Network data is ubiquitous in various scientific disciplines, including sociology, economics, and neuroscience. Latent space models are often employed in network data analysis, but the geometric effect of latent space curvature remains a significant, unresolved issue. In this work, we propose a hyperbolic network latent space model with a learnable curvature parameter. We theoretically justify that learning the optimal curvature is essential to minimizing the embedding error across all hyperbolic embedding methods beyond network latent space models. A maximum-likelihood estimation strategy, employing manifold gradient optimization, is developed, and we establish the consistency and convergence rates for the maximum-likelihood estimators, both of which are technically challenging due to the non-linearity and non-convexity of the hyperbolic distance metric. We further demonstrate the geometric effect of latent space curvature and the superior performance of the proposed model through extensive simulation studies and an application using a Facebook friendship network.

翻译：网络数据在社会学、经济学和神经科学等多个科学领域中普遍存在。潜在空间模型常被用于网络数据分析，但潜在空间曲率的几何效应仍是一个重要且未解决的问题。在本工作中，我们提出了一种具有可学习曲率参数的双曲网络潜在空间模型。我们从理论上论证，学习最优曲率对于最小化所有双曲嵌入方法（超越网络潜在空间模型）的嵌入误差至关重要。我们开发了一种采用流形梯度优化的最大似然估计策略，并建立了最大似然估计量的一致性与收敛速率。由于双曲距离度量的非线性和非凸性，这两者在技术上均具有挑战性。我们通过大量的模拟研究以及在一个Facebook友谊网络上的应用，进一步展示了潜在空间曲率的几何效应以及所提模型的优越性能。