## Class

Article

## College

College of Science

## Faculty Mentor

David Brown

## Presentation Type

Oral Presentation

## Abstract

Claude Shannon developed the concept now known as 'Shannon entropy' as a measure of uncertainty or disorder in information states. The theory of entropy has been extensively applied to quantum mechanics, where the entropy of a specific quantum state is defined as the Shannon entropy of its corresponding density matrix, in which the spectrum of its eigenvalues is viewed as that state's probability distribution. Recognizing a connection between these density matrices and normalized Laplacian matrices of undirected graphs, recent work in graph theory involves analyzing the entropy of undirected graphs. We further these ideas by applying the theory of entropy to directed graphs, in particular those that model pairwise comparisons which are known as tournament digraphs. In order to circumvent the trouble of complex eigenvalues and whatever could be meant by complex probabilities, we employ a generalized notion of the Shannon entropy, which is the Rényi α-entropy. Specifically, we determine the Rényi 2-entropy and Rényi 3-entropy of tournament digraphs using the eigenvalues of their normalized Laplacian matrices. We show that both measures depend entirely upon the degree sequence of a given tournament, and we use this characterization in order to determine tournament structures with maximal and minimal entropy.

## Location

Room 154

## Start Date

4-12-2018 9:00 AM

## End Date

4-12-2018 10:15 AM

Rényi α-Entropy of Tournament Digraphs

Room 154

Claude Shannon developed the concept now known as 'Shannon entropy' as a measure of uncertainty or disorder in information states. The theory of entropy has been extensively applied to quantum mechanics, where the entropy of a specific quantum state is defined as the Shannon entropy of its corresponding density matrix, in which the spectrum of its eigenvalues is viewed as that state's probability distribution. Recognizing a connection between these density matrices and normalized Laplacian matrices of undirected graphs, recent work in graph theory involves analyzing the entropy of undirected graphs. We further these ideas by applying the theory of entropy to directed graphs, in particular those that model pairwise comparisons which are known as tournament digraphs. In order to circumvent the trouble of complex eigenvalues and whatever could be meant by complex probabilities, we employ a generalized notion of the Shannon entropy, which is the Rényi α-entropy. Specifically, we determine the Rényi 2-entropy and Rényi 3-entropy of tournament digraphs using the eigenvalues of their normalized Laplacian matrices. We show that both measures depend entirely upon the degree sequence of a given tournament, and we use this characterization in order to determine tournament structures with maximal and minimal entropy.