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Detecting communities in networks and graphs is an important task across many disciplines such as statistics, social science and engineering. There are generally three different kinds of mixing patterns for the case of two communities: assortative mixing, disassortative mixing and core-periphery structure. Modularity optimization is a classical way for fitting network models with communities. However, it can only deal with assortative mixing and disassortative mixing when the mixing pattern is known and fails to discover the core-periphery structure. In this paper, we extend modularity in a strategic way and propose a new framework based on Unified Bigroups Standadized Edge-count Analysis (UBSea). It can address all the formerly mentioned community mixing structures. In addition, this new framework is able to automatically choose the mixing type to fit the networks. Simulation studies show that the new framework has superb performance in a wide range of settings under the stochastic block model and the degree-corrected stochastic block model. We show that the new approach produces consistent estimate of the communities under a suitable signal-to-noise-ratio condition, for the case of a block model with two communities, for both undirected and directed networks. The new method is illustrated through applications to several real-world datasets.

Detecting clusters is a critical task in various fields, including statistics, engineering and bioinformatics. Our focus is primarily on the modern high-dimensional scenario, where traditional methods often fail due to the curse of dimensionality. In this study, we introduce a non-parametric framework for clustering that is applicable to any number of dimensions. Simulation results demonstrate that this new framework surpasses existing methods across a wide range of settings. We illustrate the proposed method with real data applications in distinguishing cancer tissues from normal tissues through gene expression data.