Football Games Consist of a Self-Similar Sequence of Ball-Keeping Durations

Abstract

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.

1. Introduction

A football game (soccer) is a self-organized collective behavior that emerges from social interactions between players. Self-organizing properties resulting from cooperation and competition with the ball are supported by previous observations that the time-series data of player and ball positions in the field follow the Zipf–Mandelbrot law [1] and show self-similarity [2]. In another trial, the concept of fractals was extended to create new descriptors to quantitatively express the spatial organization of football players in the field [3]. Furthermore, the spatiotemporal pattern of passes between players involves specific types of patterns, depending on team performance, in which the pattern bifurcation can be described from the perspective of network dynamics [4]. Recently, the relationship between tactical organization and technical performance [5], as well as the positional relationships between players [6] during football games, has been explored using the recurrence plot analysis.

In the nonlinear science community, football games have long been considered complex systems. The main research objective was to understand the self-organization process in collective dynamics resulting from strong interactions between the ballplayer system. These interactions are similar to those applied in various living matters and social systems [7]. Nevertheless, room remains to theorize consistency in self-organization between two different spatiotemporal scales: the local scale (i.e., the individual player level) and the global scale (i.e., the whole-game level). In general, in complex systems, the behavior of local elements affects global spatiotemporal patterns and vice versa. In a football game, an inference can be that interactions across different spatiotemporal scales similarly strongly influence the flow of the game. A study suggests that both global-to-local and local-to-global self-organizing tendencies shape collective behavior and that effective training should harness this bi-directional interplay [8]. Therefore, clarifying the relationship between the two scales in collective dynamics is expected to lead to the discovery of an unknown hierarchical fractal structure hidden in the self-organizing properties of football games.

One way to solve the aforementioned problem is to identify self-similarity in the temporal distribution of ball-keeping duration. Throughout the game, players (or teams) in possession of the ball change over time. At the local scale, for example, once a player possesses the ball, the possession lasts for some time. However, as soon as this period ends, another player begins to possess the ball. Thus, the flow of football games from a local perspective is constructed by using the sequence of players’ ball possession durations. A duration can be long or short, depending on the player’s intentions and the game situation. A similar understanding will be held even for whole-game-level dynamics. At such a global scale, the in-play time begins with an event such as kickoff, throw-in, goal kick, or corner kick. It then ends with an event such as a foul or the ball crossing the touchline. In addition, the in-play duration varies, depending on the game situation that emerges from all players’ interactions. For example, if the ball crosses the touchline immediately after the game starts, the duration of the in-play is reduced. However, if many passes occur between players on the defensive side, the in-play duration is longer. Therefore, from a global perspective, the game flow is constructed via the sequence of in-play durations with different lengths, similar to the case at the local scale. Moreover, in both scales, short durations occur frequently, whereas long durations occur only rarely. Based on these considerations, we hypothesize that the ball-keeping durations at both the local and global scales follow similar temporal probability distributions characterized by fractal self-similarity. Specifically, we expect both distributions to exhibit heavy-tailed behavior, where short possession durations are frequent and long durations are rare. This suggests that individual local player actions and global-level game flow may be governed by the same underlying dynamical principles, reinforcing the hierarchical fractal structure of self-organization in football.

A touchstone for verifying this hypothesis is to examine whether the probability distribution of duration follows a power law. Power-law distribution often describes the stochastic properties of natural and living matter [9,10], implying the self-similarity inherent to the system. Even in human activities, the power law occurs in collective behavior such as city populations [11], block area distribution in a downtown [12], severe terrorist events [13], the behavior of the market [14], and the relationship between rank and points in football [15]. The interevent timing of many human activities driven by decision-making, more importantly, follows a power-law distribution. Examples include e-mail communications [16] and letter-correspondence patterns [17]. These facts support the validity of the hypothesis because the flow of a football game is a consequence of a series of decision-making processes performed at the individual player and team levels. While other distributions, such as exponential or Weibull models, are commonly used to analyze duration data, we primarily selected the power-law distribution because of its ability to capture heavy-tailed, scale-invariant behavior. Against this backdrop, we examined the probability distribution of the ball-keeping duration in real football games to determine whether self-similarity governs the collective dynamics of the ball player system.

