Exploring and detecting community structures hold significant importance in
genetics, social sciences, neuroscience, and finance. Especially in graphical
models, community detection can encourage the exploration of sets of variables
with group-like properties. In this paper, within the framework of Gaussian
graphical models, we introduce a novel decomposition of the underlying
graphical structure into a sparse part and low-rank diagonal blocks
(non-overlapped communities). We illustrate the significance of this
decomposition through two modeling perspectives and propose a three-stage
estimation procedure with a fast and efficient algorithm for the identification
of the sparse structure and communities. Also on the theoretical front, we
establish conditions for local identifiability and extend the traditional
irrepresentability condition to an adaptive form by constructing an effective
norm, which ensures the consistency of model selection for the adaptive
$\ell_1$ penalized estimator in the second stage. Moreover, we also provide the
clustering error bound for the K-means procedure in the third stage. Extensive
numerical experiments are conducted to demonstrate the superiority of the
proposed method over existing approaches in estimating graph structures.
Furthermore, we apply our method to the stock return data, revealing its
capability to accurately identify non-overlapped community structures.

在高斯图模型的框架下，我们引入一种新颖的分解方法，将基础图结构分解为稀疏部分和低秩对角块（非重叠社区），并提出一个三阶段估计过程以及快速高效的算法用于识别稀疏结构和社区。通过两种建模方式展示了这种分解的重要性，并在理论层面上建立了局部可辨识性的条件，并通过构造一种有效的范数将传统的不可表示性条件拓展为自适应形式，确保自适应 l1 惩罚估计器在第二阶段的模型选择的一致性。此外，我们还为第三阶段的 K 均值聚类过程提供了聚类误差界限。通过大量的数值实验，证明了所提方法在估计图结构方面的优越性。此外，我们还将该方法应用于股票回报数据，展示了其准确识别非重叠社区结构的能力。