Symmetry-driven embedding of networks in hyperbolic space

TL;DR

This paper introduces BIGUE, an MCMC algorithm for hyperbolic network embedding that quantifies uncertainty and reveals multiple plausible embeddings, enhancing understanding of network structures.

Contribution

The paper presents BIGUE, a Bayesian MCMC method that provides credible intervals for hyperbolic embeddings and uncovers multiple plausible network representations.

Findings

BIGUE produces consistent samples with existing algorithms.

It offers credible intervals for network properties.

Some networks have multiple plausible embeddings.

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.