2. Methods

2.1. Dataset

We analyzed 10 real football games held in the J1 League, the top division of the Japan Professional Football League, in 2015. The games were selected such that all 18 teams participating in the league were included at least once. In addition, the games were chosen from different points in the season to account for potential variations. The event dataset, extracted from the video data, was provided by DataStadium Inc. (Tokyo, Japan), a private company that analyzes and distributes sports data. In this study, we used event data from the time when the player or team in possession of the ball changed (i.e., the time that elapsed after the start of the game) and information on the players involved in the event (i.e., team affiliations and uniform numbers). The data resolution was set to 30 Hz.

The study was conducted in accordance with the Declaration of Helsinki. This study was approved by the Ethics Committee of the Nagoya University Research Center of Health, Physical Fitness, and Sports (Nagoya, Japan; approval no. 28-23). The company providing the dataset for this study was authorized by the Japan Professional Football League to acquire the sports data and sell it to third parties. This contract does not infringe the rights of players and teams, and it includes their consent for research use. This manuscript does not contain any information or images that could lead to the identification of a player or team.

2.2. Definition of Variables

To elucidate the fractal structural properties in the sequence of duration at the local and global scales, we recently defined two-time variables: the game variable $T_g$ and the player variable $T_p$. Figure 1 presents the schematics of these definitions.

The game variable $T_g$ represents the duration for which the game is actively in progress—that is, when the ball is in play and possession is continuously maintained by one or more players from one or both teams. It is the duration from the time interval $g_{\mathrm{start}}$ to $g_{\mathrm{end}}$. In this context, $g_{\mathrm{start}}$ denotes the moment when an event occurred, as in the following examples: kickoff, throw-in, goal kick, free kick, corner kick, penalty kick, and dropped ball. Time $g_{\mathrm{end}}$ denotes the moment at which one of the following events occurs: the ball crosses the touchline or the endline, offside, and foul. The player variable $T_p$ similarly represents the duration for which an individual player keeps a ball. Based on these definitions, a single $T_g$ is created by connecting multiple $T_p$ continuously with no gaps (Figure 1). The variable $p_{\mathrm{switch}}$ refers to the moment at which ball control is transferred from one player to another player, whether it is a teammate or an opponent. Possible examples of individual plays corresponding to this moment are traps, catches, tackles, and blocks.

In the actual data analysis, we determined the moment at which a local or global event occurred by using the aforementioned event data. We evaluated all values of $T_p$ and $T_g$ throughout the 10 games. By using these results, we calculated the probability distributions of $T_p$ and $T_g$ for each of the 10 games.

2.3. Power-Law Fitting

To examine whether the obtained probability distribution curves of $T_p$ and $T_g$ follow the power law, we used the power-law fitting method. This method is combined with the maximum likelihood fitting method, using a goodness-of-fit test [9]. This combined method is superior to ordinary least squares fitting in the sense that it can detect the minimum value in the $T_p$ (or $T_g$) range when applying power-law fitting.

First, we assumed that the probability density function (PDF) of the variable T ($T=T_g$ or $T_p$), evaluated by using real football game data, will follow, partly or as a whole, a negative power-law function $p\left(T\right)$ with a normalization constant, defined by

$$p\left(T\right)\propto T^{-\alpha }.$$

(1)

The exponent $\alpha$ had a positive value. The cumulative distribution function (CDF), denoted by $P\left(T\right)$, derived from Equation (1), should be

$$P\left(T\right)={\int }_T^{\infty }p\left(T^{\prime }\right)d{T}^{\prime }\propto T^{-\alpha +1}.$$

(2)

We thereafter assumed that the data follow the power law only in the range $T\ge T_{\min }$, in which the lower bound $T_{\min }$ of T for the power-law fitting was determined by the goodness-of-fit test, as explained later.

