as a powerful tool in statistical network analysis, particularly for network inference. In this talk, we will first review the block model family and then introduce cycle count statistics together with their fundamental asymptotic distributional results. To showcase the usefulness of cycle count statistics for network inference, we will discuss their application to network goodness-of-fit (GoF) and present GoF metrics for DCBM and DCMM with theoretical guarantees. Finally, we will demonstrate through several real data examples that DCMM provides an overall adequate fit for large real-world networks.