Abstract: With the advancement of technology, network data containing relational information among observations are prevailing across fields like sociology, biology, computer science, and economics. These networks often exhibit complex dependencies among nodes and come in varied formats, including node covariates, necessitating novel statistical models and inference methods. Latent space models are useful tools for statistical modeling and inference of network data, which aim to learn a low-dimensional positions in some latent space for each node and utilize those learned latent positions to conduct statistical inference or downstream tasks. Nevertheless, the complexity of network data continues to pose new challenges in modeling, theory, and inference.

Specifically, this thesis studies three such important problems that are related to the network latent space models. In Chapter 2, we study the problem of jointly modeling network data with high-dimensional node covariates, a multimodal data integration setting that has become increasingly popular in contemporary network applications. For this, we propose a novel joint latent space model with shared and individual factors, and then develop an estimation and inference procedure for identifying the overlapped structure of latent factors. In Chapter 3, we focus on the uncertainty quantification of then maximum likelihood estimators of network latent space models, which is a critical problem for downstream inference tasks of network data. Employing a novel theoretical analysis framework with the Lagrange-adjust Hessian analysis, we establish the consistency properties and asymptotic distributions of maximum likelihood estimators. The flexibility of this framework further enables us to study the problem with edge sparsity and dependency settings. In Chapter 4, we explore the geometric effect of latent space curvature and theoretically justify the importance of estimating the curvature when embedding network data. We thus propose a hyperbolic network latent space model with an adjustable curvature parameter, demonstrating its superiority through simulation and applied statistical learning tasks. These contributions not only advance the theoretical understanding of network data analysis but also enhance practical modeling and inference capabilities.