## 2 Literature Review
### PageRank in Sports  
The PageRank algorithm, developed by [PERSON] and [PERSON] for web search applications [22], is grounded in Markov chain theory and random walk models. It calculates the importance of a node based on both the quantity and quality of incoming links. Beyond web search, PageRank has been widely used in fields such as citation analysis, social network analysis, and recommender systems. The algorithm's logic transfers naturally to sports settings: Teams are modeled as nodes, and match outcomes are represented as directed edges pointing from the loser to the winner. A victory not only confers a direct score but also inherits part of the defeated team's strength, thereby reflecting both direct and indirect competitive performance. Empirical applications of PageRank have demonstrated its promise across a range of sports. In professional sports settings where comprehensive match data are available, PageRank has been widely employed in research on predicting game outcomes. Regarding basketball, [PERSON] and [PERSON] [23] applied a generalized PageRank model to National Basketball Association data to enhance predictive accuracy in sports rankings. Their method constructs a game graph from a Bayesian correction perspective, representing teams as nodes and assigning edge weights via a link function based on match outcomes. Parameters are optimized by minimizing the loss between predicted and actual rankings, with results showing improved performance over conventional ranking models. In baseball, [PERSON] et al. [24] analyzed 4861 Major League Baseball games from the 2017 and 2018 seasons. Their results demonstrated that PageRank provided stronger predictive validity than traditional win-loss-based ranking systems. When individual-level statistics are unavailable, network-based ranking approaches that rely on aggregated competition data, such as win-loss records or cumulative points, provide a practical alternative. PageRank has demonstrated reliable performance under such low-data conditions. As reported in [25], the PageRank rankings for male 100 m sprinters in the 2016 IAAF Diamond League exactly matched the top five positions in the official points system and outperformed other ranking models, including Keener, Colley, and Massey. Compared to methods based on win percentage or score aggregation, PageRank is better suited for handling unbalanced schedules, indirect match relationships, and non-round-robin formats such as single elimination. In the context of Olympic Taekwondo, [PERSON] and [PERSON] [26] and [PERSON] et al. [27] demonstrated the applicability of the PageRank algorithm for athlete ranking. [PERSON] and [PERSON] [26] analyzed 144 matches in the Men's -58 kg category and found that PageRank produced rankings that were more valid than those from the official Accumulative Point Index System (APIS). Similarly, [PERSON] et al. [27], using 624 matches from the 2015 to 2018 World Grand Prix series, showed that PageRank outperformed the APIs by better accounting for opponent strength and enhancing ranking interpretability. Continued research efforts have sought to improve the application of PageRank in sports ranking contexts. A recent refinement of PageRank, termed bi-directional PageRank, was proposed and tested on synthetic datasets calibrated from 18 empirical sports leagues such as soccer, basketball, baseball, and ice hockey [28]. These datasets were generated using a two-parameter variant of the Bradley-Terry model that accounts for home advantage and sport-specific randomness. The study found that standard PageRank outperforms the win ratio only in the early stages of a season and primarily in sports with low randomness. In contrast, bi-directional PageRank consistently demonstrated higher predictive accuracy by incorporating the quality of both victories and defeats. To further enhance ranking precision, [PERSON] et al. [29] proposed an artificial-intelligence-enhanced version of the PageRank algorithm.