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.