Self-similarity of Random Aggregation Trees in Hyperbolic Spaces

Abstract

Structure and function of complex networks is an intriguing area of research with numerous practical applications. It has been shown recently that several paradigmatic properties of complex networks, including power-law degree distribution and strong clustering, emerge naturally once a network is embedded into a hyperbolic space of a negative curvature. This thesis develops this general idea in application to the self-similar structure of rooted trees. We use a self-similarity framework based on the Horton-Strahler orders of tree branches and Tokunaga indices that describe aggregation of different orders. The main object of study is the nearest-neighbor aggregation in a hyperbolic metric. Extensive numerical experiments are used to formulate several hypotheses about Horton and Tokunaga self-similarity of the respective aggregation trees. The main results refer to an analytical Ring Model that describes order-based aggregation of particles in a 2-D hyperbolic space of a negative curvature -ζ^2 , ζ>0. We prove that the aggregation trees are Horton self-similar with the Horton exponent R related to the space curvature. Furthermore, we establish self-similar branching structure of the order aggregation that satisfies the Tokunaga constraint with parameters a = R – R^1/2 and c = R^1/2.