Hyperbolic Sphericity

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.