Second, we estimated the parameter ${\alpha }^{\prime }$ using the maximum likelihood estimation (MLE) method, which is commonly applied when fitting power-law distributions. Assuming that the variable T follows a continuous power-law distribution of the form, the normalized probability density function (PDF) over the domain $T\ge T_{\min }$ is given by

$$p\left(T\right)={\displaystyle \frac{\alpha -1}{T_{\min }}}{\left({\displaystyle \frac{T}{T_{\min }}}\right)}^{-\alpha },$$

(3)

where $\alpha$ is the scaling parameter and $T_{\min }$ is the minimum value at which power-law behavior holds. The log-likelihood function for a dataset is maximized when the exponent is given by

$${\alpha }^{\prime }=1+n{\left(\sum _{i=1}^{n}ln{\displaystyle \frac{T_i}{T_{\min }}}\right)}^{-1}.$$

(4)

Here, $T_i$, $i=1,2,\dots ,n,$ are the observed values of T such that $T_i\ge T_{\min }$. The prime in $\alpha$ indicates that this is an empirical estimate obtained from the data. Notably, the quantity $1/({\alpha }^{\prime }-1)$ corresponds to the arithmetic mean of the logarithmic ratios $ln(T_i/T_{\min })$. This reflects how far, on average, the observed values extend above the threshold $T_{\min }$ on a logarithmic scale. A smaller average implies a steeper distribution and, hence, a larger estimated exponent ${\alpha }^{\prime }$. For a given $T_{\min }$, the pair of values $({\alpha }^{\prime },n)$ is uniquely determined. In particular, we only needed to find the value of $T_{\min }$ that resulted in the optimal power-law fitting.

The specific value of $T_{\min }$ can be determined by calculating as follows [9]:

$$D=\underset{T\ge T_{\min }}{max}|S\left(T\right)-P\left(T\right)|.$$

(5)

The variable $S\left(T\right)$ is the CDF of the obtained data points. Thus, D quantifies the difference in the CDF curves between the measured data $S\left(T\right)$ and the fitting function $P\left(T\right)$ within a limited range of T as $T\ge T_{\min }$. Based on Kolmogorov–Smirnov test results, the optimal value of $T_{\min }$ was chosen to minimize D. Once $T_{\min }$ was determined, we substituted it into Equation (4) to obtain the corresponding maximum likelihood estimate of the power-law exponent ${\alpha }^{\prime }$. To assess the plausibility of the power-law model, we followed the method proposed by Clauset et al. [9]. The p-value is computed to be the fraction of the synthetic distances that are larger than the empirical value D. Therefore, if the p-value is larger, then the observed deviation between the empirical data and the model is consistent with statistical functions; if it is small, the model is not a plausible fit to the data [9]. Analysis of the power-law fitting was conducted by using MATLAB software (https://www.mathworks.com/products/matlab.html, accessed on 16 June 2025), which was provided by Aaron Clauset (https://aaronclauset.github.io/powerlaws/, accessed on 31 August 2020).

3. Results

3.1. Time-Series Graphs of $T_g$ and $T_p$

Figure 2a,b show the time-series changes in $T_g$ and $T_p$ in a football game labeled No. 10 among the 10 games investigated. The horizontal axis value of the data point represents the time at which a team or player newly gained possession of the ball (i.e., the time elapsed after the kickoff). The vertical axis values of the data points represent the duration that the team or player continued to possess the ball.

