## PageRank Application in Sports Performance Assessment

Source: http://www.sloansportsconference.com/wp-content/uploads/2017/02/1494.pdf

While PageRank primarily serves as the algorithm driving Google’s search engine, since its creation in 1998 there have been multiple applications in PageRank in a variety of fields, ranging from biology to statistics. One of these applications is found in sports performance assessment, specifically the evaluation of professional athletes. Player performance is often assessed using offensive statistics (ex. Points scored per game, total goals scored, etc.) and defensive statistics (steals per game, blocks per game, etc.). However, there are few instances of qualitative assessments used to evaluate player performance within the flow of the game. Shael Brown’s paper *A PageRank Model for Player Performance Assessment in Basketball, Soccer, and Hockey* describes a PageRank application that emphasizes involvement in successful plays and defensive turnovers, measures that determine the outcome of the game. Consequently, players who do not impressively fill up the stat sheet can still have high qualitative impact on the outcome of the game.

First, a directed graph is initialized and centered around a single ‘goal’ node indicating, for example, a basket made in basketball or a goal scored in soccer. There is an arc from the goal node to each player node and back. So for a three-on-three basketball game, there would be 2 arcs per player and goal, one from the goal to the player and one from the player to the goal. Additionally, there is an arc from the goal node to itself. Thus for a three-on-three basketball game we start with a total of 7 directed arcs. We then update the arcs on the graph based on several play sequences. Consequently, there could be multiple arcs from one node to another, so we update this measure as well. After traversing through the entire play sequence, an adjacency matrix is constructed, where the value at cell (i, j) represents the number of arcs from node i to node j. Then a transition matrix T is constructed, where each value Ti,j is the number of arcs from i to j divided by the total number of outgoing arcs from i. Afterwards we find the eigenvector of the transpose of T with eigenvalue 𝝺 = 1. This eigenvector v is the unique PageRank vector of the transition matrix. We can then use this PageRank vector to compute the relative ranks of each player. Because of the nature of this process, players who create plays for others (i.e. passing, assisting) and force turnovers on defense benefit the most. Although the mathematical computation is more complex, the PageRank algorithm in this sports performance context is similar to PageRank computation for search engines in the sense that player nodes are analogous to web pages and the passes or deflections are similar to endorsement links.