Football as foraging? Movements by individual players and whole teams exhibit Lévy walk dynamics

Many organisms, ranging from modern humans to extinct species, exhibit movement8 patterns that can be described by lévy walk dynamics. It has been demonstrated that such9 behavior enables optimal foraging when resource distribution is sparse. In this paper, we study a10 dataset of football player trajectories, recorded during the matches of the Japanese football11 league to elucidate the presence of statistical signatures of lévy walks; such as the heavy-tailed12 distribution of distances traveled between significant turns and the characteristic superdiffusive13 behavior. We conjecture that the competitive environment of a football game leads to movement14 dynamics reminiscent of that observed in hunter-gathering populations and more broadly in any15 biological organisms foraging for resources, whose exact distribution is unknown to them. Apart16 from analyzing individual players’ movements, we investigate the dynamics of the whole team by17 studying the movements of its center of mass (team’s centroid). Remarkably, the trajectory of the18 centroid also exhibits Lévy walk properties, which implies the presence of team-level19 coordination. Our work concludes with a comparative analysis of different teams and some20

on the relevance of our findings to sports science and science more generally.21 22 Introduction23 Play, broadly construed as an activity performed of free will for the individual’s pleasure, is widely24 considered deeply rooted in the human psyche. Homo sapiens are not the only mammals who25 engage in it and its influence on the formation of human culture and civilization can be debated26 but not neglected ( Huizinga, 2014). The commercial and societal success of team sports and ball27 games brought about a stream of scientific works seeking to elucidate factors informative of the28 successful performance of individual players and teams as a whole. In our work, we perform a29 quantitative analysis of trajectories recorded during the matches of the Japanese football league.30 Although we use some of the methods from the standard toolbox of quantitative football analysis,31 our main focus is to perform the game analysis implementing insights from the blooming field of32 research of animal movement analysis. Our conjecture is that individual player movements exhibit33 a fat-tailed step size distribution, which can be indicative of the Lévy walk dynamics.34 Lévy walk is a type of random walk, with its key characteristic being that the distribution of step35 lengths 𝑆 is drawn from a scale-free (power-law) distribution. Thus 𝑃 (𝑆) ∝ 𝑆 −𝑘 and 𝑘 lies in range36 1 < 𝑘 < 3. Particles exhibiting Lévy walk dynamics manifest superdiffusive behavior, spreading37 faster than brownian walkers. The resultant trajectory possesses fractal properties, with clusters38 of short steps interspersed with much longer sprints.39 It has been conjectured that Lévy dynamics possess several advantageous qualities to motile bi-40 1 of 11 .CC-BY-NC-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted June 13, 2024. ; https://doi.org/10.1101/2024.06.11.598528doi: bioRxiv preprint ological agents which seek to maximize their encounter rate with resources of some sort. Notably,41 it prevents the agent from revisiting already explored terrain, and combining numerous short steps42 with long leaps enables the balance of exploration and exploitation. A complete review of the rel-43 evant literature would be outside the scope of this paper; we refer the interested reader to books44 and reviews (Reynolds, 2018; Viswanathan et al., 2011 ). To briefly outline the landmarks works,45 the first quantitative evidence dates back to the ( Viswanathan et al., 1996 ), when Lévy walk be-46 havior was discovered in the movement patterns of wandering albatrosses. In a curious turn of47 events, the statistical analysis of this seminal paper was found to be inadequate( Edwards et al.,48 2007). However subsequent work, relying on improved statistical techniques showed that wander-49 ing albatrosses exhibit Lévy walk dynamics (Humphries et al., 2013). This story is not only a curious50 incident in the annals of science history, it underlines the difficulties of distinguishing between the51 Lévy walk and competing hypothesis, a topic which will be expounded further on in our work.52 Heavy-tailed step size distribution has been found in the diving pattern of aquatic predators53 (Sims et al., 2008 ), termite trajectories(Miramontes et al., 2014 ), T-cell movement (Harris et al.,54 2012), airborne seed dispersal (Reynolds, 2013 ), moving patterns of different types of terrestrial55 animals (Ríos-Uzeda et al., 2019 ; Ramos-Fernández et al., 2004 ), human mobility patterns inferred56 from cellphone data ( Gonzalez et al., 2008 ) and foraging patterns of individuals in hunter-gather57 populations (Reynolds et al., 2018 ). It is worth noting that