School of Mathematics
Georgia Institute of Technology
164 Skiles Building
Phone: 404~894~6500
Email: bwebb@math.gatech.edu
| PhD Student School of Mathematics Georgia Institute of Technology 164 Skiles Building Phone: 404~894~6500 Email: bwebb@math.gatech.edu |
| Publications and Preprints Below is a list of my publications as well as preprints in reverse chronological order. A more detailed explanation of each is given beneath this list.
Poster Presentations
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| In this paper we present a general procedure allowing for the reduction or expansion of any network (considered as a weighted graph) which maintains the spectrum of the network's adjacency matrix up to a set of eigenvalues known beforehand from its graph structure. This procedure can be used to establish new equivalence relations on the class of all weighted graphs (networks) where two graphs are equivalent if they can be reduced to the same graph. We show such transformations are useful in determining whether a dynamical network i.e. a network of interacting dynamical systems has a globally attracting fixed point. |  Isospectral Branch Reduction |
| Dynamical networks are characterized by large complex graphs of interactions. We suggest a procedure of simplifying the structure of such graphs while preserving the spectrum of their weighted adjacency matrix. As the process of isospectral graph reductions maintains the spectrum of the matrix up to some known set it is possible to estimate the spectrum of the original matrix by considering Gershgorin-type estimates associated with the reduced matrix. The main result of this paper is that eigenvalue estimates improve for all known methods as the matrix size is reduced. Moreover, our procedure of isospectral graph reductions is very flexible and in particular can be used to obtain better eigenvalue estimates of a matrix with complex valued entries to whatever degree is desired. | 
Improved Eigenvalue Estimates via Graph Reduction |
| The mathematical model that Zermelo developed for ranking by paired comparisons and that was later popularized by Bradley and Terry has several attractive theoretical properties, but computation of the associated ratings may involve solution of a system of several high-degree polynomial equations in several variables. This paper describes how to define quantities analogous to electrical resistance and conductance for certain generalized tournaments in such a way that these quantities are well-behaved with respect to certain types of decomposition of tournaments and permit comparison of the ratings of pairs of nodes. Application of this theory is illustrated through consideration of specific examples. |  Tournament with Easy to Compute Conductance |
| In the study of one dimensional dynamical systems one often assumes that the functions involved have a negative Schwarzian derivative. In this paper we consider a generalization of this condition. Specifically, we consider the interval functions of a real variable having some iterate with a negative Schwarzian derivative and show that many known results generalize to this larger class of functions. The introduction of this class was motivated by some maps arising in neuroscience. |  Function with an Eventual Negative Schwarzian Derivative |  Schwarzian Derivative of Second Iterate in Red (Mathematica Demonstration) |