# Power-Law Distribution of Natural Visibility Graphs from Reaction Times Series

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Attention and Reaction Times

#### 1.2. Gender Differences

#### 1.3. Mathematical Distribution of RTs

#### 1.4. Outline of the Work

## 2. Materials

## 3. Methods (Natural Visibility Graph)

^{th}response time and $\overline{t}$ is the mean value. Following this procedure, the integrated response times were within the fractional Brownian motion regime. In this way, NVGs provided more reliable information on the emerging networks by means of quantities like the degree distribution.

## 4. Results

#### 4.1. Description of the RTs

^{−10}). Specifically, there were differences for ages between 8 and 9 years, 9 and 10 years, and 10 and 11 years, but not between 11 and 12 years, according to the Wilcoxon test. We repeated the test after eliminating the outlier in the 12-year-old group and the result did not vary. These results are consistent with the findings on how reaction times evolve with age [38,61,62,63,64]. A linear model was adjusted taking the logarithm of RT mean as the dependent variable, and age and sex as the fixed-effect factors, obtaining the same results: significant differences across ages 8 and 9 years (10% less RT mean, p-value = 0.01); 9 and 10 years (8% less, p-value = 0.05); 10 and 11 years (15% less, p-value < 0.01); and not between sexes (p-value for sex 0.82).

#### 4.2. Power-Law Distribution of the NVGs’ Degrees

- (a)
- $E\left(x\right)$, which is the CDF of the empirical data;
- (b)
- $P\left(x\right)$, which is the CDF for the power-law fit.

#### 4.3. Comparison Between the Power-Law and the Ex-Gaussian Distribution Parameters

#### 4.4. Particular Cases

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Example of the construction of a natural visibility graph (NVG). On the left (

**a**), the integrated response time series for a sample of items. On the right (

**b**), the NVG associated with the time series. The label of each node corresponds with the number of the item in the series. Two nodes (items) are connected by an edge if they can see each other.

**Figure 4.**The respective NVGs of participants 39, 77, and 103. (

**a**) Natural visibility graph of participant 39; (

**b**) Natural visibility graph of participant 77; (

**c**) Natural visibility graph of participant 103.

**Figure 5.**Power-law fit of the degree distribution of the NVGs associated with participants 39, 77, and 103 (points represent the cumulative degree distribution function of the NVG and the red line represents the power-law that best fits the data). (

**a**) NVG degree distribution and power-law fit of participant 39; (

**b**) NVG degree distribution and power-law fit of participant 77; (

**c**) NVG degree distribution and power-law fit of participant 103.

**Figure 8.**Ex-Gaussian fit of the RTs of participants 39, 77, and 103 (continuous line represents observed data, dashed line represents fitted data). (

**a**) Ex-Gaussian fit of the RTs of participant 39; (

**b**) Ex-Gaussian fit of the RTs of participant 77; (

**c**) Ex-Gaussian fit of the RTs of participant 103.

**Figure 9.**The respective NVGs of participants 89, 120, and 84. (

**a**) Natural visibility graph of participant 89; (

**b**) Natural visibility graph of participant 120; (

**c**) Natural visibility graph of participant 84.

**Figure 10.**The respective NVGs of participants 89, 120, and 84 where black nodes represent hits or commission errors and red nodes represent omission errors. (

**a**) Natural visibility graph of participant 89; (

**b**) Natural visibility graph of participant 120; (

**c**) Natural visibility graph of participant 84.

**Figure 11.**Power-law fit of the degree distribution of the NVGs associated with participants 89, 120, and 84 (points represent the cumulative degree distribution function of the NVG and the red line represents the power-law that best fits the data). (

**a**) NVG degree distribution and power-law fit of participant 89; (

**b**) NVG degree distribution and power-law fit of participant 120; (

**c**) NVG degree distribution and power-law fit of participant 84.

**Figure 12.**Ex-Gaussian fit of RTs of participants 89, 120, and 84 (continuous line represents observed data, dashed line represents fitted data). (

