Date on Master's Thesis/Doctoral Dissertation

8-2025

Document Type

Doctoral Dissertation

Degree Name

Ph. D.

Department

Mathematics

Degree Program

Applied and Industrial Mathematics, PhD

Committee Chair

Han, Dan

Committee Member

Gill, Ryan

Committee Member

Tone, Cristina

Committee Member

Depue, Brendan

Author's Keywords

Exponential random graphs; Bayesian inferences; shrinkage priors; stochastic approximations; Bayesian regularization; social networks

Abstract

Networks are powerful tools for modeling the complexity of social interactions, biological systems, and information spread. A leading statistical frameworks for analyzing network data are Exponential Random Graph Models (ERGMs), which provide a principled approach to capturing structural dependencies. However, ERGMs remain challenging to estimate, especially in sparse or high-dimensional settings where models suffer from degeneracy and unstable parameter inference. This paper proposes a penalized Bayesian approach to ERGMs that utilizes the horseshoe prior, a sparsity-inducing global-local shrinkage prior. This prior offers robust regularization while preserving important signals, improving estimation by shrinking irrelevant parameters and reducing the impact of extreme configurations. In addition to traditional sampling methods, several stochastic gradient approximations for Bayesian ERGMs are introduced, such as RMSprop-enhanced SGLD and ADAM. The experimental results demonstrate the effectiveness of the proposed models in generating posterior distributions that align more closely with observed network features across a range of networks, outperforming existing Bayesian ERGM approaches.. While no method eliminates degeneracy, the combination of penalization and adaptive gradient methods provides a promising direction for scalable and robust Bayesian inference in network analysis.

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