Event Detail

Event Type: 
Probability Seminar
Thursday, June 4, 2015 - 12:00 to 13:00
Gilk 115

Speaker Info

University of Nevada, Reno

Nature exhibits many branching tree-like structures beyond the botanical trees. River networks, Martian drainage basins, veins of botanical leaves, lung and blood systems, and lightning can all be represented as tree graphs. In addition, time-oriented trees describe a number of dynamic processes like spread of disease or transfer of gene characteristics. This would sound like a trivial observation if not for the following fact. Despite their apparent diversity, a majority of rigorously studied branching structures exhibit simple two-parametric Tokunaga self-similarity and Horton scaling. The Horton scaling 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 solid empirical evidence suggests an existence of a universal self-similarity mechanism and prompts the question: What probability models can generate Horton/Tokunaga self-similar trees with a range of parameters?
This talk reviews the existing results and recent findings on self-similarity for tree representation of branching, coalescent processes and time series. We show that the essential models, including white noises, random walks, critical Galton-Watson branching and Kingman’s coalescent produce trees with Tokunaga and/or Horton self-similarity. Our results explain, at least partially, the omnipresence of Tokunaga and Horton structures and suggest a framework for their statistical analysis. The results are illustrated using geophysical applications.

This is a joint work with Yevgeniy Kovchegov (Oregon State U).