Scale-free Property of the Passing Behaviour in a Team Sport

Abstract

This study examined the application of network theory to skilled passing behaviour in a team sport by considering small-world and scale-free network models. Using data obtained from a 2006 soccer game in Japan between Japan and Ghana, we counted the number of passes by each player within 5-minute intervals. The structural properties of the passing behaviour, which included a characteristic path length and clustering coefficient, and the degree of distribution were analysed. This showed that the structural property of the passing behaviour represented neither a complete graph nor a random graph; rather, it reflected a small-world or scale-free network. In addition, the probabilities of outgoing and incoming passes reflected links that followed a power-law distribution. Passing behaviour in a soccer match appeared to be similar to behaviour in social networks with smaller vertices in terms of the scale-free property and a self-organising mechanism.Key word: network theory, soccer, power-law distribution Our study represents another approach to exploring the dynamics of team sports. We examined the application of network theory to skilled passing behaviour in a team sport to investigate the merits of the small-world (Watts & Strogatz, 1998) and scale-free (Barabási & Albert, 1999) network models. Brief review of network theoryNetwork theory is rooted in Euler's graph theory (Biggs et al., 1976), which was originally designed to solve the problem of the seven bridges of Königsberg, Prussia (now Kaliningrad, Russia) mathematically. The question was whether it was possible to follow a route that crossed each bridge exactly once. Euler (1736) formulated the problem by graphing Königsberg. First, he eliminated all of the features except the landmasses and the bridges connecting them; next, he replaced each landmass with a dot, called a vertex or node, and each bridge with a line, called an edge or link. The geographic relationships between the landmasses and the distances of the bridges were not considered in this model and only the connectivities of vertices deemed important.Subsequently, Euler's graph theory (Biggs et al., 1976) has been expanded to include various types of graph, including star, tree, and wheel graphs (See Figure 1). Studies by Erdös and Rényi (1959, 1960, 1961, cited by Karoński and Ruciński (1997), elaborated on these graph theories by introducing the graph generation approach. This approach uses a regular graph and a random graph; a graph is regular when all vertices have equal edges and a random graph is reconstructed from a regular graph by random linkages between vertices with a particular probability p. Random graphs can be used to examine the phase transition and percolation of graph properties. In this context, the terms graph and network are synonymous.The small-world phenomenon refers to the relatively short path between two vertices that occurs in most networks despite their frequently large size. During the 1960s, social psychologist Stanley Milgram examined this phenomenon with a sophis...

Team Play in Football: How Science Supports F. C. Barcelona’s Training Strategy

“…Such an important finding is perfectly supported by some studies examining the relationship between spatial patterns of passes and performance. By using the theory of free-scale networks, Yamamoto (2009) has demonstrated that the path of the ball is not random. This study is in resonance with Hirano and Tsumoto (2004) who showed that some pass patterns are more efficient than others, implicitly underlining the centrality of some players in passing to all other players.…”

Scaling Invariance of Sports Sex Gap

“…Scale invariance describes phenomena that are not associated with a particular or characteristic scale of length, energy, or other variables, and is mathematically equivalent to power law behavior. Many studies have used power law to analyze the law of sport ( Katz and Katz, 1994 ; Sylvan Katz and Katz, 1999 ; Vincenzo and Sandra, 2001 ; Yamamoto, 2009 ; Fernández-Revelles and García Mármol, 2019 ). Among them, Katz’s research shown that the performance of male and female superior athletes exhibits a fractal relationship between world record running and swimming times and the distance of the even, and an exceptionally good linear fit ( R 2 ≥ 0.999) was observed in the log-log plot ( Katz and Katz, 1994 ).…”

Statistical properties of position-dependent ball-passing networks in football games

“…In statistical physics, analysis for complex networks has achieved rapid development recently [15,16]. The network analysis has already been applied to football games such as a structural property of ball-passing networks [17,18], and assessment of players [19,20]. In the studies of the ball-passing networks [17,18], each node and edge of the network represent an individual player and passing of the ball, respectively.…”

“…The network analysis has already been applied to football games such as a structural property of ball-passing networks [17,18], and assessment of players [19,20]. In the studies of the ball-passing networks [17,18], each node and edge of the network represent an individual player and passing of the ball, respectively. One main conclusion in their studies is that ball-passing networks of football games have the scale-free property, namely the degree distributions follow the power law.…”