# An Improved Passing Network for Evaluating Football Team Performance
## 3 Methodology
### Model Specification
#### 3.2.3 Establishment of Coordination Index  
For players' performance on the field, researchers have built mature individual multidimensional ratings through various means [30]. Football data companies such as Whoscored and Opta provide fans worldwide with intuitive single-game ratings of players. Computer scientists can do more ground-breaking research based on player ratings [30; 31]. Compared with the relatively perfect research on individual players, the research on the whole team is obviously more challenging. A football team is a multidimensional set of players on the field, and the contribution of nodes to the network fluctuates with increasing time. Meanwhile, the weight of each node in the network is expanded or shrunk by the triangle pair formed by the combination of a few nodes. Therefore, the network mapped by the set of players is more difficult to scientifically and accurately reflect the fit between theoretical science and practical science. Compared with individual analysis, team network analysis has more practical value. Professional coaching teams can better master the individual technical and tactical abilities of team members, but lack of clear evaluation standards for the overall play of the team, most of which are judged by experience. Intuitive metrics to measure team performance can help coaching teams develop better tactical policies. Based on this, we propose a quantitative index of team coordination based on improved passing network, which is intended to provide a numerical evaluation standard for the overall team consistency. In complex network theory, there are many local network invariants or global network invariants to measure the whole network index, which can be reflected in the passing network. For example, the network average shortest path calculated by the improved Dijkstra algorithm [32] represents the average total value of passes of players in a single game, which is an effective method to measure passing performance. The closeness centrality [33] of each node in the network is measured as the average distance between the node and all other nodes, which can measure the difficulty of reaching a player by passing the ball. A higher measurement means better player integration. When the mean value of closeness centrality of all nodes is high, it means that the passing network has high acceptance for all players, and the team plays more evenly as a whole. High-quality team passing patterns require both stable ball control and high quality passing by surprise, which correspond to the stability and randomness of network attributes, respectively. The small world [34] of the network is defined as the difference between the ratio of the average clustering coefficient and the average shortest path to each random network attribute. This value determines the proportion of stability and randomness in the network, which can be used to measure the team's ball control and creativity. These network global invariants can describe the reduction degree of the passing network to the actual game in a limited way and give theoretical results. However, they have their own defects and are still difficult to use for meeting the research requirements. In order to measure the quality of the passing network, it is necessary to consider both the individual performance of the player and the overall balance of the team. In this study, the ratio of network average clustering coefficient to average betweenness centrality value is used as the coordination measure of the passing network. The betweenness centrality [35; 36] of the network is used to measure the degree to which nodes in the network are on paths between other nodes, and is usually defined as the percentage of shortest paths through a node. Higher betweenness centrality value means a player is dominant on his team, while lower value, which is more evenly distributed, means more balanced team performance.