Detecting communities in networks and graphs is an important task across many disciplines, such asstatistics, social science and engineering. There are generally three kinds of mixing patterns for the case of two communities: assortative mixing, disassortative mixing, and core-periphery structure. Modularity optimization is a classical method for fitting network models with communities, but it is limited to handling only assortative and disassortative mixing when the mixing pattern is known and cannot identify the coreperiphery structure. In this dissertation, we first propose a strategic extension of modularity and introduce a new framework called Unified Bigroups Standardized Edge-count Analysis (UBSea) in Chapter 2. This framework can address all the previously mentioned community mixing structures for K = 2 communities. In addition, this new framework is able to automatically choose the mixing type to fit the networks. Simulation studies show that UBSea performs exceptionally well in various settings under the stochastic block model and the degree-corrected stochastic block model. We demonstrate that the new approach produces consistent estimates of the communities under a suitable signal-to-noise-ratio condition, for a block model with two communities. The new method is illustrated through applications to several real-world datasets. In Chapter 3, we extend the UBSea community detection framework from two communities (K = 2) to multiple communities (K ≥ 2) and introduce the Unified Multigroups Standardized Edge-count Analysis (UMSea) framework, developed using a recursive algorithm with the γ − τ criterion established in Chapter 2. We compare its empirical performance with state-of-the-arts methods under the Binary Tree Stochastic Block Model (BTSBM). Simulation results demonstrate the superb performance of the new community detection framework under various settings.