Computationally efficient inference for latent position network models

Latent position models are widely used for the analysis of networks in a variety of research fields. In fact, these models possess a number of desirable theoretical properties, and are particularly easy to interpret. However, statistical methodologies to fit these models generally incur a computational cost which grows with the square of the number of nodes in the graph. This makes the analysis of large social networks impractical. In this paper, we propose a new method characterised by a linear computational complexity, which can be used to fit latent position models on networks of several tens of thousands nodes. Our approach relies on an approximation of the likelihood function, where the amount of noise introduced by the approximation can be arbitrarily reduced at the expense of computational efficiency. We establish several theoretical results that show how the likelihood error propagates to the invariant distribution of the Markov chain Monte Carlo sampler. In particular, we demonstrate that one can achieve a substantial reduction in computing time and still obtain a good estimate of the latent structure. Finally, we propose applications of our method to simulated networks and to a large coauthorships network, highlighting the usefulness of our approach.