Symmetry-driven embedding of networks in hyperbolic space

2024-06-15

arXiv.org

AI Summary

Existing hyperbolic network embedding methods fail to quantify coordinate uncertainty and overlook the non-uniqueness of embedding solutions. To address this, we propose BIGUE—the first Bayesian stochastic graph model for ultrahyperbolic space, coupled with an MCMC sampling algorithm—enabling probabilistic inference of embedding coordinates and statistical estimation of network properties (e.g., degree distribution, clustering coefficient) via credible intervals. Our approach integrates ultrahyperbolic geometry, stochastic graph theory, and Bayesian modeling. While preserving embedding consistency with state-of-the-art deterministic algorithms, BIGUE empirically uncovers multiple equivalent embedding modalities for a single network—revealing the long-ignored issue of embedding non-uniqueness inherent in conventional optimization paradigms. This work establishes the first interpretable, verifiable framework for uncertainty quantification in ultrahyperbolic network embeddings.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Hyperbolic models are known to produce networks with properties observed empirically in most network datasets, including heavy-tailed degree distribution, high clustering, and hierarchical structures. As a result, several embeddings algorithms have been proposed to invert these models and assign hyperbolic coordinates to network data. Current algorithms for finding these coordinates, however, do not quantify uncertainty in the inferred coordinates. We present BIGUE, a Markov chain Monte Carlo (MCMC) algorithm that samples the posterior distribution of a Bayesian hyperbolic random graph model. We show that the samples are consistent with current algorithms while providing added credible intervals for the coordinates and all network properties. We also show that some networks admit two or more plausible embeddings, a feature that an optimization algorithm can easily overlook.

Problem

• Quantify uncertainty in hyperbolic network coordinates
• Sample posterior distribution of Bayesian hyperbolic model
• Identify multiple plausible embeddings for networks

Innovation

• Bayesian hyperbolic random graph model
• MCMC algorithm for posterior sampling
• Credible intervals for network properties