Bayesian Nonparametric Models for Network Data

Network data arise in various fields, depicting relationships between nodes in a network through edges between them. The latent position model (LPM) is a well established model for the analysis of network data. The LPM posits that the edge formation process is influenced by relations between nodes in a p-dimensional latent space. While the number of dimensions p is an important parameter of the LPM, its choice is a difficult problem. The first contribution develops the latent shrinkage position model (LSPM) for the analysis of network data with binary or count edges. The LSPM employs a Bayesian nonparametric shrinkage prior with the LPM, facilitating automatic inference on the number of latent dimensions with inference performed via an adaptive MCMC algorithm. The second contribution, the latent shrinkage position cluster model (LSPCM) builds on the LSPM by considering the problem of clustering nodes in a network. The LSPCM employs a sparse finite mixture of LPSMs, facilitating simultaneous inference on both the number of clusters and the number of latent dimensions. The third contribution provides a variational inference approach to inferring the LSPM, which reduces computational demand, offering a more practical modelling tool for networks with large numbers of nodes. The performance of the proposed models and inferential approaches are illustrated throughout using simulation studies and real-world network applications, and open source code is available to facilitate implementation.