Adaptive Regularization via Extreme Value Distributions for Gaussian Graphical Models

Abstract

Edge selection in Gaussian graphical models is fundamentally a variable selection problem where pairwise relationships determine construct validity and variable importance in psychological networks. In psychology, network estimation relies predominantly on \(\ell_1\) regularization where uniform shrinkage systematically underestimates edge and centrality parameters. Alternative penalties overcome this bias but rely on fixed hyperparameters that do not adapt to the signal in the data. We develop a family of data-adaptive regularization penalties grounded in extreme value theory. Across 290 empirical psychological datasets, we show that absolute partial correlations are well-described by the Weibull distribution. Using this empirical regularity, we derive Weibull, Gumbel, and Exponential penalties that approximate \(\ell_0\) penalization and calibrate their hyperparameters to each dataset's noise floor. We formally prove the asymptotic properties of their static forms and conduct a large-scale simulation spanning two network topologies and various sample sizes (\(N\) = 100--10,000), demonstrating that their adaptive forms maintain high specificity while accumulating sensitivity as sample size increases with low parameter bias and high rank-order centrality congruence relative to field standards. Empirically, method choice alone determined centrality rankings at sample sizes typical in psychology. Of the three adaptive penalties, Weibull is recommended given the interpretability of its parameters.