  High-dimensional data analysis typically focuses on low-dimensional
structure, often to aid interpretation and computational efficiency. Graphical
models provide a powerful methodology for learning the conditional independence
structure in multivariate data by representing variables as nodes and
dependencies as edges. Inference is often focused on individual edges in the
latent graph. Nonetheless, there is increasing interest in determining more
complex structures, such as communities of nodes, for multiple reasons,
including more effective information retrieval and better interpretability. In
this work, we propose a multilayer graphical model where we first cluster nodes
and then, at the second layer, investigate the relationships among groups of
nodes. Specifically, nodes are partitioned into "supernodes" with a
data-coherent size-biased tessellation prior which combines ideas from Bayesian
nonparametrics and Voronoi tessellations. This construct allows accounting also
for dependence of nodes within supernodes. At the second layer, dependence
structure among supernodes is modelled through a Gaussian graphical model,
where the focus of inference is on "superedges". We provide theoretical
justification for our modelling choices. We design tailored Markov chain Monte
Carlo schemes, which also enable parallel computations. We demonstrate the
effectiveness of our approach for large-scale structure learning in simulations
and a transcriptomics application.
