Hyperbolic Latent Space Models for Network Embedding: Model Specification and Bayesian Inference

2026-05-11

AI Summary

This work addresses the challenge that existing network embedding methods struggle to effectively capture the hierarchical tree-like structures and heavy-tailed degree distributions commonly observed in real-world networks. The authors propose a hyperbolic continuous latent space Bayesian model that, for the first time, treats the temperature parameter as a learnable variable, revealing its critical role in shaping network tree-like topology. The model performs probabilistic embedding on negatively curved Riemannian manifolds and employs Hamiltonian Monte Carlo for accurate posterior inference, while also introducing an autoencoded variational Bayesian algorithm to scale to large networks. Experimental results demonstrate that the proposed method significantly outperforms baselines with fixed temperature and Euclidean geometry in graph reconstruction tasks, confirming both the efficacy of a learnable temperature and the enhanced representational capacity of the model.

Abstract

Many real-world networks exhibit hierarchical, tree-like structure and heavy-tailed degree distributions, phenomena not readily captured by standard statistical models for network data. Extensions of the popular continuous latent space modeling framework have been proposed to accommodate such networks. Drawing on insights from statistical physics, continuous latent space models with underlying hyperbolic geometry have been proposed as a natural framework, probabilistically embedding nodes in a latent Riemannian manifold with constant negative curvature. Most statistical implementations, however, simplify the original physics-based model by omitting the ``temperature parameter," which controls the sharpness of the latent distance-to-probability mapping. We argue this omission is critical. We demonstrate that temperature is the fundamental parameter governing a network's tree-like topology, and that failing to infer it weakens model expressiveness. We formalize a Bayesian hyperbolic continuous latent space model with an unknown, learnable temperature parameter. We then develop two inferential procedures: a Hamiltonian Monte Carlo approach for rigorous posterior characterization and a scalable auto-encoding variational Bayes algorithm for large-scale networks. Through simulation and real data examples, we show that our model outperforms models with fixed temperature and misspecified Euclidean geometries in graph reconstruction tasks in most settings, confirming temperature is a crucial and inferable feature of complex networks.

Problem

• hyperbolic geometry
• latent space model
• temperature parameter
• network embedding
• tree-like topology

Innovation

• hyperbolic geometry
• temperature parameter
• Bayesian inference
• network embedding
• latent space model