## 4 Empirical Results and Discussion  
### Overview of PageRank Rankings  
Based on data from the 2013 to 2023 tournaments, average PageRank scores were computed for all teams across seasons. Figure 1 presents the top 10 teams by their average PageRank, along with their corresponding win percentages (win rate), which are defined as wins divided by the total games played.  
Teams such as Pingzhen High School (HS), Kupao Home Economics and Commerce HS, and Kaoyuan Vocational HS consistently ranked among the top three across both metrics. Their PageRank values more clearly distinguish them from other teams, reflecting sustained competitive dominance. Notably, Pingzhen HS leads with a long-term average PageRank of 0.609 and secured the most Black Panther Cup titles over the study period.  
Several teams exhibit notable discrepancies between their PageRank and win-rate rankings. For instance, Taoyuan Agricultural & Industrial HS ranks fourth in PageRank but fifth by win rate, suggesting stronger opponents defeated and a higher performance value that is not reflected in the raw win percentage. In contrast, Yingge Vocational HS ranks higher by win rate but lower in PageRank, indicating that their success may have come against weaker teams.  
Similarly, Shiyuan HS ranks 6 th in PageRank but only 10 th in win rate, likely due to impactful wins over high-strength opponents despite a modest overall record. Conversely, Mailiao HS ranks 6 th in win rate but 10 th in PageRank, reflecting inflated results from favorable matchups. These cases demonstrate PageRank's capacity to capture performance quality beyond win-loss totals, offering a more nuanced measure of team strength in structurally imbalanced tournaments.  
To further analyze longitudinal performance patterns, all 17 teams that reached the quarterfinals at least once during the 2013-2023 seasons were identified, and their annual PageRank scores were visualized in a heatmap (see Figure 2).  
The heatmap illustrates a clear contrast between teams exhibiting sustained excellence and those demonstrating episodic success. Teams such as Pingzhen HS, Kupao Home Economics & Commerce HS, and Kaoyuan Vocational HS consistently display high PageRank values across multiple seasons, underscoring their long-term dominance and structural centrality within the tournament network. These stable patterns reflect not only frequent advancement into later rounds but also victories over strong opponents, which PageRank effectively captures through its recursive weighting mechanism.  
Conversely, teams such as Pumen HS, Yingge Vocational HS, and NCHU Affiliated Agricultural & Industrial Vocational HS show marked spikes in PageRank scores during isolated years, suggesting that transient competitive surges are likely tied to cohort effects, favorable draws, or exceptional individual players. This episodic behavior contrasts with their generally lower presence in other seasons, indicating weaker overall program stability.  
Furthermore, the matrix suggests that only about 3 to 4 teams consistently attain high PageRank scores across multiple years, while most other teams remain low-ranked. This concentration indicates a relatively uneven competitive landscape, with dominance clustered among a few programs. Nevertheless, the occasional emergence of 6 to 8 different teams reaching high scores in select years also reflects a degree of mobility and turnover, characteristic of youth-level tournaments where talent development and cohort variability play significant roles.  
Figure 1: Comparison of the top 10 teams by average PageRank score and win rate (2013–2023).