The sport of football is played between two teams of eleven players each using a spherical ball. Each team strives to score by driving the ball into the opposing goal as the result of skillful interactions among players. Football can be regarded from the network perspective as a competitive relationship between two cooperative networks with a dynamic network topology and dynamic network node. Many complex large-scale networks have been shown to have topological properties in common, based on a small-world network and scale-free network models. However, the human dynamic movement pattern of this network has never been investigated in a real-world setting. Here, we show that the power law in degree distribution emerged in the passing behavior in the 2006 FIFA World Cup Final and an international “A” match in Japan, by describing players as vertices connected by links representing passes. The exponent values are similar to the typical values that occur in many real-world networks, which are in the range of , and are larger than that of a gene transcription network, . Furthermore, we reveal the stochastically switched dynamics of the hub player throughout the game as a unique feature in football games. It suggests that this feature could result not only in securing vulnerability against intentional attack, but also in a power law for self-organization. Our results suggest common and unique network dynamics of two competitive networks, compared with the large-scale networks that have previously been investigated in numerous works. Our findings may lead to improved resilience and survivability not only in biological networks, but also in communication networks.

Participation in interpersonal competitions, such as fencing or Japanese martial arts, requires players to make instantaneous decisions and execute appropriate motor behaviors in response to various situations. Such actions can be understood as complex phenomena emerging from simple principles. We examined the intentional switching dynamics associated with continuous movement during interpersonal competition in terms of their emergence from a simple syntax. Linear functions on return maps identified two attractors as well as the transitions between them. The effects of skill differences were evident in the second- and third-order state-transition diagrams for these two attractors. Our results suggest that abrupt switching between attractors is related to the diverse continuous movements resulting from quick responses to sudden changes in the environment. This abrupt-switching-quick-response behavior is characterized by a joint action syntax. The resulting hybrid dynamical system is composed of a higher module with discrete dynamics and a lower module with continuous dynamics. Our results suggest that intelligent human behavior and robust autonomy in real-life scenarios are based on this hybrid dynamical system, which connects interpersonal coordination and competition.

Complex human behavior, including interlimb and interpersonal coordination, has been studied from a dynamical system perspective. We review the applications of a dynamical system approach to a sporting activity, which includes continuous, discrete, and switching dynamics. Continuous dynamics identified switching between in- and anti-phase synchronization, controlled by an interpersonal distance of 0.1 m during expert kendo matches, using a relative phase analysis. As discrete dynamics, return map analysis was applied to the time series of movements during kendo matches, and six coordination patterns were classified. Furthermore, state transition probabilities were calculated based on the two states, which clarified the coordination patterns and switching behavior. We introduced switching dynamics with temporal inputs to clarify the simple rules underlying the complex behavior corresponding to switching inputs in a striking action as a non-autonomous system. As a result, we determined that the time evolution of the striking action was characterized as fractal-like movement patterns generated by a simple Cantor set rule with rotation. Finally, we propose a switching hybrid dynamics to understand both court-net sports, as strongly coupled interpersonal competition, and weakly coupled sports, such as martial arts.