The number of datapoints for $T_g$ and $T_p$, denoted by $n_g$ and $n_p$, respectively, are $n_g=111$ and $n_p=1199$ for the case of No.10. The value of $n_p$ is an order of magnitude greater than $n_g$ because, as is clear from Figure 1, a single duration $T_g$ includes the multiple durations of $T_p$. The average values of $n_g$ and $n_p$ for all 10 games with standard deviations of $n_g=109\pm 13$ and $n_p=1238\pm 121$, respectively, are listed in

Table 1. List of the fitting parameters obtained for each of the 10 football games investigated. The parameter values in bold font indicate that the p-value associated with them is 0.1 or higher and thus is statistically significant. The mean value and standard deviation (SD) of $T_{\min }$, $\alpha$, and $n_{\mathrm{tail}}$ were calculated by employing only values with a p-value of 0.1 or higher.

Match No.	Observed Number		Lower Bound		Exponent		Analyzed Number		p-Value
	${\mathit{n}}_{\mathit{g}}$	${\mathit{n}}_{\mathit{p}}$	${\mathit{T}}_{{\mathbf{m}\mathbf{i}\mathbf{n}}_{\mathbf{g}}}$	${\mathit{T}}_{{\mathbf{m}\mathbf{i}\mathbf{n}}_{\mathbf{p}}}$	${\mathit{\alpha }}_{\mathit{g}}$	${\mathit{\alpha }}_{\mathit{p}}$	${\mathit{n}}_{{\mathbf{t}\mathbf{a}\mathbf{i}\mathbf{l}}_{\mathbf{g}}}$	${\mathit{n}}_{{\mathbf{t}\mathbf{a}\mathbf{i}\mathbf{l}}_{\mathbf{p}}}$	${\mathit{p}}_{\mathit{g}}$	${\mathit{p}}_{\mathit{p}}$
1	123	1311	29.3	4.8	3.08	4.19	47	138	0.44	0.32
2	112	1317	18.9	4.4	2.30	4.54	57	161	0.07	0.56
3	120	1246	39.9	3.5	3.32	3.54	26	254	0.85	0.78
4	108	1258	9.4	5.0	1.93	3.81	83	86	0.00	0.93
5	95	1369	39.7	4.1	2.63	4.22	34	235	0.00	0.13
6	126	1052	37.2	3.3	3.18	3.46	22	267	0.42	0.03
7	101	1342	67.9	4.4	4.44	4.36	19	176	0.23	0.35
8	81	1286	66.4	5.0	3.52	5.20	20	146	0.90	0.59
9	114	1005	30.8	3.0	3.10	3.41	35	284	0.26	0.38
10	111	1199	35.9	4.1	3.80	4.02	33	201	0.85	0.16
Mean	109.1	1238.5	43.9	4.2	3.4	4.1	28.9	186.8
SD	13.70	121.44	16.29	0.67	0.49	0.54	10.1	62.7

. Comparing the relationship between those average values of $n_g$ and $n_p$, we find that the number of game variables is approximately 11 times (Mean $n_p/n_g=11.35$) that of player variables. Since $n_g$ represents the number of in-play actions in a game, this suggests that, on average, about 11 players are involved per in-play. Although some in-play actions may involve players from both teams, this trend may reflect the fact that there are 11 players on a team.

The deviation of both variables follows from Figure 2a,b, which shows that both time-series graphs exhibited severe fluctuations, whereas the vertical axis values of the data points rarely became significantly large.

3.2. Histogram and CDF Curves

Figure 2c,d show the occurrence frequency of $T_g$ and $T_p$ for case No.10 in histogram representations. The widths of the bins were set to 10 s for $T_g$ and 1 s for $T_p$. In both graphs, the higher the value of T, the lower the occurrence frequency. Nevertheless, even if it is very small, the frequency of the occurrence of a large T is not zero. This finding implied a heavy tail in the CDF curves obtained from the histogram data. A similar trend existed in all histograms evaluated from the other game data (No. 1–No. 9). Figure 2e,f show the CDF curves of $T_g$ and $T_p$ for case No.10. The function decreases from zero; this form is commonly used in the analysis of power-law or heavy-tailed distributions [9], as it appears as a straight line on a logarithmic plot when the data follows a power law. As expected, both curves showed a rapid decay at small T values and long tails at large T values.

