Graphical model inference with external network data

A frequent challenge when using graphical models in practice is that the sample size is limited relative to the number of parameters. They also become hard to interpret when the number of variables p gets large. We consider applications where one has external data, in the form of networks between variables, that can improve inference and help interpret the fitted model. An example of interest regards the interplay between social media and the co-evolution of the COVID-19 pandemic across USA counties. We develop a spike-and-slab prior framework that depicts how partial correlations depend on the networks, by regressing the edge probabilities, average partial correlations, and their variance on the networks. The goal is to detect when the network data relates to the graphical model and, if so, explain how. We develop computational schemes and software in R and probabilistic programming languages. Our applications show that incorporating network data can improve interpretation, statistical accuracy, and out-of-sample prediction.