Estimation of the complexity of a network under a Gaussian graphical model

2026-02-25

AI Summary

This study addresses the problem of accurately estimating the proportion of non-zero partial correlation edges in high-dimensional Gaussian graphical models to quantify the complexity of conditional dependence structures. By reframing graph complexity estimation as a large-scale multiple hypothesis testing problem, the approach leverages edge-specific p-values combined with Storey’s method to estimate the proportion of true null hypotheses, enabling robust inference of the precision matrix’s sparsity structure under false discovery rate control. Theoretically, the work establishes convergence of the empirical p-value distribution under high-dimensional weak dependence settings—common in genetic association studies—and reveals the asymptotic upward bias of the Schweder–Spjøtvoll estimator, leading to a more accurate complexity estimator. Simulations demonstrate that the proposed method effectively and robustly recovers graph complexity across diverse high-dimensional scenarios.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

The proportion of edges in a Gaussian graphical model (GGM) characterizes the complexity of its conditional dependence structure. Since edge presence corresponds to a nonzero entry of the precision matrix, estimation of this proportion can be formulated as a large-scale multiple testing problem. We propose an estimator that combines p-values from simultaneous edge-wise tests, conducted under false discovery rate control, with Storey's estimator of the proportion of true null hypotheses. We establish weak dependence conditions on the precision matrix under which the empirical cumulative distribution function of the p-values converges to its population counterpart. These conditions cover high-dimensional regimes, including those arising in genetic association studies. Under such dependence, we characterize the asymptotic bias of the Schweder--Spjøtvoll estimator, showing that it is upward biased and thus slightly underestimates the true edge proportion. Simulation studies across a variety of models confirm accurate recovery of graph complexity.

Problem

• Gaussian graphical model
• graph complexity
• edge proportion
• precision matrix
• multiple testing

Innovation

• Gaussian graphical model
• graph complexity
• false discovery rate
• Storey's estimator
• high-dimensional inference