Inevitable self-similar topology of binary trees and their diverse hierarchical density |
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Authors: | K Paik P Kumar |
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Institution: | (1) Environmental Hydrology and Hydraulic Engineering, Department of Civil and Environmental Engineering, University of Illinois, Urbana, IL 61801, USA |
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Abstract: | Self-similar topology, which can be characterized
as power law size distribution, has been found in diverse tree
networks ranging from river networks to taxonomic trees. In this
study, we find that the statistical self-similar topology is an
inevitable consequence of any full binary tree organization. We show
this by coding a binary tree as a unique bifurcation string. This
coding scheme allows us to investigate trees over the realm from
deterministic to entirely random trees. To obtain partial random
trees, partial random perturbation is added to the deterministic
trees by an operator similar to that used in genetic algorithms. Our
analysis shows that the hierarchical density of binary trees is more
diverse than has been described in earlier studies. We find that the
connectivity structure of river networks is far from strict
self-similar trees. On the other hand, organization of some social
networks is close to deterministic supercritical trees. |
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Keywords: | 89 75 Da Systems obeying scaling laws 89 75 Hc Networks and genealogical trees 89 75 Fb Structures and organization in complex systems 05 45 Df Fractals |
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