there is evidence to suggest that such58 movement pattern is evolutionarily ancient, as it is evidenced by the discovery (Sims et al., 2014) of59 Lévy walks in the fossilized trails of sea urchins, which are around 50 million years old. Currently,60 much of the research, concerning Lévy movement patterns in animal foraging, accepts its de facto61 presence, while the main focus is on elucidating its generative mechanism, with hypothesizes in-62 cluding collective effects ( Reynolds and Ouellette, 2016), interactions between the foraging agent63 and its environment (de Jager et al., 2011 ; Sims et al., 2008) and inherent neural generators ( Sims64 et al., 2019).65 Concurrently with the exploration of animal foraging strategies quantitative analysis of football66 player movements has also blossomed in recent years (Sarmento et al., 2018). Several main venues67 of analysis emerged - examining the behavior of teams’ center of mass (centroid) ( Frencken et al.,68 2012, 2011), investigating correlations between player pairs ( Marcelino et al., 2020 ), and applying69 graph theoretic measures to complex networks derived from players’ interaction (Cotta et al., 2013;70 Buldú et al., 2019). Summarily, these works demonstrate that football teams exhibit a high level of71 cohesion and the motion of individual players is highly correlated, with their mates as well as with72 their opponents and centroid position, as well as player dispersion are informative of the games’73 dynamics. This research remains mostly confined to the sports science field, with virtually no inter-74 section with animal movement ecology and other quantitative behavioral science endeavors. We75 hope to address this discrepancy in our current work.76 Results77 We use high-resolution (25 fps, centimeter precision) trajectories of players recorded during the78 matches of the Japanese Football League (J-league) during the 2022 season. Data was acquired79 from the league’s official data company, DataStadium, Inc. In this work the study properties of80 player’s trajectory, most notably their step-size distribution, both for individual players and for81 the whole teams. The rest of this paper is structured as follows: in the first section, we analyze82 single-match while in the second section, we present summary statistics computed using data for83 all games in the 2022 season.84 Analysis of a Single match85 For the illustrative analysis presented below, we choose a game played between Hokkaido Con-86 sadole Sapporo and Nagoya Grampus, on the 30th of July 2022. The game was played on the pith87 of dimension 105 × 68, which is standard for J-league. The aforementioned teams are hereafter88 referred to as Team 1 and Team 2. This game ended in a draw with a score of 2:2. Trajectories of89 2 of 11 .CC-BY-NC-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted June 13, 2024. ; https://doi.org/10.1101/2024.06.11.598528doi: bioRxiv preprint 40 20 0 20 40 x cor meters 20 0 20 y cor meters A 60 40 20 0 20 40 x cor meters 20 0 20 y cor meters B 0 20 40 60 80 100 Time minutes 0 25 0 25 C 0 20 40 60 80 100 Time minutes 0 25 0 25 D 0 20 0 20 Figure 1. Pane A shows the ball’s trajectory during the first 5 minutes of the game. Pane B presents the trajectories of all players from both teams depicted with semitransparent dotted lines and trajectories of their respective centers of masses shown with solid lines. The first 5 minutes of the game are shown. The green color identifies Team 1, and the blue color identifies Team 2. Pane C depicts the mean distance 𝜇 of the player from their team’s center of mass for the game’s duration, while pane D shows the standard deviation from the center of mass position 𝜎. Inset plots in the right corner of panes B and D show normalized histograms of 𝜇 and 𝜎 respectively. all players for both teams have been used for analysis, whenever the player was present for the90 whole duration of the game or was substituted during its course. Both teams substituted players91 in the second half of the game, with team 1 substituting 3 players and team 2, 4 players.92 Smothering with a Gaussian kernel with 𝜎 = 3 is applied separately for the x and y components93 of the trajectories to remove possible recording artifacts. Sensitivity analysis was conducted to94 show that varying 𝜎 within reasonable bounds does not alter the results. To obtain a systems’ level95 description of the teams’ activity we computed the center of mass trajectory, for each team, as96 ⃗ 𝑟𝐶𝑀 = 1 𝑁 ∑𝑁 1 ⃗ 𝑟𝑖, 𝑁 = 11. To ascertain to what degree players tend to cluster during the game we97 computed the mean distance of players from the center of mass of their team 𝜇 = ⟨||⃗ 𝑟− ⃗ 𝑟𝐶𝑀 ||⟩98 and its standard deviation 𝜎 as functions of time. These results are presented in Figure 1. Note,99 that both those quantities exhibit significant variability over time suggesting alternation of contrac-100 tion and relaxation dictated by the dynamics of the game. Positions of the