  We investigate the effect of cadmium (a toxic environmental pollutant) on the
correlation structure of a number of urinary metabolites using Gaussian
graphical models (GGMs). The inferred metabolic associations can provide
important information on the physiological state of a metabolic system and
insights on complex metabolic relationships. Using the fitted GGMs, we
construct differential networks, which highlight significant changes in
metabolite interactions under different experimental conditions. The analysis
of such metabolic association networks can reveal differences in the underlying
biological reactions caused by cadmium exposure. We consider Bayesian inference
and propose using the multiplicative (or Chung-Lu random graph) model as a
prior on the graphical space. In the multiplicative model, each edge is chosen
independently with probability equal to the product of the connectivities of
the end nodes. This class of prior is parsimonious yet highly flexible; it can
be used to encourage sparsity or graphs with a pre-specified degree
distribution when such prior knowledge is available. We extend the
multiplicative model to multiple GGMs linking the probability of edge inclusion
through logistic regression and demonstrate how this leads to joint inference
for multiple GGMs. A sequential Monte Carlo (SMC) algorithm is developed for
estimating the posterior distribution of the graphs.
