Latent space models are widely used in statistical network analysis and are often fit by Markov chain Monte Carlo. However, posterior summaries of latent coordinates are not canonical because the likelihood depends only on pairwise distances and is invariant under rigid motions of the latent space. Standard post hoc alignment can aid visualization, but the resulting summaries depend on an arbitrary reference configuration. We propose a quotient-based posterior analysis for Euclidean latent space models using the centered Gram map, which represents identifiable latent structure while removing nonidentifiability. This yields intrinsic posterior summaries of mean structure and uncertainty that can be computed directly from posterior samples, together with basic theoretical guarantees including canonicality, existence, and stability. Through simulations and analyses of the Florentine marriage network and a statisticians' coauthorship network, the proposed framework clarifies when alignment-based summaries are stable, when they become reference-sensitive, and which nodes or relationships are weakly identified. These results show how coherent posterior analysis can reveal latent relational structure beyond a single embedding.

翻译：潜空间模型在统计网络分析中被广泛应用，通常通过马尔可夫链蒙特卡洛方法进行拟合。然而，潜坐标的后验概括不具备典范性，因为似然函数仅依赖于点对之间的距离，且在潜空间的刚体运动下保持不变。标准的后验对齐方法有助于可视化，但所得概括依赖于任意参考构型。我们提出一种基于商空间的欧几里得潜空间模型后验分析方法，该方法利用中心化格拉姆映射，在消除不可识别性的同时表征可识别的潜结构。这产生了可直接从后验样本计算的内在后验均值结构与不确定性概括，并具备基本理论保证，包括典范性、存在性与稳定性。通过模拟实验以及对佛罗伦萨婚姻网络和统计学家合著网络的分析，本研究框架揭示了何时基于对齐的概括是稳定的、何时对参考构型敏感，以及哪些节点或关系识别较弱。研究结果表明，连贯的后验分析可以揭示超越单一嵌入的潜在关系结构。