Inference of rankings planted in random tournaments

TL;DR

This paper studies the problem of inferring a hidden ranking from a correlated random tournament, establishing thresholds for detection and recovery, and proposing simple algorithms that are near-optimal, revealing a detection-recovery gap.

Contribution

It introduces the first analysis of detection and recovery thresholds in ranking inference from random tournaments, highlighting a new detection-recovery gap and providing simple, near-optimal algorithms.

Findings

Detection requires weaker correlations than recovery.

Simple algorithms achieve near-optimal performance.

Detection-recovery gap is demonstrated in ranking inference.

Abstract

We consider the problem of inferring an unknown ranking of $n$ items from a random tournament on $n$ vertices whose edge directions are correlated with the ranking. We establish, in terms of the strength of these correlations, the computational and statistical thresholds for detection (deciding whether an observed tournament is purely random or drawn correlated with a hidden ranking) and recovery (estimating the hidden ranking with small error in Spearman's footrule or Kendall's tau metric on permutations). Notably, we find that this problem provides a new instance of a detection-recovery gap: solving the detection problem requires much weaker correlations than solving the recovery problem. In establishing these thresholds, we also identify simple algorithms for detection (thresholding a degree 2 polynomial) and recovery (outputting a ranking by the number of "wins" of a tournament vertex, i.e., the out-degree) that achieve optimal performance up to constants in the correlation strength. For detection, we find that the above low-degree polynomial algorithm is superior to a natural spectral algorithm. We also find that, whenever it is possible to achieve strong recovery (i.e., to estimate with vanishing error in the above metrics) of the hidden ranking, then the above "Ranking By Wins" algorithm not only does so, but also outputs a close approximation of the maximum likelihood estimator, a task that is NP-hard in the worst case.