  Exponential random graph models are a class of widely used exponential family
models for social networks. The topological structure of an observed network is
modelled by the relative prevalence of a set of local sub-graph configurations
termed network statistics. One of the key tasks in the application of these
models is which network statistics to include in the model. This can be thought
of as statistical model selection problem. This is a very challenging
problem---the posterior distribution for each model is often termed "doubly
intractable" since computation of the likelihood is rarely available, but also,
the evidence of the posterior is, as usual, intractable. The contribution of
this paper is the development of a fully Bayesian model selection method based
on a reversible jump Markov chain Monte Carlo algorithm extension of Caimo and
Friel (2011) which estimates the posterior probability for each competing
model.
