# Mathematical Models to Measure the Variability of Nodes and Networks in Team Sports

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## Abstract

**:**

## 1. Introduction

## 2. Variability of Nodes and Networks

**Definition**

**1.**

**Remark**

**1.**

**Proposition**

**1.**

**Proof.**

**Remark**

**2.**

**Definition**

**2.**

**Remark**

**3.**

**Proposition**

**2.**

**Proof.**

**Remark**

**4.**

**Proposition**

**3.**

**Proof.**

**Remark**

**5.**

**Proposition**

**4.**

**Proof.**

**Remark**

**6.**

**Proposition**

**5.**

**Proof.**

**Remark**

**7.**

## 3. Experimental Results

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

${\mathit{R}}_{\mathit{i}}^{\mathit{o}\mathit{u}\mathit{t}}\left({\mathit{n}}_{\mathit{i}}\right)$ | ${\mathit{R}}_{\mathit{i}}^{\mathit{i}\mathit{n}}\left({\mathit{n}}_{\mathit{i}}\right)$ | |||
---|---|---|---|---|

Player | Chelsea | Manchester City | Chelsea | Manchester City |

1 | 0.241 | 0.230 | 0.143 | 0.089 |

2 | 0.226 | 0.224 | 0.351 | 0.314 |

3 | 0.232 | 0.220 | 0.436 | 0.569 |

4 | 0.226 | 0.230 | 0.181 | 0.409 |

5 | 0.233 | 0.231 | 0.303 | 0.400 |

6 | 0.253 | 0.228 | 0.246 | 0.547 |

7 | 0.228 | 0.233 | 0.340 | 0.149 |

8 | 0.239 | 0.254 | 0.310 | 0.319 |

9 | 0.248 | 0.250 | 0.157 | 0.140 |

10 | 0.242 | 0.250 | 0.241 | 0.170 |

11 | 0.250 | 0.243 | 0.355 | 0.273 |

12 | 0.224 | 0.250 | 0.202 | 0.132 |

13 | 0.258 | 0.260 | 0.079 | 0.054 |

14 | 0.247 | 0.248 | 0.140 | 0.118 |

$\mathit{I}\mathit{n}\mathit{d}{\mathit{R}}_{\mathit{i}}^{\mathit{o}\mathit{u}\mathit{t}}\left({\mathit{n}}_{\mathit{i}}\right)$ | $\mathit{I}\mathit{n}\mathit{d}{\mathit{R}}_{\mathit{i}}^{\mathit{i}\mathit{n}}\left({\mathit{n}}_{\mathit{i}}\right)$ | |||
---|---|---|---|---|

Player | Chelsea | Manchester City | Chelsea | Manchester City |

1 | 0.063 | 0.061 | 0.038 | 0.023 |

2 | 0.059 | 0.059 | 0.092 | 0.082 |

3 | 0.061 | 0.058 | 0.115 | 0.149 |

4 | 0.059 | 0.061 | 0.048 | 0.108 |

5 | 0.061 | 0.061 | 0.080 | 0.105 |

6 | 0.066 | 0.060 | 0.065 | 0.144 |

7 | 0.060 | 0.061 | 0.089 | 0.039 |

8 | 0.063 | 0.067 | 0.081 | 0.084 |

9 | 0.065 | 0.066 | 0.041 | 0.037 |

10 | 0.064 | 0.066 | 0.063 | 0.045 |

11 | 0.066 | 0.064 | 0.093 | 0.072 |

12 | 0.059 | 0.066 | 0.053 | 0.035 |

13 | 0.068 | 0.068 | 0.021 | 0.014 |

14 | 0.065 | 0.065 | 0.037 | 0.031 |

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Chelsea | Manchester City | |
---|---|---|

${C}_{N}^{out}$ | 0.256 | −0.112 |

${C}_{N}^{in}$ | 0.407 | 0.293 |

$Ind{C}_{N}^{out}$ | 0.067 | 0.029 |

$Ind{C}_{N}^{in}$ | 0.107 | 0.077 |

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**MDPI and ACS Style**

Martins, F.; Gomes, R.; Lopes, V.; Silva, F.; Mendes, R.
Mathematical Models to Measure the Variability of Nodes and Networks in Team Sports. *Entropy* **2021**, *23*, 1072.
https://doi.org/10.3390/e23081072

**AMA Style**

Martins F, Gomes R, Lopes V, Silva F, Mendes R.
Mathematical Models to Measure the Variability of Nodes and Networks in Team Sports. *Entropy*. 2021; 23(8):1072.
https://doi.org/10.3390/e23081072

**Chicago/Turabian Style**

Martins, Fernando, Ricardo Gomes, Vasco Lopes, Frutuoso Silva, and Rui Mendes.
2021. "Mathematical Models to Measure the Variability of Nodes and Networks in Team Sports" *Entropy* 23, no. 8: 1072.
https://doi.org/10.3390/e23081072