Simultaneous Identification of Sparse Structures and Communities in Heterogeneous Graphical Models

TL;DR

This paper introduces a novel method for simultaneously identifying sparse structures and non-overlapping communities in Gaussian graphical models, with theoretical guarantees and practical applications demonstrated through experiments and stock data analysis.

Contribution

It proposes a three-stage estimation procedure with theoretical conditions for model selection consistency and clustering accuracy, advancing community detection in heterogeneous graphical models.

Findings

The method outperforms existing approaches in structure estimation.

Theoretical conditions ensure model selection consistency.

Successfully applied to stock return data for community detection.

Abstract

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.