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Abstract:
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Network data are prevalent in various scientific and engineering fields, including sociology, economics, neuroscience, and so on. While latent space models are widely used in analyzing network data, the geometric effect of latent space remains an important but unsolved problem. In this work, we propose a hyperbolic network latent space model with a learnable curvature parameter, which allows the proposed model to fit network data with the most suitable latent space. We theoretically justify that learning the optimal curvature is essential to minimize the embedding error for all hyperbolic embedding methods beyond network latent space models. We also establish consistency rates for maximum-likelihood estimators and develop an estimation approach with manifold gradient optimization, both of which are technically challenging due to the non-linearity and non-convexity of hyperbolic distance metric. We further illustrate the superiority of the proposed model and the geometric effect of latent space with extensive simulation studies followed by a Facebook friendship network application.
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