  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.
