  The latent position cluster model is a popular model for the statistical
analysis of network data. This model assumes that there is an underlying latent
space in which the actors follow a finite mixture distribution. Moreover,
actors which are close in this latent space are more likely to be tied by an
edge. This is an appealing approach since it allows the model to cluster actors
which consequently provides the practitioner with useful qualitative
information. However, exploring the uncertainty in the number of underlying
latent components in the mixture distribution is a complex task. The current
state-of-the-art is to use an approximate form of BIC for this purpose, where
an approximation of the log-likelihood is used instead of the true
log-likelihood which is unavailable. The main contribution of this paper is to
show that through the use of conjugate prior distributions it is possible to
analytically integrate out almost all of the model parameters, leaving a
posterior distribution which depends on the allocation vector of the mixture
model. This enables posterior inference over the number of components in the
latent mixture distribution without using trans- dimensional MCMC algorithms
such as reversible jump MCMC. Our approach is compared with the
state-of-the-art latentnet (Krivitsky & Handcock 2015) and VBLPCM
(Salter-Townshend & Murphy 2013) packages.
