Tournament limits: Degree distributions, score functions and self-converseness

TL;DR

This paper characterizes the degree distributions and score functions of tournament limits, establishing conditions for their realizability and exploring unique and self-converse cases in the context of tournament kernels.

Contribution

It provides a complete characterization of degree distributions and score functions for tournament limits, including conditions for uniqueness and self-converseness.

Findings

Only the uniform distribution is uniquely realizable as an outdegree distribution.

The paper characterizes all degree distributions and score functions that can appear in tournament limits.

Defines and characterizes self-converse tournament limits and kernels.

Abstract

Motivated by known results for finite tournaments, we define and study the score functions of tournament kernels and the degree distributions of tournament limits. Our main theorem completely characterises those distributions that appear as the degree distribution of some tournament limit and those functions that appear as the score function of some tournament kernel. We also show that only the uniform distribution can be realised as the outdegree distribution of a unique tournament limit. Finally we define self-converse tournament limits and kernels and characterise their degree distributions and score functions.