Latent space models are powerful statistical tools for modeling and understanding network data. While the importance of accounting for uncertainty in network analysis has been well recognized, the current literature predominantly focuses on point estimation and prediction, leaving the statistical inference of latent space models an open question. This work aims to fill this gap by providing a general framework to analyze the theoretical properties of the maximum likelihood estimators. In particular, we establish the uniform consistency and asymptotic distribution results for the latent space models under different edge types and link functions. Furthermore, the proposed framework enables us to generalize our results to the dependent-edge and sparse scenarios. Our theories are supported by simulation studies and have the potential to be applied in downstream inferences, such as link prediction and network testing problems.

翻译：潜变量空间模型是建模和理解网络数据的强大统计工具。尽管网络分析中不确定性建模的重要性已得到广泛认可，但现有文献主要集中于点估计和预测，而潜变量空间的统计推断问题仍是一个开放性问题。本文旨在填补这一空白，通过提供一个通用框架来分析极大似然估计量的理论性质。具体而言，我们建立了不同边类型和链接函数下潜变量空间模型的一致性和渐近分布结果。此外，该框架使我们能够将结论推广至依赖边情形和稀疏网络场景。我们的理论通过仿真研究得到验证，并有望应用于下游推断，如链接预测和网络检验问题。