Football Games Consist of a Self-Similar Sequence of Ball-Keeping Durations - Keiko Yokoyama; Hiroyuki Shima; Akifumi Kijima; Yuji Yamamoto (Fractal and Fractional, 2025)
In football, local interactions between players generate long-term game trends at the global scale, and vice versa—the global trends also influence individual decisions and actions. The harmonization of local and global scales often creates self-organizing spatiotemporal patterns in the movements of players and the ball. In this study, we confirmed that, in real football games, the probability distribution of the ball-keeping duration tends to obey negative power-law behavior, exhibiting hierarchical fractal self-similarity at both the local scale (i.e., individual-player level) and at the global scale (i.e., whole-game level). Furthermore, we found that the probability distribution functions transitioned from an exponential distribution to a power-law distribution at a certain characteristic time and that the characteristic time was equal to the upper limit of the time during which the trend of the game was maintained.