# Hierarchy and ranking in pairwise sports contests

## TL;DR

This paper models athlete competitions as evolving networks to analyze hierarchical structures, revealing differences between tournament types and enabling outcome predictions comparable to traditional ranking systems.

## Contribution

It introduces a network-based approach to analyze and compare hierarchies in sports contests, providing insights into their structure and predictive capabilities.

## Key findings

- Elimination tournaments produce less hierarchical networks with more cycles.
- Network metrics can predict match outcomes with accuracy similar to Elo ratings.
- Hierarchical structures differ between round-robin and elimination formats.

## Abstract

Ranking athletes by their performance in competitions and tournaments is common in every popular sport and has significant benefits that contribute to both the organization and strategic aspects of competitions. Although rankings are perhaps the most concise and most straightforward representation of the relative strength among the competitors, beyond this one-dimensional characterization, it is also possible to capture the relationships between athletes in greater detail. Following this approach, our study examines the networks between athletes in individual sports such as tennis and fencing, where the nodes are associated with the contestants and the edges are directed from the winner to the loser. We demonstrate that the connections formed through matches arrange themselves into a time-evolving hierarchy, with the top players positioned at its apex. The structure of the resulting networks exhibits detectable differences depending on whether they are constructed purely from round-robin data or from purely elimination-style tournaments. We find that elimination tournaments lead to networks with a smaller level of hierarchy and thus, importantly, to an increased probability of circular win-loss situations (cycles). The position within the hierarchy, along with other network metrics, can be used to predict match outcomes. In the systems studied, these methods provide predictions with an accuracy comparable to that of forecasts based on official sports ranking points or the Elo rating system. A deeper understanding of the delicate aspects of the networks of pairwise contests enhances our ability to model, predict, and optimize the behaviour of many complex systems, whether in sports tournaments, social interactions, or other competitive environments.