# Tournament Entropy

Article

## College

College of Science

David Brown

## Presentation Type

Oral Presentation

## Abstract

Across disciplines, the term 'entropy' is used to quantify the complexity of a system. In particular, Rényi and Von Neumann entropies measure the structural complexity of discrete binary structures (undirected graphs). Here, the spectrum of the graph's normalized Laplacian matrix is treated as a discrete probability distribution by which the entropy functionals operate. This works well for undirected graphs because the normalized Laplacian matrices are positive semidefinite, so each eigenvalue is a real number between 0 and 1, and because of the normalization, the eigenvalues add to 1. In the case of directed graphs, little research has been done because eigenvalues are often complex. We find that by extending the entropy definitions to complex numbers, Rényi and Von Neumann entropies still give insight into structure of the graph. We focus our attention on a certain class of directed graphs called tournaments, which consist of pairwise comparisons on a set. We find that Rényi and Von Neumann entropies can be analyzed combinatorially and that tournaments with high transitivity tend to have lower entropy, while tournaments with more chaotic structures tend to have higher entropy.

Room 421

## Start Date

4-12-2018 12:00 PM

## End Date

4-12-2018 1:15 PM

## Share

COinS

Apr 12th, 12:00 PM Apr 12th, 1:15 PM

Tournament Entropy

Room 421

Across disciplines, the term 'entropy' is used to quantify the complexity of a system. In particular, Rényi and Von Neumann entropies measure the structural complexity of discrete binary structures (undirected graphs). Here, the spectrum of the graph's normalized Laplacian matrix is treated as a discrete probability distribution by which the entropy functionals operate. This works well for undirected graphs because the normalized Laplacian matrices are positive semidefinite, so each eigenvalue is a real number between 0 and 1, and because of the normalization, the eigenvalues add to 1. In the case of directed graphs, little research has been done because eigenvalues are often complex. We find that by extending the entropy definitions to complex numbers, Rényi and Von Neumann entropies still give insight into structure of the graph. We focus our attention on a certain class of directed graphs called tournaments, which consist of pairwise comparisons on a set. We find that Rényi and Von Neumann entropies can be analyzed combinatorially and that tournaments with high transitivity tend to have lower entropy, while tournaments with more chaotic structures tend to have higher entropy.