3.3. Exponential Distribution and Power-Law Distribution

For a better understanding of the decaying behavior, we displayed the semi-logarithmic plots (Figure 3a,b) and the logarithmic plots (Figure 3c,d) for case No. 10. The semi-logarithmic plot revealed that the datapoints tended to follow a straight line downward to the right for $T_g$ and $T_p$ (Figure 3a,b). This finding indicated that $P\left(T\right)$ at small T decays exponentially.

Regarding the logarithmic plot (Figure 3c,d), $P\left(T\right)$ tended to exhibit a negative power-law behavior at a large T value. Of the 10 game datasets investigated in this study, seven games for $P\left(T_g\right)$ and nine games for $P\left(T_p\right)$ exhibited a negative power-law with significant significance (see Table 1). With regard to the exponent $\alpha$, the mean values and standard deviations derived from $P\left(T_g\right)$ and $P\left(T_p\right)$, denoted by ${\alpha }_g$ and ${\alpha }_p$, respectively, were evaluated as $3.4\pm 0.4$ and $4.1\pm 0.5$. The corresponding 95% confidence intervals were [3.19, 3.79] for ${\alpha }_g$ [3.8, 4.48] for ${\alpha }_p$. The vertical dotted lines in Figure 3 represent the position of the lower bound $T_{\min }$ for the power-law fitting of $P\left(T_g\right)$ and $P\left(T_p\right)$.

3.4. Transition of the Power-Law Range Between $P\left(T_g\right)$ and $P\left(T_p\right)$

An intriguing observation is that when both $P\left(T_g\right)$ and $P\left(T_p\right)$ follow the power law, the following two positions almost coincide with each other: the right-end position of the power curve of $P\left(T_p\right)$ (i.e., the maximum value of $T_p$) and the left-end position from which $P\left(T_g\right)$ begins on the power-law curve (i.e., the position of $T_{\min }$ with respect to $P\left(T_g\right)$). For instance, in data No. 1, the power-law behavior of $P\left(T_p\right)$ ends at approximately 30 s, which is nearly equal to the lower bound $T_{\min }\simeq 30$ s of the power-law behavior of $P\left(T_g\right)$, as shown in Figure 4. The same tendency occurred for Nos. 3, 9, and 10, despite the few exceptions in Nos. 7 and 8 (see Supplementary Figure S1). The reason for the transition in the power-law range between $P\left(T_p\right)$ and $P\left(T_g\right)$ and the almost identical value for the transition position, which was nearly equal to 30 s, is discussed later in this paper.

4. Discussion

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 at the local scale (i.e., individual-player level) and at the global scale (i.e., whole-game level). This finding indicated that the time sequence of the ball-keeping duration had a self-similar structure with no characteristic timescale, whereas the scaling factor (i.e., the exponent $\alpha$) differed slightly between the local and global scales. The fractal nested structures of $T_g$ and $T_p$ are shown in Figure 1. One conclusion is that the flow of football games has a two-level hierarchical structure, reflecting fractal self-similarity across two distinct spatiotemporal scales [18].

The transition time from the $P\left(T_p\right)$ power-law regime to the $P\left(T_g\right)$ power-law regime was estimated to be 30 s, based on the plots shown in Figure 4. This finding is consistent with our previous work on self-similarity in football dynamics [2]. In Ref. [2], we analyzed the time-series variations in the team-turf boundary position and the ball location. The analysis revealed that they were governed by fractional Brownian motion (fBm) only within a timescale shorter than 30 s. This feature was also discussed in Ref. [2], in which correlations between team-turf boundary positions (or ball positions) at any two different times $t_1$ and $t_2$ in the time-series data became considerably smaller in the region $|t_1-t_2|>30$ s. A salient implication of our results for a transition time of 30 s is that the flow of a football game at a given moment is influenced by the movements of the players and the ball that had taken place over the past 30 s up to that moment. Thus, individual performances separated by more than 30 s, which is the so-called “memory limit,” exhibit little correlation with each other. In the present work, we revealed that the power law of $P\left(T_g\right)$ only occurs on timescales longer than the memory limit of 30 s. This quantitative consistency suggested that the transition from the power law of $P\left(T_p\right)$ below 30 s to that of $P\left(T_g\right)$ above 30 s is driven by the transition of the self-organizing process from that caused by the fBm mechanism, describing individual player performance, to those caused by a different unknown mechanism on the global scale.

