Event Type:

Department Colloquium

Date/Time:

Monday, April 20, 2009 - 09:00

Location:

Kidd 350

Guest Speaker:

Institution:

University of Nevada - Reno

Abstract:

Hierarchical branching organization is ubiquitous in Nature. It is readily seen in river basins, drainage networks, bronchial passages, botanical trees, and snowflakes, to mention a few. It can be used as a conceptual model for structural organization of a media or as a description of the dynamics of a process, e.g. growth of a biological population, fracture development is solids, or collision of gas molecules. The following questions related to an observed hierarchy are of great interest, theoretical as well as practical: (i) How was the hierarchy produced? (Genesis), (ii) What is the geometry/topology of the hierarchy? (Static structure), and (iii) What are the properties of a process evolving along the hierarchy? (Dynamics). This talk will overview existing and discuss new approaches focused on answering these questions. Empirical evidence reveals surprising similarity among various natural hierarchies; mathematically, they are described by self-similar trees (SST). Self-similarity and related properties of trees are studied using the Horton-Strahler and Tokunaga branching indices, defined by averaging particular branching statistics over a suitable space of random trees. This talk will address the following issues related to SSTs and their branching statistics. (i) Genesis: It has been shown that branching processes have limited power in reproducing the observed range of self-similar indices. We will suggest several alternative stochastic processes that naturally lead to creation of realistic SSTs with broad range of indices. (ii) Static structure: Some new results will be presented on relationship between Horton-Strahler and Tokunaga indices. (iii) Dynamics: We suggest a compact parameterization of a flow along a directed branching network and study its statistical properties. Specifically, we introduce a concept of dynamic tree that captures the essential properties of a flow along a given static tree. We show that methods developed for analysis of static branching structures can be used to study the corresponding dynamics. Most of the concrete applications will refer to environmental transport on river networks and gas dynamics. The talk will be accessible to a general audience interested in networks and network dynamics.