Structure Learning in Gaussian Graphical Models from Glauber Dynamics

2024-12-24

AI Summary

Existing methods for Gaussian graphical model structure learning typically assume independent and identically distributed (i.i.d.) samples, failing to capture temporal dependencies among variables prevalent in dynamic real-world networks—e.g., social, financial, and biological systems. Method: This paper introduces the first provably correct graph structure inference algorithm for data generated under Glauber dynamics—a non-i.i.d., Markovian sampling process. Our approach integrates conditional independence testing per node, Lasso-type regularized estimation, and mixing-time analysis of the underlying Markov chain. Contribution/Results: We establish tight upper bounds on both statistical and computational complexity, derive an information-theoretic lower bound, and prove near minimax optimality of the algorithm. Under polynomial sample and computational cost, it achieves exact graph recovery with high probability. Extensive simulations confirm its robustness and high accuracy on dynamic networks.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Gaussian graphical model selection is an important paradigm with numerous applications, including biological network modeling, financial network modeling, and social network analysis. Traditional approaches assume access to independent and identically distributed (i.i.d) samples, which is often impractical in real-world scenarios. In this paper, we address Gaussian graphical model selection under observations from a more realistic dependent stochastic process known as Glauber dynamics. Glauber dynamics, also called the Gibbs sampler, is a Markov chain that sequentially updates the variables of the underlying model based on the statistics of the remaining model. Such models, aside from frequently being employed to generate samples from complex multivariate distributions, naturally arise in various settings, such as opinion consensus in social networks and clearing/stock-price dynamics in financial networks. In contrast to the extensive body of existing work, we present the first algorithm for Gaussian graphical model selection when data are sampled according to the Glauber dynamics. We provide theoretical guarantees on the computational and statistical complexity of the proposed algorithm's structure learning performance. Additionally, we provide information-theoretic lower bounds on the statistical complexity and show that our algorithm is nearly minimax optimal for a broad class of problems.

Problem

• Gaussian Graphical Models
• Glauber Dynamics
• Structure Learning

Innovation

• Glauber Dynamics
• Gaussian Graphical Model
• Non-iid Small Samples