Projective, Sparse, and Learnable Latent Position Network Models

When modeling network data using a latent position model, it is typical to assume that all nodes' latent positions are independently and identically distributed. However, this assumption implies the average node degree grows linearly with the number of nodes, which is inappropriate when the graph is thought to be sparse. We propose an alternative assumption--- that the latent positions are generated according to a Poisson point process--- and show that it is compatible with various levels of sparsity. Unlike other sparse latent position models, our approach also defines a projective family of probability distributions, ensuring statistical inference and prediction are well-defined for networks of different sizes. We establish conditions for consistently inferring the latent positions in a latent position network model, and compare our results to those of existing frameworks for modeling sparse graphs.