Entropy of Tournament Digraphs

The Renyi 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 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 among all tournaments on a fixed number of vertices H-2 and H-3 are maximized by any regular tournament on that number of vertices. 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. (C) 2019 Elsevier Inc. All rights reserved.