# Entropy of Tournament Digraphs

## TL;DR

This paper investigates the Rényi $lpha$-entropy of tournament digraphs, revealing how entropy varies with regularity and identifying maximum entropy configurations among regular tournaments.

## Contribution

It characterizes the behavior of Re9nyi entropy for tournaments, optimizing for specific lpha values and linking entropy to graph regularity and structure.

## Key findings

- H_lpha is maximized on doubly regular tournaments for lpha=2,3.
- Regular tournaments have higher entropy with increased regularity.
- A calculation related to von Neumann entropy applies to all graphs.

## Abstract

The R\'{e}nyi $\alpha$-entropy $H_{\alpha}$ of complete antisymmetric directed graphs (i.e., tournaments) is explored. We optimize $H_{\alpha}$ when $\alpha = 2$ and $3$, and find that as $\alpha$ increases $H_{\alpha}$'s sensitivity to what we refer to as `regularity' increases as well. A regular tournament on $n$ vertices is one with each vertex having out-degree $\frac{n-1}{2}$, but there is a lot of diversity in terms of structure among the regular tournaments; for example, a regular tournament may be such that each vertex's out-set induces a regular tournament (a doubly-regular tournament) or a transitive tournament (a rotational tournament). As $\alpha$ increases, on the set of regular tournaments, $H_{\alpha}$ has maximum value on doubly regular tournaments and minimum value on rotational tournaments. The more `regular', the higher the entropy. We show, however, that $H_2$ and $H_3$ are maximized, among all tournaments on any number of vertices by any regular tournament. We also provide a calculation that is equivalent to the von Neumann entropy, but may be applied to any directed or undirected graph and shows that the von Neumann entropy is a measure of how quickly a random walk on the graph or directed graph settles.