# Spatial and Temporal Entropies in the Spanish Football League: A Network Science Perspective

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## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. Datasets

#### 2.2. Spatial Analysis: Quantifying Spatial Entropy

#### 2.3. Temporal Entropy and Complexity of Football Passing Networks

## 3. Discussion

## 4. Methods

#### 4.1. Definition of Network Metrics

#### 4.1.1. Centroid Coordinates and Dispersion

#### 4.1.2. Clustering Coefficient

#### 4.1.3. Shortest Path Length

#### 4.1.4. Largest Eigenvalue of the Adjacency Matrix

#### 4.1.5. Eigenvector Centrality: Maximum Value and Dispersion

#### 4.1.6. 50-Pass Network Time

#### 4.2. Temporal Entropy and Complexity

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Figure A1.**Temporal Complexity $S{C}_{t}$ vs. average number of $\langle n\rangle $. Only network parameters with a correlation ${r}^{2}>0.6$ are shown; namely, the x-coordinate of the network centroid ${X}_{CM}$ (

**a**), the clustering coefficient C (

**b**), the largest eigenvector centrality of a player $e{c}_{max}$ (

**c**) and the dispersion of the centrality of all players of the team ${\sigma}_{ec}$ (

**d**).

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**Figure 1.**Spatial entropy parameter ${H}_{s}$. (

**a**–

**c**) Three examples of $n=517$ artificially generated passes for which the distances between locations where passes were made were increased. (

**a**) Passes form clusters, leading to a low spatial entropy parameter (${H}_{s}=0.14$). (

**b**) The location of the passes is randomly generated, resulting in a spacial entropy very close to one (${H}_{s}=1.01$). (

**c**) The distance between nearest neighbors is close to its optimal, leading to a regular spatial distribution of passes and a spatial entropy (${H}_{s}=1.73$) close to the highest possible value in 2-dimensional distributions. Finally, (

**d**) the real location of all passes made by Real Madrid during its match against F.C. Barcelona (season 2017/2018). We obtained a value of (${H}_{s}=0.88$), revealing that the actual distribution of passes is close to the random case.

**Figure 2.**Values of the spatial entropy parameter ${H}_{s}$ for all Spanish teams during the season 2017/2018. Error bars are the standard deviations of the means. Atlético de Madrid was the team with the highest spatial entropy parameter (∼0.91), in contrast with Real Betis, which had the lowest one (∼0.89). Note that all values are slightly lower than the unity, indicating a high randomness for the locations of the passes.

**Figure 3.**Spatial entropy ${H}_{s}$ vs. the location of the players. (

**a**) Each dot depicts the spatial entropy for the average x-coordinate of the passes made by players during each match. (

**b**) Here, the absolute entropy is plotted as a function of the y-coordinate of players. (

**c**) Average spatial entropy $\langle {H}_{s}\rangle $ in 20 subdivisions of the field (with a width of 5 field units) along the whole season. Errors bars are the standard deviations of each subdivision. (

**d**). Same as (

**c**) but considering the Y axis.

**Figure 4.**Spatial entropy parameter $\langle {H}_{s}\rangle $ vs. location on the pitch. (

**a**) Spatial entropy parameter $\langle {H}_{s}\rangle $ averaged over all players. (

**b**) The corresponding standard deviations of the values shown in (

**a**).

**Figure 5.**Evolution of passing networks and their corresponding parameters. (

**a**) The structure of Real Madrid 50-pass network at three different moments of the match (${t}_{1}$ = 15:00, ${t}_{2}$ = 38:00 and ${t}_{3}$ = 60:00). (

**b**) An example of how a network parameter fluctuates during the match. Specifically, we plotted the ${X}_{CM}$ of Real Madrid (blue) and F.C. Barcelona (red), the latter playing at home.

