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

TL;DR

This paper introduces a new method for estimating the density of edges in Gaussian graphical models by combining p-value based testing with Storey's estimator, applicable in high-dimensional settings like genetics.

Contribution

It proposes a novel estimator for the edge proportion in GGMs that accounts for dependence among tests and provides theoretical bias analysis and empirical validation.

Findings

Estimator accurately recovers graph complexity in simulations.

Under dependence, the estimator exhibits slight upward bias.

Method is applicable to high-dimensional genetic data.

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--Spj{\o}tvoll 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.