  A Bayesian approach to conduct network model selection is presented for a
general class of network models referred to as the congruence class models
(CCMs). CCMs form a broad class that includes as special cases several common
network models, such as the Erd\H{o}s-R\'{e}nyi-Gilbert model, stochastic block
model and many exponential random graph models. Due to the range of models able
to be specified as a CCM, investigators are better able to select a model
consistent with generative mechanisms associated with the observed network
compared to current approaches. In addition, the approach allows for
incorporation of prior information. We utilize the proposed Bayesian network
model selection approach for CCMs to investigate several mechanisms that may be
responsible for the structure of patient-sharing networks, which are associated
with the cost and quality of medical care. We found evidence in support of
heterogeneity in sociality but not selective mixing by provider type nor
degree.
