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

TL;DR

This paper introduces a Bayesian hyperbolic latent space model for networks that incorporates a learnable temperature parameter, improving the modeling of hierarchical and heavy-tailed networks.

Contribution

It formalizes a Bayesian hyperbolic model with an inferable temperature parameter and develops scalable inference algorithms, enhancing network embedding accuracy.

Findings

Model with learnable temperature outperforms fixed-temperature models.

Hyperbolic geometry better captures hierarchical network structures.

Inference methods effectively recover network features in simulations and real data.

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.