Tokunaga and Horton self-similarity for level set trees of Markov chains |
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Authors: | Ilia Zaliapin Yevgeniy Kovchegov |
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Institution: | 1. Department of Mathematics and Statistics, University of Nevada, Reno, NV 89557-0084, USA;2. Department of Mathematics, Oregon State University, Corvallis, OR 97331-4605, USA;1. Saint Petersburg Mining University, 2, 21st Line, St Petersburg 199106, Russia;2. Moscow Region State Educational Institution for Higher Education University ‘Dubna’, Universitetskaya 19, Moscow 141982, Russia;1. Department of Physics, Puer University, Simao 665000, China;2. Department of Physics, Yunnan University, Kunming 650091, China |
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Abstract: | The Horton and Tokunaga branching laws provide a convenient framework for studying self-similarity in random trees. The Horton self-similarity is a weaker property that addresses the principal branching in a tree; it is a counterpart of the power-law size distribution for elements of a branching system. The stronger Tokunaga self-similarity addresses so-called side branching. The Horton and Tokunaga self-similarity have been empirically established in numerous observed and modeled systems, and proven for two paradigmatic models: the critical Galton–Watson branching process with finite progeny and the finite-tree representation of a regular Brownian excursion. This study establishes the Tokunaga and Horton self-similarity for a tree representation of a finite symmetric homogeneous Markov chain. We also extend the concept of Horton and Tokunaga self-similarity to infinite trees and establish self-similarity for an infinite-tree representation of a regular Brownian motion. We conjecture that fractional Brownian motions are also Tokunaga and Horton self-similar, with self-similarity parameters depending on the Hurst exponent. |
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