centers of masses for101 competing teams are visibly correlated.102 First insights into the nature of biological movement can be often obtained from computing103 the Mean Square Displacement (MSD)(1). Due to the small (N=11) number of agents in the field104 simultaneously, we used methods to approximate MSD from a single trajectory ( Michalet, 2010).105 For each player, we partitioned the entire trajectory of its movements into 20-second intervals and106 computed the squared average displacement.107 𝑀𝑆𝐷 (𝑡) = ⟨( ⃗ 𝑟0 − ⃗ 𝑟𝑡)2⟩ (1) As it is known from statistical mechanics ( Berg, 1993; Viswanathan et al., 2011 ) the exponent108 𝑎 relating the square of the displacement and time is informative of the type of motion: while in109 the case of Brownian motion, the variance of displacement is directly proportional to time, 𝑎 = 1110 higher or lower values indicate superdiffusive or subdiffusive motion respectively. Lévy walks are111 superdiffusive, therefore their exponents lie in the range 1 < 𝑎 < 2, between the Brownian motion112 3 of 11 .CC-BY-NC-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted June 13, 2024. ; https://doi.org/10.1101/2024.06.11.598528doi: bioRxiv preprint 100 101 Step size 10 3 10 2 10 1 100 P(Step size) k1 = 1.88 k2 = 1.43 A 100 0 100 E Goalkeper 101 Delay(s) 101 102 MSD m2 a = 1.4 I 100 101 Step size 10 3 10 2 10 1 100 P(Step size) k1 = 1.61 k2 = 1.08 B 100 0 100 F Player ID = 1 101 Delay(s) 101 102 MSD m2 a = 1.55 G 100 101 Step size 10 3 10 2 10 1 100 P(Step size) k1 = 1.65 k2 = 1.19 C 100 0 100 G Player ID = 4 101 Delay(s) 101 102 MSD m2 a = 1.55 K 100 101 Step size 10 3 10 2 10 1 100 P(Step size) k1 = 1.58 k2 = 1.03 D 100 0 100 H Player ID = 8 101 Delay(s) 101 102 MSD m2 a = 1.62 L Figure 2. Analysis of four players from team 1. Panes A, B, C, and D demonstrate the step size distribution. The solid red line corresponds to the truncated power-law fit, the magenta line shows the power-law fit and the dashed blue line shows the exponential distribution. Hollow black circles represent the normalized distribution histogram; logarithmic binning is used. Panes E, F, G, and H show the distribution of turning angles. Panes I, G, K, and L present the time vs MSD relation. The dashed red line shows the linear fit to the data. and the ballistic limit. As the third column of Figure 2 shows, values of 𝑎 computed for individual113 players show that their motion is superdiffusive.114 Super diffusive behavior is a crucial characteristic of the Lévy walk, however, it can exhibited by115 other modes of motion. For the definitive test, it is necessary to examine the step-size distribution.116 As the players’ trajectories are continuous and two-dimensional, we implement the methodology117 developed by (Humphries et al., 2013 ) for such scenarios. This approach uses the symmetry of118 Lévy walks: when projected to a lower dimension Lévy walk retains its properties and remains119 distinguishable from other modes of movement, such as Brownian motion. Considering the 1D120 component of trajectory significant turns can be unequivocally determined as reversions of motion121 in one dimension. Motion in X and Y dimensions are considered separately, as we assume that the122 rectangular field would impose different constraints on player mobility in vertical and horizontal123 directions and therefore different values of Lévy walk exponents. Different Lévy exponents for124 different axes of motion have been previously reported for airborne dispersal of seeds ( Reynolds,125 4 of 11 .CC-BY-NC-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted June 13, 2024. ; https://doi.org/10.1101/2024.06.11.598528doi: bioRxiv preprint 100 101 Step size 10 4 10 3 10 2 10 1 100 P(Step size) A k1 = 1.63 k2 = 1.08 Team 1 100 101 Step size 10 4 10 3 10 2 10 1 100 P(Step size) B k1 = 1.6 k2 = 1.0 Team 2 0 20 40 0.0 0.5 1.0 1.5 0 20 40 0.0 0.5 1.0 1.5 Figure 3. Step size distribution for centers’ of mass. Panes A and B depict the normalized distribution of step sizes for trajectories of the centers of mass of Teams 1 and 2, respectively. Black circles correspond to the real data, hollow circles represent synthetic data. The solid red line indicates the truncated power-law distribution, the dashed magenta line shows the power-law distribution and the blue line corresponds to the exponential fit. Insets show the the same data in the linear coordinates. Note the pronounced absence of the longer steps in the synthetic data. 2013), concerning horizontal and vertical dimensions.126 The first column of Figure 2 presents the distribution of step sizes for the Y dimension for127 the goalkeeper (Pane A) and 3 field players with different roles (remaining panes). We used the128 Maximum-likelihood estimator(MLE) to fit the power-law and truncated power-law to the data ( Al-129 stott et al., 2014 ) as well as to conduct log-likelihood tests to compare truncated power-law with130 other candidate long-tailed distributions, such as