Complexity emerges in measures of the marking dynamics in football games

In this article, we study the dynamics of marking in football matches. To do this, we survey and analyze a database containing the trajectories of players from both teams on the field of play during three professional games. We describe the dynamics through the construction of temporal bipartite networks of proximity. Based on the introduced concept of proximity, the nodes are the players, and the links are defined between opponents that are close enough to each other at a given moment. By studying the evolution of the heterogeneity parameter of the networks during the game, we characterize a scaling law for the average shape of the fluctuations, unveiling the emergence of complexity in the system. Moreover, we propose a simple model to simulate the players' motion in the field from where we obtained the evolution of a synthetic proximity network. We show that the model captures with a remarkable agreement the complexity of the empirical case, hence it proves to be helpful to elucidate the underlying mechanisms responsible for the observed phenomena.

Figure 1

Defining the range of interaction distances. For this visualization, we use the game recorded in $DS1$. (a) Players' positions at $t\approx 20$. The heat map in the background shows the explored zones for the player highlighted with a star. (b) Average position of the players from the center of mass frame of reference. Parameter $\delta$ indicates the distance between nearest opponents, and the ellipse shows the action zone of the player star.

Figure 2

Describing the marking as proximity networks for the three games recorded in $DS1, DS2$, and $DS3$. (a) Range of interaction distances used to define the threshold range. (b) Relation between the mean value of the heterogeneity parameter $\langle \kappa \rangle$ and the threshold $\theta$. (c) Relation between the mean value of the fraction of nodes on the giant component $\langle n0\rangle$ and the threshold $\theta$. (d) Time evolution of the heterogeneity parameter during the game. (e–g) Proximity networks displayed in the field at different times for data sets $DS1, DS2$, and $DS3$, respectively.

Figure 3

Statistics of the successive increments $I$. We compare the results obtained for $DS1, DS2$, and $DS3$ with a series of Gaussian noise in both plots. (a) Distribution $P\left(I\right)$, and (b) power spectrum density $s_f\left[I\right]$.

Figure 4

Self-similarity in the series of $\kappa$. (a and b) Probability distributions of avalanche lifetime $P\left(T\right)$ and avalanche size $P\left(S\right)$. (c) Relation between avalanches lifetime $T$ and the mean value of the size of the avalanches $\langle S\rangle$. (d) Several examples of avalanches with a different lifetime. (e) Collapse of the avalanches produced by rescaling. Black solid lines in (a)–(c) and (e) show the result of nonlinear fit in the drawn regions (see main text for further details).

Figure 5

Outcomes of the simulations. (a and b) Comparison of empirical players' action zones of both teams with those obtained from the simulations. (c) Evolution of the heterogeneity parameter $\kappa$. In the main plot, we show a small time window of $500\text{s}$ where we can visualize some avalanches, whereas in the inset we show the entire time series. (d) Power spectrum density $s_f\left[I\right]$ and the distribution of $I$ in the inset.

Figure 6

Self-similarity in the series of $\kappa$ obtained from simulations. (a and b) Probability distributions of avalanche lifetime $P\left(T\right)$ and avalanche size $P\left(S\right)$. (c) Relation between avalanche lifetime $T$ and the mean value of the size of the avalanche $\langle S\rangle$. (d) Several examples of avalanches with a different lifetime. (e) Collapse of the avalanches produced by rescaling. Black solid lines in (a)–(c) and cyan solid lines in (e) show the result of nonlinear fit in the drawn regions. Dashed line in (e) depicts the nonlinear fit previously shown in Fig. 4 and is used to compare the results with the empirical case.