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Self-similarity of Random Aggregation Trees in Hyperbolic Spaces
Mathematics and Statistics
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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.