Hyperbolic Sphericity

2026-10-01

AI Summary

This study addresses the significant influence of radius on graph sphericity in hyperbolic space, where its divergence from Euclidean counterparts and dimensional properties remain unclear, thereby hindering theoretical advances in low-dimensional graph embeddings. By integrating geometric graph theory with hyperbolic geometric analysis, this work systematically defines hyperbolic sphericity across arbitrary dimensions for the first time, deriving dimensional upper bounds and analytically characterizing radius effects through constructive proofs. It rigorously establishes that under variable radii, hyperbolic sphericity never exceeds its Euclidean value, whereas at fixed radii it surpasses the Euclidean bound by at most one. Furthermore, the oscillatory behavior of sphericity with respect to radius is revealed and instantiated. By clarifying the fundamental distinctions between hyperbolic and Euclidean sphericity, this research provides critical theoretical foundations for low-dimensional graph representations in machine learning.

Abstract

The sphericity of a graph is the minimum dimension d such that the graph has an intersection representation of d-dimensional balls of equal radius. While sphericity has been studied in Euclidean space, we initiate the study of hyperbolic sphericity. The hyperbolic sphericity of a graph can be significantly smaller than its Euclidean counterpart, but, contrary to the Euclidean setting, depends strongly on the radius of the balls. We show that, if the radius of the balls can be chosen depending on the graph, the hyperbolic sphericity is upper bounded by the Euclidean sphericity. This extends a previous result for 2-dimensional hyperbolic space, i.e., uniform disk graphs, to arbitrary dimensions. Moreover, our proof is significantly simpler. If we fix the radius, i.e., do not make it dependent on the graph, we show that hyperbolic sphericity can be larger than Euclidean sphericity, but by at most 1. Additionally, we study how hyperbolic sphericity changes with the ball radius. We show that choosing a larger radius can substantially decrease the sphericity while increasing it by at most 1. We also provide a construction of a graph where the sphericity oscillates between different values as the radius increases. Besides being theoretically interesting, we note that these results are relevant for graph embeddings in machine learning, where one is interested in low-dimensional numeric representations of symbolic data like graphs.

Problem

• Hyperbolic sphericity
• Graph intersection representation
• Euclidean sphericity
• Graph embeddings
• Ball radius

Innovation

• Hyperbolic sphericity
• Graph embeddings
• Intersection representation
• Euclidean sphericity
• Machine learning