Variational Regularized Bilevel Estimation for Exponential Random Graph Models

TL;DR

This paper introduces a variational regularized estimation algorithm for ERGMs that improves parameter estimation accuracy, especially for triangles, addressing intractability and degeneracy issues in network modeling.

Contribution

It presents a novel variational mean-field approach with $\, ext{l}_2$ regularization for ERGMs, providing convergence analysis and enhanced triangle parameter estimation.

Findings

Achieves perfect sign recovery for triangle parameters in simulations.

Addresses intractability and degeneracy in ERGM estimation.

Provides practical sensitivity analysis for hyperparameters.

Abstract

I propose an estimation algorithm for Exponential Random Graph Models (ERGM), a popular statistical network model for estimating the structural parameters of strategic network formation in economics and finance. Existing methods often produce unreliable estimates of parameters for the triangle, a key network structure that captures the tendency of two individuals with friends in common to connect. Such unreliable estimates may lead to untrustworthy policy recommendations for networks with triangles. Through a variational mean-field approach, my algorithm addresses the two well-known difficulties when estimating the ERGM, the intractability of its normalizing constant and model degeneracy. In addition, I introduce $\ell_2$ regularization that ensures a unique solution to the mean-field approximation problem under suitable conditions. I provide a non-asymptotic optimization convergence rate analysis for my proposed algorithm under mild regularity conditions. Through Monte Carlo simulations, I demonstrate that my method achieves a perfect sign recovery rate for triangle parameters for small and mid-sized networks under perturbed initialization, compared to a 50% rate for existing algorithms. I provide the sensitivity analysis of estimates of ERGM parameters to hyperparameter choices, offering practical insights for implementation.