log-normal and exponential (Figure 2). For all131 players, these tests have shown unambiguously (p»0.05) that the truncated power-law is the best132 fit for the data. Power-law without truncation gives a better fit than the exponential distribution133 but is superseded by both the log-normal distribution and the truncated power law. Power-law134 scaling extends more than two orders of magnitude in the Y-axis and is close to two orders of135 magnitude on the X-axis, therefore being close to fulfilling the proposed ( Stumpf and Porter, 2012)136 "rule on thumb" for elucidating scale-free phenomena in biological systems. The curtailed scale on137 the X-axis is caused by the size of the football pith. Values of 𝑘1 and 𝑘2 correspond to the Lévy walk138 exponent of regular and truncated power-laws; they are different for different players; most pro-139 nounced is the distinction between the goalkeeper and the field players. As is expected, power-law140 exponents 𝑘1 and 𝑘2 are inversely related to the MSD exponent 𝑎.141 Another characteristic of the true Lévy walk is that the distribution of turning angles between142 consecutive bouts of motion is close to uniform. We use the points previously identified as rever-143 sals of 1D motion in the Y dimension to compute the angle direction of movement before and144 compute players’ turning angles at these points. The turning angle distribution is presented in the145 central column of Figure 2. Observed distributions could be interpreted as follows: small turn-146 ing angles are rarely encountered, as the nature of the projection operation eliminated them from147 data, for 𝜃 > 0 the distribution is close to uniform. Uniform turning angle distribution distinguishes148 Lévy walk from the correlated random walk, which has similar superdiffusive properties. Such turn-149 ing angle distribution has been previously reported for marsupials ( Ríos-Uzeda et al., 2019) and150 wandering albatrosses (Humphries et al., 2013 ). Both these species have been found to exhibit151 Lévy walk behavior with their step size distribution best approximated by truncated power-law.152 5 of 11 .CC-BY-NC-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted June 13, 2024. ; https://doi.org/10.1101/2024.06.11.598528doi: bioRxiv preprint Additionally, we study the distribution of the step sizes for movements of the centers of masses153 of both teams. We compute the center of mass positions as described previously and then com-154 pute the step size distributions for movements in Y-dimension. As a control, we construct surrogate155 data by performing a circular random shift of individual players’ position timieseries before com-156 puting the team’s center of mass position. These results are presented in Figure 3. An inspection157 of data presented on logarithmic and linear axes (inset) yields that longer steps are completely158 absent from the surrogate data. Max-likelihood fits prove that the step size distribution of the real159 data is best fitted with the truncated power-law distribution ( 𝑝 < 0.05).162 1.4 1.6 1.8 2.0 k1 20 30 40Dball R2 = 0.42 A 1.4 1.6 1.8 2.0 k1 10 20 30Dteam R2 = 0.24 B 20 30 40 Dball 10 20 30Dteam R2 = 0.61 C 20 30 40 Dball 0.00 0.02 0.04 0.06 0.08 0.10 D 10 20 30 Dteam 0.00 0.02 0.04 0.06 0.08 0.10 E 1.4 1.6 1.8 2.0 k1 0 2 4 6 8 F 40 45 0.0 0.2 30 35 0.0 0.2 1.75 2.00 0 5 Figure 4. Lévy walk exponents and players’ performance. Pane A depicts the relationship between the 𝑘1 exponent of the step size distribution and mean distance to the ball ⟨𝐷𝑏𝑎𝑙𝑙⟩, while pane B shows how players Lévy walk exponent is related to the average distance to the center of his team ⟨𝐷𝑡𝑒𝑎𝑚⟩ during the game. Pane C presents the relationship between the ⟨𝐷𝑏𝑎𝑙𝑙⟩ and ⟨𝐷𝑡𝑒𝑎𝑚⟩. The regression slope is given with a solid red line. All distances are expressed in meters. Hollow green triangles correspond to the data of goalkeepers. Panes D, E, and F show the distributions (normalized histograms) of ⟨𝐷𝑏𝑎𝑙𝑙⟩, ⟨𝐷𝑡𝑒𝑎𝑚⟩, and 𝑘1 for all players analyzed. Distributions of these quantities for goalkeepers are presented in the inset plots. Analysis of multiple games163 Additionally, we analyze all games contained in the 2022 dataset to elucidate if the Lévy walk expo-164 nent is in some way related to the player’s performance or his role in the team. To acquire insight165 into the game’s dynamics, for each player we compute two parameters: the mean distance of the166 player from the ball ⟨𝐷𝑏𝑎𝑙𝑙⟩ and his mean distance from the center of mass of the team ⟨𝐷𝑡𝑒𝑎𝑚⟩.167 The former is meant to characterize players’ efficiency in getting to target while the latter seeks168 to quantify their level of involvement in team dynamics. Due to the distinct rules governing the169 goalkeepers’ movements, we choose to omit their data from the linear regression model, they are170 presented on the plots for the visual reference only.171 Panes A and B in Figure 4 demonstrate that a statistically significant relationship exists