Strain-Minimizing Hyperbolic Network Embeddings with Landmarks

TL;DR

L-hydra is a landmark-based hyperbolic embedding method that efficiently scales to large graphs, offering a trade-off between speed and accuracy, with an improved version L-hydra+ achieving better quality.

Contribution

The paper introduces L-hydra, a landmark heuristic for hyperbolic embeddings, and proposes L-hydra+ to enhance embedding quality while maintaining efficiency.

Findings

L-hydra is an order of magnitude faster than existing methods.

L-hydra scales linearly with the number of nodes.

L-hydra+ outperforms existing methods in both runtime and embedding quality.

Abstract

We introduce L-hydra (landmarked hyperbolic distance recovery and approximation), a method for embedding network- or distance-based data into hyperbolic space, which requires only the distance measurements to a few 'landmark nodes'. This landmark heuristic makes L-hydra applicable to large-scale graphs and improves upon previously introduced methods. As a mathematical justification, we show that a point configuration in d-dimensional hyperbolic space can be perfectly recovered (up to isometry) from distance measurements to just d+1 landmarks. We also show that L-hydra solves a two-stage strain-minimization problem, similar to our previous (unlandmarked) method 'hydra'. Testing on real network data, we show that L-hydra is an order of magnitude faster than existing hyperbolic embedding methods and scales linearly in the number of nodes. While the embedding error of L-hydra is higher than the error of existing methods, we introduce an extension, L-hydra+, which outperforms existing methods in both runtime and embedding quality.