Random self-similar trees: models, statistical inference, and applications

Ilya Zaliapin, University of Nevada, Reno - Department of Mathematics and Statistics
Hierarchical branching organization is ubiquitous in nature. It is readily seen in river basins, drainage networks, bronchial passages, botanical trees, lightening, and snowflakes, to mention but a few. Notably, empirical evidence reveals a surprising similarity among natural hierarchies – many of them are closely approximated by so-called self-similar trees (SSTs). This talk will focus on the Horton and Tokunaga self-similarity that provide easily parameterized constraints on random tree graphs. 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 system’s elements. The stronger Tokunaga self-similarity addresses so-called side branching; it ensures that different levels of a hierarchy have the same probabilistic structure (in a sense that can be rigorously defined). The Horton and Tokunaga self-similarity have been empirically established in numerous observed and modeled systems. This hints at the existence of a universal underlying self-similarity mechanism and prompts the question: What basic probability models can generate Horton/Tokunaga self-similar trees with a range of parameters? We review the existing results and present recent findings on self-similarity for tree representation of branching and coalescent processes, random walks, and white noises. In particular, we establish the equivalence of tree representation for selected coalescent processes and time series models. We also describe a statistical framework for testing self-similarity in a finite tree and estimating the related parameters. Our results suggest at least a partial explanation for the omnipresence of Tokunaga self-similar structures in natural branching systems. The results are illustrated using applications in hydrology, seismology, and billiard dynamics. This is a joint work with Yevgeniy Kovchegov (Oregon State U) and Alejandro Tejedor (U of Minnesota).
October, 8 2013 | 12:30 - 2:00 | 230E Gross Hall

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