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：潜变量空间模型是网络数据建模与分析的有力统计工具。尽管网络分析中不确定性考量已获充分重视，但现有文献主要聚焦于点估计与预测，而潜变量空间模型的统计推断问题仍是悬而未决的课题。本文旨在填补这一研究空白，通过构建一般性理论框架，系统分析最大似然估计量的理论性质。具体而言，我们针对不同连边类型与链接函数下的潜变量空间模型，建立了估计量的一致相合性与渐近分布特征。此外，该理论框架可拓展至相关连边场景与稀疏网络情形。模拟研究验证了理论结果的有效性，且该框架可应用于下游推断任务，包括链接预测与网络检验等。