  In the last two decades many random graph models have been proposed to
extract knowledge from networks. Most of them look for communities or, more
generally, clusters of vertices with homogeneous connection profiles. While the
first models focused on networks with binary edges only, extensions now allow
to deal with valued networks. Recently, new models were also introduced in
order to characterize connection patterns in networks through mixed
memberships. This work was motivated by the need of analyzing a historical
network where a partition of the vertices is given and where edges are typed. A
known partition is seen as a decomposition of a network into subgraphs that we
propose to model using a stochastic model with unknown latent clusters. Each
subgraph has its own mixing vector and sees its vertices associated to the
clusters. The vertices then connect with a probability depending on the
subgraphs only, while the types of edges are assumed to be sampled from the
latent clusters. A variational Bayes expectation-maximization algorithm is
proposed for inference as well as a model selection criterion for the
estimation of the cluster number. Experiments are carried out on simulated data
to assess the approach. The proposed methodology is then applied to an
ecclesiastical network in Merovingian Gaul. An R code, called Rambo,
implementing the inference algorithm is available from the authors upon
request.
