Spherical latent space models for social networks: Geometry-aware inference and comparison across latent geometries

Abstract This article studies latent space models for social network data in which actors are embedded on a hypersphere and link probabilities depend on angular similarity. In contrast to Euclidean embeddings, the spherical formulation provides a compact parameter space, stabilizes the linear predictor through bounded inner products, and offers a natural representation of directional and cyclic structure. For inference, we combine maximum likelihood estimation, used to obtain initial values for latent positions and model parameters, with geometry-aware Bayesian methods based on Metropolis–Hastings and Hamiltonian Monte Carlo algorithms, including a geodesic Hamiltonian scheme for manifold-constrained parameters. We conduct a systematic empirical comparison between Euclidean and spherical latent space models on a benchmark social network dataset, evaluating model fit, predictive performance, and interpretability. The results show that spherical representations provide competitive performance while offering a more constrained and geometrically interpretable structure. Overall, the paper clarifies the role of latent space geometry in network modeling and highlights the importance of geometry-aware inference in statistical analysis of relational data.