between172 the power-law exponent of the Lévy walk distribution and the player’s distance to the ball, as well173 6 of 11 .CC-BY-NC-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted June 13, 2024. ; https://doi.org/10.1101/2024.06.11.598528doi: bioRxiv preprint as with his distance to the center of his team (𝑝 < 0.05) in both cases. Pane C shows the relationship174 between the ⟨𝐷𝑏𝑎𝑙𝑙⟩ and ⟨𝐷𝑡𝑒𝑎𝑚⟩. As the ball drives the dynamics of the football game this connection175 is to be expected. We have found that although the truncated power-law provides a better fit to176 the data, its exponent 𝑘2 has a lesser connection to the player’s dynamics, therefore 𝑘1 was used.177 Lower panes D, E, and F show the distribution of the ⟨𝐷𝑏𝑎𝑙𝑙⟩,⟨𝐷𝑡𝑒𝑎𝑚⟩, and 𝑘1 for all players, excluding178 the goalkeepers. All said quantities are distributed normally, as the Kolmogorov-Smirnoff test for179 normality confirms.180 Additional insight can be acquired by computing, for each player a convex hull - a minimal181 polygon that encompasses all points in the trajectory. Its area is informative of the portion of the182 field explored by each player during the game. Convex hulls are computed for each player, two183 separate values are computed for two parts of the game, their average is used. As it is illustrated184 by Figure 5 there is a statistically significant ( 𝑝 < 0.05) relationship between the hull area and 𝑘1,185 which implies that trajectories containing a higher portion of longer steps tend to encompass a186 larger area. An analogous relationship exists between the area of the hull and ⟨𝐷𝑏𝑎𝑙𝑙⟩. It should be187 noted that the portion of explained variance is relatively small in both cases.188 50 0 50 x cor 40 20 0 20 40 y cor A 1.4 1.6 1.8 2.0 k1 2000 4000 6000Area m2 R2 = 0.18 B 20 30 40 Dball 2000 4000 6000Area m2 R2 = 0.14 C Figure 5. Lévy walks exponents and the area coverage. Pane A exhibits two illustrative trajectories for players from one team for the first half of the game. Red and green dashed lines show concave hulls encompassing the players’ trajectory. The area of the green hull is ≈ 7304 𝑚2 and the red ≈ 3889 𝑚2. Lévy exponents 𝑘1 computed for the whole duration of the game for these players are 1.6 and 1.61, respectively. Pane B presents the relationship between the players’ hull and 𝑘1 and pane C shows the relationship between the players’ hull and his mean distance to the ball ⟨𝐷𝑏𝑎𝑙𝑙⟩. Solid red lines show linear regression fits, blue circles correspond to field players, and green triangles to goalkeepers. As discussed later Lévy walk exponent can be seen as an adaptive property for the organisms,189 therefore its slope could be influenced by both the player’s skill and their role in the team. For190 example, the goalkeepers’, whose movements are constrained by the rules the latter is the case.191 At the same time, previous research indicates ( Cabrera and Milton, 2004 ), that in human subjects192 the exponent changes as the subject becomes more proficient in the task. At this stage, we make193 no claims whenever the observed relationship between the players’ metric and 𝑘1 is determined194 solely by their role in the team or if it is affected by their skill level as well.195 Discussion196 Lévy walk-like behavior has been repetitively identified in human and animal locomotion. Although197 some of these findings have been challenged, due to improper data analysis most of them with-198 stood the most rigorous statistical test. To the best of our knowledge, Lévy flight has not yet been199 confirmed in any of the sport’s practitioners, despite the fact the quantitative analysis of trajecto-200 ries of professional players is becoming a common practice (Sarmento et al., 2018 ). Currently, the201 discourse surrounding the Lévy walk research is starting to shift from a mere debate regarding202 its plausibility in human and animal locomotion to a more nuanced discussion of its origin and203 purpose (Reynolds, 2018).204 7 of 11 .CC-BY-NC-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted June 13, 2024. ; https://doi.org/10.1101/2024.06.11.598528doi: bioRxiv preprint As it is implied by the wide body of research, the Lévy walk is an optimal foraging strategy205 when the target resource is sparsely distributed. Analytical methods and numerical experiments206 demonstrated (Viswanathan et al., 1999) that 𝑘 ≈ 2 is the optimal value for patchy resource distri-207 bution. Studies of diving patterns of sea mammals suggest that scale-free movement distribution208 is prompted by the scale-free distribution of prey (Sims et al., 2008) density. Such a view is con-209 firmed by numerical experiments which show that particles performing random walks in a fractal210 environment exhibit Lévy flight behavior and by studies of hunter-gatherers who resort to the Lévy211 behavior to navigate complex fractal-like landscapes ( Reynolds et al., 2018). Thus, one could argue212 that Lévy walk is an emergent phenomenon