What do albatrosses searching for food, stock market fluctuations, and the dispersal patterns of seeds in the wind have in common?
They all exhibit a type of movement pattern called Lévy walk, which is characterized by a flurry of short, localized movements interspersed with occasional, long leaps. For living organisms, this is an optimal strategy for balancing the exploitation of nearby resources with the exploration of new opportunities when the distribution of resources is sparse and unknown.
Originally described in the context of particles drifting through liquid, Lévy walk has been found to accurately describe a very wide range of phenomena, from cold atom dynamics to swarming bacteria. And now, a new study published in Complexity has for the first time found Lévy walk in the movements of competing groups of organisms: football teams. "Football is a game about scarcity of resources: to win, a team requires possession of the ball, and there is only one ball in play," says Professor Tom Froese, senior author of the study and leader of the Embodied Cognitive Science Unit at OIST. "And so, it makes sense for individual players to move in a way that balances exploration and exploitation, ensuring that they do not stay in the same place too long while increasing their chances of getting the ball at each point. We found that the teams as a whole act in exactly the same way."
From particles in liquid to players on a field
The concept of Lévy walks originated with the French mathematician Paul Lévy, who developed a statistical model of heavy-tailed probability distributions that was later applied by Mandelbrot to describe random movements with occasional long jumps. Lévy walks found their first practical application in physics, where researchers used them to model the movement of particles in turbulent flows, finding that particles following Lévy walk patterns were superdiffusive, meaning that they spread faster due to the occasional long jumps. The concept made a significant leap into biology in 1996, when a study demonstrated that wandering albatrosses exhibit Lévy walks while foraging.
The OIST researchers used data from a match in the top Japanese football league, J-League, which included exact locations of each player and the ball at a centimeter-level precision throughout the game. "We found through statistical tests that the players exhibited Lévy walk when hunting for the ball - much like animals foraging for food. As soon as they got the ball, they deviated from Lévy walk, possibly due to the constraints of interacting with the ball. We saw the same behavior in the team's centroid-the average position of all players-which suggests that the teams act as an individual agent when seeking possession of the ball," explains Ivan Shpurov, first author and PhD student in the unit.
In addition to testing the movements of players and teams for Lévy walk, the researchers also found a strong correlation between the degree to which players exhibit Lévy walk and their mean distance to the ball as well as to their team's centroid, finding that the players with a more prominent Lévy walk pattern were overall closer to both the ball and the team's centroid. "While we cannot conclude that Lévy walk is the hallmark of a good football player, it does suggest that players who strongly exhibit Lévy walk movement are more active and generally closer to the ball, and contribute better to the team dynamic," says Shpurov. Differences in the degree of Lévy walk could also provide insights into the distinct player roles. For one, the goalkeeper has a very different movement pattern and role.
One for the team
In finding that not only individual football players but also entire teams act to optimize the balance between exploration and exploitation, the researchers have demonstrated the explanatory power of Lévy walks and provided insights into how we - and other pack-hunting animals - work together to achieve shared goals. "While we haven't yet investigated the underlying mechanisms of team coherence, previous research has suggested that behavioral cooperation in pairs is often associated with increased interbrain synchrony - the finding that two people can essentially link up their minds to perform a shared activity," explains Prof. Froese.
The novel finding that a group of individuals can come together and collectively exhibit Lévy walk is evidence of such synchrony, potentially suggesting that we can project our cognition beyond our immediate selves into a collective self that can behave as a single, integrated agent. As Prof. Froese puts it, "the footballers do not all act in exactly the same way, as that would be a highly inefficient tactic. Instead, their individual actions complement one another in response to the game, with the behavior of the team emerging from the individual behaviors of the teammates."
This collective action could also suggest why team sports are so popular. Given the universality of Lévy walks, which has been observed in 50-million-year-old fossilized trails of sea urchins as well as in the movement patterns of contemporary hunter-gatherers, it's possible that our fascination with games like football stems from their enactment of very ancient foraging patterns. The ability of individuals to coordinate and adapt their behavior to act as a unified agent in pursuit of a shared goal could also suggest that the fundamental principles of cooperation and efficiency resonate with us at a primal level.
The research not only highlights the observed prevalence of Lévy walks across particles, organisms, and now group behavior, but also opens new avenues for understanding collective dynamics that challenge traditional, single-brain centered views of cognition, showing that players can come together in team that behaves as a collective self.
