Strongly Hyperbolic Unit Disk Graphs

TL;DR

This paper introduces strongly hyperbolic unit disk graphs, a new class with hierarchical structures, and demonstrates their usefulness for efficient greedy routing algorithms, contrasting with Euclidean unit disk graphs.

Contribution

The paper defines strongly hyperbolic unit disk graphs, explores their properties, and shows they enable more efficient routing algorithms than traditional Euclidean models.

Findings

Hierarchical structures in strongly hyperbolic graphs

Efficient greedy routing schemes for these graphs

Better routing performance compared to general networks

Abstract

The class of Euclidean unit disk graphs is one of the most fundamental and well-studied graph classes with underlying geometry. In this paper, we identify this class as a special case in the broader class of hyperbolic unit disk graphs and introduce strongly hyperbolic unit disk graphs as a natural counterpart to the Euclidean variant. In contrast to the grid-like structures exhibited by Euclidean unit disk graphs, strongly hyperbolic networks feature hierarchical structures, which are also observed in complex real-world networks. We investigate basic properties of strongly hyperbolic unit disk graphs, including adjacencies and the formation of cliques, and utilize the derived insights to demonstrate that the class is useful for the development and analysis of graph algorithms. Specifically, we develop a simple greedy routing scheme and analyze its performance on strongly hyperbolic unit disk graphs in order to prove that routing can be performed more efficiently on such networks than in general.