that arises due to the interaction between the organ-213 ism and its environment.214 At the same time, recent research ( Sims et al., 2019) demonstrated that Lévy walk behavior is215 found in Drosophila larvae when their synaptic brain activity is chemically inactive and their ability216 to perceive environmental cues is thus absent. This implies the intrinsic cause of Lévy walk distribu-217 tion, rooted in autonomous nervous activity. Examination of fossil traces revealed Lévy patterns in218 the trajectories of extinct isopods ( Sims et al., 2014), implying that such behavior is ancient in the219 evolutionary sense. Another hypothesis implies that Lévy walks arise due to collective interactions:220 proportion of longer steps in trajectories of individual midges increases as the swarm becomes221 bigger (Reynolds and Ouellette, 2016).222 In our view, the game of football could be viewed as the type of foraging behavior in which223 agents seek to locate and engage the "resource" - the ball. This conjecture is supported by the224 observed relationship between the ⟨𝐷𝑏𝑎𝑙𝑙⟩ and the power-law exponent 𝑘1. On other hand, football225 is a team endeavor in which individuals seek to coordinate their activity with their teammates and226 opponents. Some elements of the apparent coordination can be glimpsed from Figure 1, where we227 see periods of contraction and relaxation as well as evident interdependence between the trajecto-228 ries of centroids of opposing teams. Curiously, intermittent dynamics of contraction and relaxation229 have been discovered in grazing sheep herd ( Ginelli et al., 2015) where they serve to balance con-230 flicting imperatives: optimal area coverage provided by the dispersed state and safety in numbers231 enabled by dense clustering. It a is plausible conjecture that these oscillations serve a similar role232 in teams’ dynamics during the game.233 Despite evident similarities between the behavior of football players and that of foraging ani-234 mals, direct analogy is implausible due the the multiple added levels of behavioral complexity and235 high level of coordination between different players. Furthermore, presumed resource - ball is it-236 self mobile and its movements are influenced by the players activity. Therefore it stands to reason237 that the Lévy exponents computed for football players are quite far from the optimal value of 𝑘 = 2238 found in simulation studies for sparse resource distribution - football players’ trajectories contain239 a higher portion of long steps. It is to early to suggest a definitive generative mechanism for the240 observed behavior, however, we conjecture that it arises from the interplay of the foraging needs241 and collective effects, both of which are present during the football game.242 Said collective effects manifests themselves when we study the centers of masses dynamics,243 which exhibit Lévy walk dynamics as well as individual players. First of all, this finding appears244 unusual in the context of Lévy walk research, as this field of study invariantly focuses on individuals.245 However, then a spectrum of work regarding sports team dynamics is considered such behavior246 becomes expected, as high correlation between different team members is considered a hallmark247 of team behavior ( Marcelino et al., 2020) and is indicative of its performance in the match. We248 interpret these findings as evidence of foraging activity performed not by a single individual but by249 a group, acting to maximize its collective performance. The "resource" foraged for, would be not250 a material object but a spatial configuration optimal to the needs of the moment. In our view, it251 would be instructive to investigate if similar phenomena could be observed in animals who practice252 collective hunting, such as wolves and hyenas.253 One way to consider the broader significance of the uncovered phenomena is by invoking the254 work of Levin (Levin, 2019, 2023). One of his conjectures is that the impetus to understating the255 8 of 11 .CC-BY-NC-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted June 13, 2024. ; https://doi.org/10.1101/2024.06.11.598528doi: bioRxiv preprint functioning of complex systems can be gained from expanding the definition of "Self": leaving the256 idea of selfhood as something reserved to a single embodied and cognizant entity, such as a human257 or animal and entertaining the possibility of expanded or reduced selves, examples of which can258 be collectives or subsystems of an organism. "Selves" in such a paradigm should be treated as259 integrated entities capable of pursuing goals by modifying their behavior.260 Once such a standpoint is taken concerning the Football team dynamics presence of Lévy walk261 behavior which was previously only found in individuals in a collective becomes something that262 could easily be anticipated: a football team is an entity that has a goal and can pursue it by modify-263 ing its behavior, both