  This article is motivated by studying the interaction of magnetic moments in
high entropy alloys (HEAs), which plays an important role in guiding HEA
designs in materials science. While first principles simulations can capture
magnetic moments of individual atoms, explicit models are required to analyze
their interactions. This is essentially an inverse covariance matrix estimation
problem. Unlike most of the literature on graphical models, the inverse
covariance matrix possesses inherent structural constraints encompassing node
types, topological distance of nodes, and partially specified conditional
dependence patterns. The present article is, to our knowledge, the first to
consider such intricate structures in graphical models. In particular, we
introduce graph constraints to formulate these structures that arise from
domain knowledge and are critical for interpretability, which leads to a
Bayesian conditional autoregressive model with graph constraints (CARGO) for
structured inverse covariance matrices. The CARGO method enjoys efficient
implementation with a modified Gauss-Seidel scheme through proximity operators
for closed-form posterior exploration. We establish algorithmic convergence for
the proposed algorithm under a verifiable stopping criterion. Simulations show
competitive performance of CARGO relative to several other methods and confirm
our convergence analysis. In a novel real data application to HEAs, the
proposed methods lead to data-driven quantification and interpretation of
magnetic moment interactions with high tractability and transferability.
