Rényi ordering of tournaments

Abstract

We develop an ordering function on the class of tournament digraphs (complete antisymmetric digraphs) that is based on the Rényi \(\alpha\)-entropy. This ordering function partitions tournaments on \(n\) vertices into equivalence classes that are approximately sorted from transitive (the arc relation is transitive — the score sequence is \((0, 1, 2, \ldots, n-1)\)) to regular (score sequence like \((\frac{n-1}{2}, \ldots, \frac{n-1}{2})\)). However, the diversity among regular tournaments is significant; for example, there are 1,123 regular tournaments on 11 vertices and 1,495,297 regular tournaments on 13 vertices up to isomorphism, which is captured to an extent.