## 1 Introduction  
It is often necessary to determine the importance of an alternative compared to others in a group. The process of ordering a list of alternatives based on their relative strengths is referred to as ranking. In this paper, we focus on ranking methods with applications to sports. Ranking of individual players or teams in sports, both professional and amateur, is a tool for entertaining fans and developing sports business. Moreover, it is of great significance to provide relevant personnel with reference to the strengths of competitors.  
Recent years have seen increasing interest in ranking elite athletes and teams in professional sports leagues and in predicting the outcomes of games. Several methods have been proposed since the last century. Examples of these ranking methods include the [PERSON] method [1], the [PERSON] method [2], the [PERSON] method [3], the [PERSON] method [4], the logistic regression/Markov chain (LRMC) method [5, 6], and the network-based method [7]. More recently, due to the extraordinary success of Google's PageRank, an algorithm originally developed for ranking web pages based on their importance [8], graph-based algorithms have gained more ground in the topic of sports ranking problems. The graph-based methods, such as PageRank and its extensions, have been adopted to rank individual players or teams in tennis [9], soccer [10], basketball [11], football [12], rugby [13], and hockey [14].  
Although PageRank considers the indirect effect between games through the graph model, like most ranking models, these efforts, which just use game points and win-loss information, pay little attention to obtaining a comprehensive representation to measure the marginal effect of each game. The performance of players and teams is influenced by a variety of causes [15]. For example, the effects of home wins and away wins should be treated differently [16]. As another example, in basketball games, in addition to points, assists and rebounds are also important factors used to measure the strength of a team [17]. To solve this problem, the idea of learning to rank can be used to learn a comprehensive representation of each game.  
Learning to rank is first used in information retrieval where machine learning algorithms are used to learn to rank candidate documents in a query according to their relevance to the query. In learning to rank, ranking is considered as a supervised learning problem, and several machine learning algorithms have been applied to it. The work in [18] provided a detailed overview of learning to rank. Learning to rank on a graph is ranking the results of queries that are executed against graphical data, such as social media, linked data, semantic web data, and a semantic bibliographic network [19]. In these graphs, nodes represent entities, and edges mimic interactions or relationships. The work in [20] extended the well-known PageRank authority flow-based ranking method to entity relationship graphs in order to learn an authority flow weight vector.  
The sound data index system has been developed for basketball games, providing the possibility and demand for a more comprehensive ranking system. In this paper, we focus on the ranking methods in basketball. A general framework is proposed to learn the setting parameters in a game graph with various game information and rank the teams by an extended PageRank method. Moreover, due to the model setting, our model has the power to realize FutureRank, predicting the rank of teams in the future. The rest of this paper is organized as follows. Section 2 presents the framework of learning to rank sports teams on a graph. In Section 3, the methodology is applied to the data for the National Basketball Association. More aspects about the empirical results are discussed in Section 4. Concluding remarks and future research are summarized in Section 5.