**Figure 6.**Complexity-entropy diagram of all network parameters. Each point represents the average value of the entropy $\overline{H}\left[p\right]$ and complexity $\overline{SC}\left[p\right]$ of each network parameter p for all the 20 teams of “La Liga” during the season 2017/2018. Error bars are the standard deviation of the means. Specifically, network parameters are: the x-coordinate of the network centroid ${X}_{CM}$, the y-coordinate of the network centroid ${Y}_{CM}$, the spatial dispersion ${\sigma}_{CM}$ of players around the network centroid, the clustering coefficient C, the average shortest path length $sp$, the largest eigenvalue of the adjacency matrix ${\lambda}_{1}$, the largest eigenvector centrality of a player $e{c}_{max}$ and the dispersion of the centrality of all players of the team ${\sigma}_{ec}$.

**Figure 7.**Temporal entropy ${H}_{t}$ vs. average number of $\langle n\rangle $. Only network parameters with a correlation ${r}^{2}>0.6$ are shown; namely, the x-coordinate of the network centroid ${X}_{CM}$, the clustering coefficient C, the largest eigenvector centrality of a player $e{c}_{max}$ and the dispersion of the centrality of all players of the team ${\sigma}_{ec}$. (

**a**) The average of ${X}_{CM}$ for each team along the whole season has a negative correlation with the number of passes (${R}^{2}=0.74$). (

**b**) The average clustering coefficient C has a positive correlation with ${R}^{2}=0.79$, as it is the case of $e{c}_{max}$ (

**c**) and ${\sigma}_{ec}$ (

**d**).

**Figure 8.**Obtaining ordinal patterns from time series. (

**a**) An example of a set of ordinal patterns $\pi $ of $D=3$. Some of them are repeated more frequently than others. (

**b**) $D!=6$ different ordinal patterns can be possible for $D=3$, leading to a probability distribution $p\left(\pi \right)$ with 6 possible elements.

**Table 1.**Structure of the datasets. Time, in seconds, corresponds to the moment when a pass was made. Player 1 and Player 2 are, respectively, the sender and receiver of the pass, while ${x}_{1,2}$ and ${y}_{1,2}$ are the coordinates of both players, in field units (bounded, at both axis, between 0 and 100).

Time (Seconds) | Team | Player 1 | ${\mathit{x}}_{1}$ | ${\mathit{y}}_{1}$ | Player 2 | ${\mathit{x}}_{2}$ | ${\mathit{y}}_{2}$ |
---|---|---|---|---|---|---|---|

… | … | … | … | … | … | … | … |

128 | Real Madrid | Ramos | 33.47 | 58.35 | Modrić | 42.30 | 58.75 |

130 | Real Madrid | Modrić | 59.36 | 70.00 | Benzema | 60.10 | 74.90 |

136 | Real Madrid | Benzema | 60.15 | 80.15 | Vinícius | 65.40 | 86.50 |

… | … | … | … | … | … | … | … |

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**MDPI and ACS Style**

Martínez, J.H.; Garrido, D.; Herrera-Diestra, J.L.; Busquets, J.; Sevilla-Escoboza, R.; Buldú, J.M.
Spatial and Temporal Entropies in the Spanish Football League: A Network Science Perspective. *Entropy* **2020**, *22*, 172.
https://doi.org/10.3390/e22020172

**AMA Style**

Martínez JH, Garrido D, Herrera-Diestra JL, Busquets J, Sevilla-Escoboza R, Buldú JM.
Spatial and Temporal Entropies in the Spanish Football League: A Network Science Perspective. *Entropy*. 2020; 22(2):172.
https://doi.org/10.3390/e22020172

**Chicago/Turabian Style**

Martínez, Johann H., David Garrido, José L Herrera-Diestra, Javier Busquets, Ricardo Sevilla-Escoboza, and Javier M. Buldú.
2020. "Spatial and Temporal Entropies in the Spanish Football League: A Network Science Perspective" *Entropy* 22, no. 2: 172.
https://doi.org/10.3390/e22020172