during the game and before it in the training process. It comes as no surprise264 then, that the movement strategy that the "Team self" adapts to optimize its performance shares265 statistical properties with the movement strategies adopted by individual "Selves" who experience266 similar environmental pressures.267 Conclusion268 In our work, we showed that football players’ trajectories during the game manifest Lévy walk269 dynamics. Furthermore, Lévy walks exponents, which characterize the preponderance of longer270 steps in the distribution are related to the players’ role during the game, players with lower ex-271 ponents are on average closer to the ball and explore a larger portion of the field. Furthermore,272 power-law distribution of step sizes is present in the collective description of players’ activity (cen-273 ter of mass trajectory), an observation novel to the Lévy flight discourse, as it has been centered274 on the individual trajectory properties. At this stage we make no definitive claims regarding the275 generative mechanism of observed behavior, however, we theorize that the scale-free distribution276 of step sizes is influenced by both the collective nature of the game and its inherent "foraging"277 premise. Further work should include modeling to elucidate generative mechanism as well as ad-278 ditional inquiry into the relationship between the Lévy walk exponent and other trajectory metrics279 and players’ performance. An alternative road, for other disciplines might be to study if the ele-280 ments of foraging present in the football game might explain its widespread popularity.281

and Materials282 We obtained match recordings of Japan’s top football league, J-League, by licensing two datasets283 from the league’s official data company, DataStadium, Inc. Each dataset contains time-stamped284 trajectories of players from both teams for the sequence of games in which the season’s winning285 team participated. The data was acquired using the TRACAB’s optical video tracking systems Gen IV286 (Linke et al., 2020). In brief, this technology relies on gathering optical data from two multicamera287 units, located at both sides of the midfield line, which is then used to reconstruct players’ trajecto-288 ries. Ball position is acquired in a separate process: when a player has the ball, his position is taken289 as a proxy for the ball’s coordinates, in between these events the position of the ball is obtained290 by linear interpolation.291 The original data format is 25 HZ and the spatial resolution of the recording is in cm. Coordi-292 nates of all players from both teams were recorded, as well as the ball trajectory. To investigate293 if the temporal resolution of the data affects the results we have created several downsampled294 datasets with temporal resolution ranging from 1HZ to 25HZ and repeated our analysis. No statis-295 tically significant differences were observed between different downsamplings of the data.296 All computations for the paper have been performed using custom scripts written in python297 programming language and are available from authors upon reasonable request. For fitting the298 power-law distribution to the data and performing log-likelihood powerlaw python package was299 used (Alstott et al., 2014). 𝑥𝑚𝑖𝑛 value was set to 0.5 meters, to exclude potential spurious fluctua-300 tions.301 9 of 11 .CC-BY-NC-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted June 13, 2024. ; https://doi.org/10.1101/2024.06.11.598528doi: bioRxiv preprint Acknowledgments302 This work was supported by JST Grant Number JPMJPF2205303 References304 Alstott J, Bullmore E, Plenz D. powerlaw: a Python package for analysis of heavy-tailed distributions. PloS one.305 2014; 9(1):e85777.306 Berg HC. Random walks in biology. Princeton University Press; 1993.307 Buldú JM, Busquets J, Echegoyen I, Seirul lo F. Defining a historic football team: Using Network Science to308 analyze Guardiola’s FC Barcelona. Scientific reports. 2019; 9(1):13602.309 Cabrera JL, Milton JG. Human stick balancing: tuning Lévy flights to improve balance control. Chaos: An310 Interdisciplinary Journal of Nonlinear Science. 2004; 14(3):691–698.311 Cotta C, Mora AM, Merelo JJ, Merelo-Molina C. A network analysis of the 2010 FIFA world cup champion team312 play. Journal of Systems Science and Complexity. 2013; 26(1):21–42.313 Edwards AM, Phillips RA, Watkins NW, Freeman MP, Murphy EJ, Afanasyev V, Buldyrev SV, da Luz MG, Raposo314 EP, Stanley HE, et al. Revisiting Lévy flight search patterns of wandering albatrosses, bumblebees and deer.315 Nature. 2007; 449(7165):1044–1048.316 Frencken W, Lemmink K, Delleman N, Visscher C. Oscillations of centroid position and surface area of soccer317 teams in small-sided games. European journal of sport science. 2011; 11(4):215–223.318 Frencken W, Poel Hd, Visscher C, Lemmink K. Variability of inter-team distances associated with match events319 in elite-standard soccer. Journal of sports sciences. 