  We consider the problem of jointly estimating a collection of graphical
models for discrete data, corresponding to several categories that share some
common structure. An example for such a setting is voting records of
legislators on different issues, such as defense, energy, and healthcare. We
develop a Markov graphical model to characterize the heterogeneous dependence
structures arising from such data. The model is fitted via a joint estimation
method that preserves the underlying common graph structure, but also allows
for differences between the networks. The method employs a group penalty that
targets the common zero interaction effects across all the networks. We apply
the method to describe the internal networks of the U.S. Senate on several
important issues. Our analysis reveals individual structure for each issue,
distinct from the underlying well-known bipartisan structure common to all
categories which we are able to extract separately. We also establish
consistency of the proposed method both for parameter estimation and model
selection, and evaluate its numerical performance on a number of simulated
examples.
