  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.
