Hyperbolic space provides a natural latent geometry for complex networks, as random geometric graphs constructed in this space reproduce many empirical network properties. The inverse problem—recovering the hyperbolic coordinates that best represent a given graph—enables efficient network routing and has been used for link prediction. Although this is a challenging optimization task due to the abundance of local maxima, many performant algorithms have been developed. However, these methods ignore uncertainty in the embeddings, producing a single solution without quantifying error or acknowledging alternative configurations.

The first part of this talk introduces the principles of hyperbolic geometry and Bayesian inference, two disciplines rarely combined. The second part of the talk discusses BIGUE, a Markov chain Monte Carlo algorithm that samples from the posterior of a Bayesian model for hyperbolic random graphs. BIGUE leverages the space symmetries to achieve better mixing than both random walk and a Hamiltonian Monte Carlo methods. The resulting credible intervals align with existing embedding methods, while also revealing the potential for multimodal posteriors, which we demonstrate using a synthetic graph model.