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Tokunaga and Horton self-similarity for level set trees of Markov chains

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  • Zaliapin, Ilia
  • Kovchegov, Yevgeniy

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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  • Zaliapin, Ilia & Kovchegov, Yevgeniy, 2012. "Tokunaga and Horton self-similarity for level set trees of Markov chains," Chaos, Solitons & Fractals, Elsevier, vol. 45(3), pages 358-372.
    • Handle: RePEc:eee:chsofr:v:45:y:2012:i:3:p:358-372
    DOI: 10.1016/j.chaos.2011.11.006
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    References listed on IDEAS

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    1. Turcotte, D.L. & Malamud, B.D. & Morein, G. & Newman, W.I., 1999. "An inverse-cascade model for self-organized critical behavior," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 268(3), pages 629-643.
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    1. Kovchegov, Yevgeniy & Zaliapin, Ilya, 2019. "Random self-similar trees and a hierarchical branching process," Stochastic Processes and their Applications, Elsevier, vol. 129(7), pages 2528-2560.

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