Speaker: Leonid Bunimovich, Regents' Professor of Mathematics, Georgia Tech

Title:

How to compress networks while keeping their important characteristics
Abstract:

One of the fundamental concerns in the study of networks is
understanding the relation between a network's structure ("topology") and
its dynamics (evolution in time). However, the networks we often encounter
in either nature or engineering are typically very large. It is therefore
tempting to want to reduce such networks by excluding a part of their
elements while preserving some important characteristic(s) of the original
network. Such fundamental characteristic is the spectrum (collection of all
eigenvalues) of the network's weighted adjacency matrix. Moreover, this
matrix often contains all the known information about a network. Can one
hope to reduce a network while maintaining its spectrum? It seems that
there is no hope because of the Fundamental Theorem of Algebra which says
that the spectrum of MxM matrix contains M eigenvalues. Therefore the
spectrum of a smaller matrix (corresponding to a smaller network) contains
fewer eigenvalues. However, it is possible to do and the recently developed
theory of isospectral networks' reduction suggests new ways of networks'
analysis and synthesis. In particular, a new equivalence relation in the
class of all networks was found. Besides (as a "byproduct") the theory of
isospectral networks transformations allowed to advance some classical
areas of Mathematics as e.g. estimation of matrices' spectra. Another good
news is that numerical implementation of the procedure of isospectral
reduction is very simple and straightforward.

Bio:

Prof. Leonid Bunimovich is a Regents' Professor of Mathematics at
Georgia Tech. His research concerns a wide array of problems at the
intersection of dynamical systems and statistics. He is known for his
discovery of focusing chaotic billiards (the "Bunimovich stadium") and for
the Bunimovich mushroom, a billiard with mixed regular and chaotic
dynamics. He became a Regents’ Professor and given the Exemplary Senior
Faculty Award, in 2000 receiving the Outstanding Faculty Research Author
Award. He was made a Fellow of the Institute of Physics in 2004, and was
named a Chartered Physicist and Fellow of the UK Institute of Physics in
1999. He received the Humboldt Prize in 2003. Prof. Bunimovich received his
PhD from the University of Moscow (1973) and a Doctorate of Sciences from
the Institute for Theoretical Physics of the Academy of Sciences of USSR,
Kiev (1986).