2012; 30(12):1207–1213.320 Ginelli F, Peruani F, Pillot MH, Chaté H, Theraulaz G, Bon R. Intermittent collective dynamics emerge from con-321 flicting imperatives in sheep herds. Proceedings of the National Academy of Sciences. 2015; 112(41):12729–322 12734.323 Gonzalez MC, Hidalgo CA, Barabasi AL. Understanding individual human mobility patterns. nature. 2008;324 453(7196):779–782.325 Harris TH, Banigan EJ, Christian DA, Konradt C, Tait Wojno ED, Norose K, Wilson EH, John B, Weninger W, Luster326 AD, et al. Generalized Lévy walks and the role of chemokines in migration of effector CD8+ T cells. Nature.327 2012; 486(7404):545–548.328 Huizinga J. Homo ludens ils 86. Routledge; 2014.329 Humphries NE , Weimerskirch H, Sims DW. A new approach for objective identification of turns and steps330 in organism movement data relevant to random walk modelling. Methods in Ecology and Evolution. 2013;331 4(10):930–938.332 de Jager M, Weissing FJ, Herman PM, Nolet BA, van de Koppel J. Lévy walks evolve through interaction between333 movement and environmental complexity. Science. 2011; 332(6037):1551–1553.334 Levin M. The computational boundary of a “self”: developmental bioelectricity drives multicellularity and scale-335 free cognition. Frontiers in psychology. 2019; 10:493866.336 Levin M. Darwin’s agential materials: evolutionary implications of multiscale competency in developmental337 biology. Cellular and Molecular Life Sciences. 2023; 80(6):142.338 Linke D, Link D, Lames M. Football-specific validity of TRACAB’s optical video tracking systems. PloS one. 2020;339 15(3):e0230179.340 Marcelino R, Sampaio J, Amichay G, Gonçalves B, Couzin ID, Nagy M. Collective movement analysis reveals341 coordination tactics of team players in football matches. Chaos, Solitons & Fractals. 2020; 138:109831.342 Michalet X. Mean square displacement analysis of single-particle trajectories with localization error: Brownian343 motion in an isotropic medium. Physical Review E. 2010; 82(4):041914.344 Miramontes O, DeSouza O, Paiva LR, Marins A, Orozco S. Lévy flights and self-similar exploratory behaviour345 of termite workers: beyond model fitting. PloS one. 2014; 9(10):e111183.346 10 of 11 .CC-BY-NC-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted June 13, 2024. ; https://doi.org/10.1101/2024.06.11.598528doi: bioRxiv preprint Ramos-Fernández G, Mateos JL, Miramontes O, Cocho G, Larralde H, Ayala-Orozco B. Lévy walk patterns in347 the foraging movements of spider monkeys (Ateles geoffroyi). Behavioral ecology and Sociobiology. 2004;348 55:223–230.349 Reynolds AM, Ouellette NT. Swarm dynamics may give rise to Lévy flights. Scientific reports. 2016; 6(1):30515.350 Reynolds A, Ceccon E, Baldauf C, Karina Medeiros T, Miramontes O. Lévy foraging patterns of rural humans.351 PlOS one. 2018; 13(6):e0199099.352 Reynolds AM. Beating the odds in the aerial lottery: passive dispersers select conditions at takeoff that maxi-353 mize their expected fitness on landing. The American Naturalist. 2013; 181(4):555–561.354 Reynolds AM. Current status and future directions of Lévy walk research. Biology open. 2018; 7(1):bio030106.355 Ríos-Uzeda B, Brigatti E, Vieira M. Lévy like patterns in the small-scale movements of marsupials in an unfa-356 miliar and risky environment. Scientific reports. 2019; 9(1):2737.357 Sarmento H, Clemente FM, Araújo D, Davids K, McRobert A, Figueiredo A. What performance analysts need358 to know about research trends in association football (2012–2016): A systematic review. Sports medicine.359 2018; 48:799–836.360 Sims DW, Humphries NE, Hu N, Medan V, Berni J. Optimal searching behaviour generated intrinsically by the361 central pattern generator for locomotion. Elife. 2019; 8:e50316.362 Sims DW, Reynolds AM, Humphries NE, Southall EJ, Wearmouth VJ, Metcalfe B, Twitchett RJ. Hierarchical ran-363 dom walks in trace fossils and the origin of optimal search behavior. Proceedings of the National Academy364 of Sciences. 2014; 111(30):11073–11078.365 Sims DW, Southall EJ, Humphries NE, Hays GC, Bradshaw CJ, Pitchford JW, James A, Ahmed MZ, Brierley AS,366 Hindell MA, et al. Scaling laws of marine predator search behaviour. Nature. 2008; 451(7182):1098–1102.367 Stumpf MP, Porter MA. Critical truths about power laws. Science. 2012; 335(6069):665–666.368 Viswanathan GM, Afanasyev V, Buldyrev SV, Murphy EJ, Prince PA, Stanley HE. Lévy flight search patterns of369 wandering albatrosses. Nature. 1996; 381(6581):413–415.370 Viswanathan GM, Da Luz MG, Raposo EP, Stanley HE. The physics of foraging: an introduction to random371 searches and biological encounters. Cambridge University Press; 2011.372 Viswanathan GM, Buldyrev SV, Havlin S, da Luz MG, Raposo EP, Stanley HE. Optimizing the success of random373 searches. nature. 1999; 401(6756):911–914.374 .CC-BY-NC-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted June 13, 2024. ; https://doi.org/10.1101/2024.06.11.598528doi: bioRxiv preprint