# An Improved Passing Network for Evaluating Football Team Performance
## 2 Related Work  
Graph or network analysis plays a role in a wide range of problems, from sociology to sports science [9; 10; 11]. Mathematicians have developed rich theories around network science, aiming to study the connections among the components of a network in a more efficient and reliable way. The passing movement is an ideal data source to measure team performance because of its repeated times and high behavior discrimination in football matches [12; 13]. The passing map of a football match has high similarity to the data structure of a computer network graph, and the application of network science to map the passing line has high analyzability, which can intuitively reveal the information in the passing data flow [14]. Football is characterized by a large range of ball flow and high frequency of convection. Traditional statistics such as goals, assists, and ball possession rate are not enough to measure individual and overall performance of a football team [15; 16]. By using network science to analyze the phenomenon of football, which has better theoretical support, the complex movement behavior set can be abstracted into a more intuitive network topology structure. A professional football team with a distinctive style can display high levels of tactical discipline and coordination during matches, but images presented by television cameras alone are difficult to quantify and cannot support computer analysis. Therefore, in this study, we use network science to focus on the overall coordination index of team performance. In order to transform the passing routes into a network structure, we collected the total data from 38 rounds of Everton's Premier League matches in 2017-2018 from a public free data platform, and mapped the passing image distribution for every Everton's match into a weighted oriented network, with each node corresponding to the players in the match. Directional lines between nodes represent passes between players, and there are usually two opposite edges between nodes corresponding to two players. Before this study, some scholars had built a similar network [17], but their network simply takes the times of players passing each other as the weight of the edges between nodes. Obviously, this weight calculation method has a higher contribution proportion for the players who frequently pass each other in the middle or back court, but not so friendly to the front court players who do not pass the ball as often and have a high turnover rate. Therefore, we optimize the weight calculation method and build an improved passing network.  
At present, centrality analysis is commonly used for measurement of the passing network [18]. By using degree centrality [19], closeness centrality [20], eigenvector centrality [21], and their variants, researchers can analyze player interactions in transfer sequences and further classify teams' tactical styles [22; 23; 24]. However, it is difficult for a single centrality measure to contribute high quality indicators to the analysis of complex passing networks [25]. We take the ratio of network global clustering coefficient to average betweenness centrality as the network consistency index. Furthermore, the correlation test and sensitivity analysis are carried out between the index and Everton's score of attack and defense efficiency and the score of control power in each match, and relatively ideal results are obtained, which prove the validity and reliability of the proposed index. This index evaluates the coordination of the overall performance of football teams from a quantitative perspective, and can improve the difficult problem of football team evaluation. We try to promote this index to help coaches have a clearer and intuitive understanding of their team's performance.