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Abstract:
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Random graph models have been a heated topic in statistics and machine learning, as well as a broad range of application areas. In this work, we focus on a class of low-rank random graph models, namely, the random dot product graphs. We propose a Bayesian approach, called the posterior spectral embedding, for estimating the latent positions, and prove its optimality. Unlike the classical spectral-based methods, the posterior spectral embedding is a fully likelihood-based graph estimation method taking advantage of the likelihood information of the graph model. A minimax lower bound for estimating the latent positions is established, and we show that the posterior spectral embedding achieves this lower bound in the following two senses: the posterior contraction rate is minimax-optimal, and it produces a point estimator achieving the minimax lower bound asymptotically. The practical performance of the proposed methodology is illustrated through extensive synthetic examples and the analysis of a Wikipedia network data.
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