Entropy of tournament digraphs

Abstract

The Rényi α-entropy Hα of complete antisymmetric directed graphs (i.e., tournaments) is explored. We optimize Hα when α=2 and 3, and find that as α increases Hα'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−12, 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 α increases, on the set of regular tournaments, Hα 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 H2 and H3 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.