Hyperbolic Trees in Complex Networks

Abstract

The two-dimensional hyperbolic space turned out to be an efficient geometry for generative models of complex networks. The networks generated with this hyperbolic metric space share their basic structural properties (like small diameter or scale-free degree distribution) with several real networks. In this paper, we present a new model for generating trees in the two-dimensional hyperbolic plane. The generative model is not based on known hyperbolic network models: the trees are not inferred from the existing links of any network; instead, the hyperbolic tree is generated from scratch purely based on the hyperbolic coordinates of nodes. We show that these hyperbolic trees have scale-free degree distributions and are present to a large extent both in synthetic hyperbolic complex networks and real ones (Internet autonomous system topology, US flight network) embedded in the hyperbolic plane.