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Maps of Tournaments: Distances, Experiments, and Data

Filip Nikolow, Piotr Faliszewski, Stanisław Szufa

TL;DR

The paper addresses visualizing and comparing tournament graphs by importing the map-of-elections framework to create 2D maps where distances between tournaments reflect structural similarity. It introduces two distance notions—Graph Edit Distance $d_{GED}$ and Katz-distance $d_{katz}$—along with characteristic tournaments (e.g., $T_{ord}$ and $T_{rps}$) and both synthetic and real-life data sources (NBA and Bridge) to populate the maps. Through extensive experiments on size-12 and size-7 data, the work analyzes how different generator models and real datasets distribute on the map, assesses embedding quality via distortion, and demonstrates how maps illuminate winner determination in round-robin and knockout settings. The results show that synthetic models near $T_{ord}$ capture hierarchical structures found in NBA data, while election-based data can occupy distinct regions, guiding future model selection and methodological improvements. Overall, the map framework provides a rigorous, visual playground to study tournament spaces, compare generation processes, and understand how structural properties influence computational tasks such as winner determination and manipulation analysis.

Abstract

We form a "map of tournaments" by adapting the map framework from the world of elections. By a tournament we mean a complete directed graph where the nodes are the players and an edge points from a winner of a game to the loser (with no ties allowed). A map is a set of tournaments represented as points on a 2D plane, so that their Euclidean distances resemble the distances computed according to a given measure. We identify useful distance measures, discuss ways of generating random tournaments (and compare them to several real-life ones), and show how the maps are helpful in visualizing experimental results (also for knockout tournaments).

Maps of Tournaments: Distances, Experiments, and Data

TL;DR

The paper addresses visualizing and comparing tournament graphs by importing the map-of-elections framework to create 2D maps where distances between tournaments reflect structural similarity. It introduces two distance notions—Graph Edit Distance and Katz-distance —along with characteristic tournaments (e.g., and ) and both synthetic and real-life data sources (NBA and Bridge) to populate the maps. Through extensive experiments on size-12 and size-7 data, the work analyzes how different generator models and real datasets distribute on the map, assesses embedding quality via distortion, and demonstrates how maps illuminate winner determination in round-robin and knockout settings. The results show that synthetic models near capture hierarchical structures found in NBA data, while election-based data can occupy distinct regions, guiding future model selection and methodological improvements. Overall, the map framework provides a rigorous, visual playground to study tournament spaces, compare generation processes, and understand how structural properties influence computational tasks such as winner determination and manipulation analysis.

Abstract

We form a "map of tournaments" by adapting the map framework from the world of elections. By a tournament we mean a complete directed graph where the nodes are the players and an edge points from a winner of a game to the loser (with no ties allowed). A map is a set of tournaments represented as points on a 2D plane, so that their Euclidean distances resemble the distances computed according to a given measure. We identify useful distance measures, discuss ways of generating random tournaments (and compare them to several real-life ones), and show how the maps are helpful in visualizing experimental results (also for knockout tournaments).
Paper Structure (46 sections, 1 theorem, 10 equations, 8 figures, 1 table)

This paper contains 46 sections, 1 theorem, 10 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Components of a Map.
  3. Evaluation.
  4. Supplementary Material.
  5. Preliminaries
  6. Graphs and Tournaments.
  7. Distances.
  8. Distances Between Vectors.
  9. Distances Between Tournaments.
  10. Centrality Measures.
  11. Generating and Collecting Tournaments
  12. Characteristic Tournaments
  13. Statistical Models of Tournaments
  14. Collected Real-Life Data
  15. NBA.
  16. ...and 31 more sections

Key Result

Proposition 4.1

If $n$ is an odd integer representing the number of players, then $d_{{\mathrm{GED}}}(T_{{\mathrm{ord}}},T_{{\mathrm{rps}}}) = 1/8(n^2-1)$.

Figures (8)

  • Figure 1: Basic insights regarding tournaments. Plot (a) shows a histogram of GED distances between $T_{{\mathrm{ord}}}$ and all RPS tournaments with 11 players, plot (b) compares the GED and Katz distances in the Size-12 dataset (each point is a pair of tournaments, its $x$ coordinate is their GED distance and its $y$ coordinate is the Katz distance), and plot (c) shows PCC of the GED and Katz distances between $T_{{\mathrm{ord}}}$ and $100$ tournaments generated using the uniform model, depending on the number of players (for each number of players we repeated the experiment $10$ times; the shaded area shows standard deviation).
  • Figure 2: Maps of different datasets computed using GED (upper row) and Katz distance (lower row).
  • Figure 3: Maps of the MoV dataset.
  • Figure 4: Distortion of the the Size-7 and Size-12 maps.
  • Figure 5: The numbers of winners under (a) the Copeland rule, (b) top cycle, and (c) Slater, as well as (d) the average running time to compute Slater winners.
  • ...and 3 more figures

Theorems & Definitions (6)

  • Remark 3.1
  • Remark 3.2
  • Proposition 4.1
  • Definition 4.1
  • Remark 4.1
  • Remark 5.1