Computationally efficient inference for latent position network models
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Latent position models are nowadays 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 that infer latent position models generally require 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 may be used to fit latent position models on networks with several tens of thousands of nodes. Our approach relies on an approximation of the likelihood function, where the amount of noise introduced 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 illustrate that one can achieve a substantial reduction in computing time and still obtain a reasonably good estimation of the latent structure. Finally, we propose applications of our method to simulated networks and to a large coauthorships network, demonstrating the usefulness of our approach.
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