Bayesian model selection for the latent position cluster model for social networks

The latent position cluster model is a popular model for the statistical analysis of network data. This approach 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 tend 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 very 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. A consequence of this is that it is possible to carry out posterior inference over the number of components in the latent mixture distribution without using trans-dimensional MCMC algorithms such as reversible jump MCMC. Moreover, our algorithm allows for more reasonable computation times for larger networks than the standard methods using the latentnet package (Krivitsky and Handcock 2008; Krivitsky and Handcock 2013).