## 3 Methodology
### Evaluation Procedures  
To evaluate the performance of the PageRank rankings, three empirical tests were conducted:  
* Ranking Consistency Analysis  
To quantify the ordinal agreement between the PageRank output and the official tournament standings, [PERSON]'s tau rank correlation coefficient (hereafter, Kendall's tau) was employed. As an established and widely adopted non-parametric metric, [PERSON]'s tau remains highly suitable for this analysis because it effectively captures pairwise ranking reversals and provides a robust indicator of structural similarity between rankings. Higher [PERSON]'s tau values indicate stronger consistency, with values approaching 1 reflecting near-perfect concordance. The coefficient is calculated as follows:  
\[\tau=\frac{C-D}{\frac{1}{2}n(n-1)} \tag{2}\]  
where \(C\) and \(D\) are the numbers of concordant and discordant pairs among \(n\) teams. Statistical significance is assessed via a two-tailed \(p\)-value, with \(p<0.05\) indicating a meaningful correlation.  
* Simulation-Based PredictionForecast accuracy is tested by simulating each season's matches. For every game, the team with the higher pre-match PageRank score is predicted to win. Forecast accuracy is then computed as follows:  
\[\text{Accuracy}=\frac{\text{Number of correctly predicted matches}}{\text{Total number of matches}} \tag{3}\]  
This metric is calculated on a per-season basis to assess the model's out-of-sample predictive reliability.  
To evaluate the predictive utility of PageRank in estimating team strength, a series of empirical tests was conducted using historical match data. An \(n\)-step-ahead prediction framework was implemented, in which PageRank scores were calculated prior to each game based on match results from the current season and up to five preceding seasons.  
Each tournament season follows a single-elimination format comprising four stages: preliminary (P), quarterfinals (Q), semifinals (S), and final (F). After the preliminary round, eight teams advance to four quarterfinal matchups, followed by two semifinal games (three in 2013 and 2014 due to format variations) and culminating in the championship final. To illustrate the n-step-ahead prediction framework, a one-year historical window was employed to forecast the 2015 quarterfinal outcomes. PageRank scores were computed using all matches from the 2014 season (including the preliminary, quarterfinal, semifial, and final stages), along with the 2015 preliminary results. Similarly, to predict the semifinals, both the complete 2014 season data and the 2015 preliminary and quarterfinal matches were used to estimate PageRank scores prior to those games.  
* Temporal Robustness Assessment  
To evaluate temporal robustness, two complementary analyses were performed using data from the 2013 to 2023 seasons. First, Kendall's tau was calculated between PageRank rankings and official standings in consecutive years to assess the year-to-year consistency of estimated team strengths. Second, predictive accuracy was evaluated across different historical data horizons, as defined in the simulation framework. These assessments demonstrate the stability and reliability of the PageRank rankings across tournament seasons.