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 $(0,1,2,\dots ,n-1)$) to regular (score sequence like $(\frac{n-1}{2},\dots ,\frac{n-1}{2})$). 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.