  Labelled networks are an important class of data, naturally appearing in
numerous applications in science and engineering. A typical inference goal is
to determine how the vertex labels (or features) affect the network's
structure. In this work, we introduce a new generative model, the feature-first
block model (FFBM), that facilitates the use of rich queries on labelled
networks. We develop a Bayesian framework and devise a two-level Markov chain
Monte Carlo approach to efficiently sample from the relevant posterior
distribution of the FFBM parameters. This allows us to infer if and how the
observed vertex-features affect macro-structure. We apply the proposed methods
to a variety of network data to extract the most important features along which
the vertices are partitioned. The main advantages of the proposed approach are
that the whole feature-space is used automatically and that features can be
rank-ordered implicitly according to impact.
