Latent space models assume that network ties are more likely between nodes that are closer together in an underlying latent space. Euclidean space is a popular choice for the underlying geometry, but hyperbolic geometry can mimic more realistic patterns of ties in complex networks. To identify the underlying geometry, past research has applied non-Euclidean extensions of multidimensional scaling (MDS) to the observed geodesic distances: the shortest path lengths between nodes. The difference in stress, a standard goodness-of-fit metric for MDS, across the geometries is then used to select a latent geometry with superior model fit (lower stress). The effectiveness of this method is assessed through simulations of latent space networks in Euclidean and hyperbolic geometries. To better account for uncertainty, we extend permutation-based hypothesis tests for MDS to the latent network setting. However, these tests do not incorporate any network structure. We propose a parametric bootstrap distribution of networks, conditioned on observed geodesic distances and the Gaussian Latent Position Model (GLPM). Our method extends the Davidson-MacKinnon J-test to latent space network models with differing latent geometries. We pay particular attention to large and sparse networks, and both the permutation test and the bootstrapping methods show an improvement in detecting the underlying geometry.

翻译：潜在空间模型假设，在底层潜在空间中距离更近的节点之间更可能产生网络连接。欧几里得空间是底层几何结构的常用选择，但双曲几何能够更好地模拟复杂网络中更真实的连接模式。为识别底层几何结构，以往研究将多维缩放（MDS）的非欧几里得扩展应用于观测到的测地距离（即节点间最短路径长度）。随后通过比较不同几何结构下的应力值（MDS的标准拟合优度指标），选择具有更优模型拟合度（更低应力）的潜在几何结构。通过在欧几里得与双曲几何中模拟潜在空间网络，评估了该方法的有效性。为更好地考量不确定性，我们将基于置换的MDS假设检验扩展至潜在网络场景。然而，这些检验未纳入任何网络结构信息。我们提出一种基于参数化自助法的网络分布，其条件为观测到的测地距离与高斯潜在位置模型（GLPM）。本方法将Davidson-MacKinnon J检验扩展至具有不同潜在几何结构的潜在空间网络模型。我们特别关注大规模稀疏网络，置换检验与自助法均显示在检测底层几何结构方面有所改进。