  Graphical models are commonly used to discover associations within gene or
protein networks for complex diseases such as cancer. Most existing methods
estimate a single graph for a population, while in many cases, researchers are
interested in characterizing the heterogeneity of individual networks across
subjects with respect to subject-level covariates. Examples include assessments
of how the network varies with patient-specific prognostic scores or
comparisons of tumor and normal graphs while accounting for tumor purity as a
continuous predictor. In this paper, we propose a novel edge regression model
for undirected graphs, which estimates conditional dependencies as a function
of subject-level covariates. Bayesian shrinkage algorithms are used to induce
sparsity in the underlying graphical models. We assess our model performance
through simulation studies focused on comparing tumor and normal graphs while
adjusting for tumor purity and a case study assessing how blood protein
networks in hepatocellular carcinoma patients vary with severity of disease,
measured by HepatoScore, a novel biomarker signature measuring disease
severity.