Why the probability distribution of the ball possession duration is followed by a negative power law is difficult to explain, particularly on a global scale. However, it is possible to explain phenomenologically why the probability distribution exhibited a sharp decay with respect to time. One possible reason for the latter issue is diversity in the player’s role and the game’s situation throughout the game flow. At the local scale, each player has a unique role, depending on the position. In particular, the duration of a player’s ball-keeping is strongly affected by the degree of pressure from the opponent players. For instance, players such as goalkeepers or defenders are under less pressure from the opponent team; therefore, they sometimes keep the ball for a longer time than offensive players. However, these situations do not occur frequently. By contrast, offensive players are in a high-pressure situation. Therefore, the duration of ball-keeping is short, and the frequency of occurrence of this situation is more pronounced than that of the defender.

On a global scale, for instance, when events at the end of an in-play, such as the ball crossing the touchline or endline, offside, and fouls, occur at a high frequency, the duration of the game decreases. If these events do not end, the game duration will be longer. However, these situations do not often occur because the two teams compete for the ball. Both $P\left(T_p\right)$ and $P\left(T_g\right)$ consequently decay rapidly over time, as demonstrated in the present study.

In addition to the power-law behavior of $P\left(T\right)$ at large T value, a comment on the exponential decay of $P\left(T\right)$ at small T values is warranted. As illustrated in Figure 3, in the semi-logarithmic plot, the data points seem to follow a straight line downward to the right for $T_g$ and $T_p$. This finding indicated that $P\left(T\right)$ at small T values decays exponentially. Such an exponential distribution is widely used to describe the intervals between events with a constant average incidence. This finding is also a probability distribution, representing the occurrence of an event per unit of time or distance. For example, an exponential distribution can be used to determine the time until the next car accident occurs at an intersection [19], the time until the next (second) shooting star is seen in the night sky [20], or the distance between adjacent potholes [12]. In the present case, the exponential decay of $P\left(T\right)$ at small T values is a consequence of immediate turnover of the ball between teams or players occurring in real football games, with a constant average probability of occurrence. Such quick turnovers occur stochastically, regardless of the intention of the players or teams. For $P\left(T\right)$ to follow an exponential distribution in the region where T is small is reasonable.

Finally, we would like to discuss the future directions and limitations of the present study. This study focused on the structural similarity between local and global scales. Therefore, in contrast to standard techniques commonly used in sports analytics [21], the current results are not directly applicable to immediate practical applications such as coaching strategies or team performance evaluation. However, if this approach is applied to a large number of games involving players of varying skill levels and age groups—and if the differences across these groups can be clearly identified—it may provide a framework for understanding how local-level practices contribute to global-level game behavior. For instance, it could offer insights into the effective time ranges for ball possession in individual decision-making or optimal attacking durations tailored to different skill levels and developmental stages.

5. Conclusions

In this study, we analyzed the probability distribution of ball-keeping durations using data from 10 real football games, examining both the local scale (i.e., individual-player level) and the global scale (i.e., whole-game level). The results revealed negative power-law behavior at both scales, indicating the presence of hierarchical fractal self-similarity in football. These findings contribute to a deeper understanding of fractal-based, self-organized human collective behavior that emerges from social interactions and advance the understanding of complex systems in sports sciences.