**a**) Ex-Gaussian fit of RTs of participant 89; (

**b**) Ex-Gaussian fit of RTs of participant 120; (

**c**) Ex-Gaussian fit of RTs of participant 84.

Age | Females | Males | Total |
---|---|---|---|

8 | 23 (63.9%) | 13 (36.1%) | 36 |

9 | 15 (48.4%) | 16 (51.6%) | 31 |

10 | 13 (41.9%) | 18 (58.1%) | 31 |

11 | 8 (42.1%) | 11 (57.9%) | 19 |

12 | 7 (53.8%) | 6 (46.2%) | 13 |

Overall | 66 (50.8%) | 64 (49.2%) | 130 |

Age | Sex | Mean No. of Hits | Mean No. of Commission Errors | Mean No. of Omission Errors |
---|---|---|---|---|

8 | Females | 116.1 | 11.6 | 2.1 |

8 | Males | 112.6 | 15.3 | 1.8 |

9 | Females | 121.5 | 6.8 | 1.5 |

9 | Males | 120.5 | 8.4 | 0.8 |

10 | Females | 122.0 | 7.1 | 0.6 |

10 | Males | 115.4 | 12.8 | 1.3 |

11 | Females | 126.1 | 3.7 | 0.1 |

11 | Males | 121.5 | 7.9 | 0.4 |

12 | Females | 124.9 | 4.8 | 0.3 |

12 | Males | 118.2 | 7.3 | 4.5 |

Overall | Females | 120.6 | 7.9 | 1.2 |

Overall | Males | 117.4 | 10.9 | 1.4 |

Age | Sex | $\mathbf{Mean}\left(\mathit{\alpha}\right)$ | $\mathbf{SD}\left(\mathit{\alpha}\right)$ |
---|---|---|---|

8 | Females | 4.24 | 2.06 |

8 | Males | 4.03 | 1.27 |

9 | Females | 4.55 | 2.09 |

9 | Males | 4.17 | 1.01 |

10 | Females | 4.66 | 2.26 |

10 | Males | 4.54 | 2.12 |

11 | Females | 3.61 | 0.74 |

11 | Males | 5.54 | 1.83 |

12 | Females | 5.14 | 2.56 |

12 | Males | 4.09 | 1.22 |

Overall | Females | 4.41 | 2.04 |

Overall | Males | 4.47 | 1.64 |

Age | Sex | Mean (h) | SD (h) |
---|---|---|---|

8 | Females | 3.65 | 0.39 |

8 | Males | 3.50 | 0.31 |

9 | Females | 3.49 | 0.29 |

9 | Males | 3.47 | 0.41 |

10 | Females | 3.44 | 0.40 |

10 | Males | 3.62 | 0.41 |

11 | Females | 3.64 | 0.38 |

11 | Males | 3.54 | 0.62 |

12 | Females | 3.51 | 0.34 |

12 | Males | 3.39 | 0.23 |

Overall | Females | 3.56 | 0.36 |

Overall | Males | 3.53 | 0.42 |

Age | Sex | $\mathbf{Mean}\left(\mathit{\mu}\right)$ | $\mathbf{SD}\left(\mathit{\mu}\right)$ | $\mathbf{Mean}\left(\mathit{\sigma}\right)$ | $\mathbf{SD}\left(\mathit{\sigma}\right)$ | $\mathbf{Mean}\left(\mathit{\tau}\right)$ | $\mathbf{SD}\left(\mathit{\tau}\right)$ |
---|---|---|---|---|---|---|---|

8 | Females | 643.4 | 60.55 | 105.34 | 22.88 | 304.10 | 79.17 |

8 | Males | 650.7 | 93.61 | 133.18 | 32.14 | 316.00 | 124.5 |

9 | Females | 587.5 | 60.97 | 98.62 | 33.99 | 279.89 | 125.6 |

9 | Males | 614.9 | 95.29 | 122.33 | 41.63 | 248.00 | 103.0 |

10 | Females | 554.3 | 64.48 | 103.58 | 28.96 | 221.67 | 79.67 |

10 | Males | 552.6 | 73.77 | 122.91 | 65.99 | 255.10 | 125.9 |

11 | Females | 524.9 | 68.43 | 81.06 | 24.12 | 131.90 | 43.79 |

11 | Males | 507.4 | 42.39 | 82.43 | 31.46 | 185.54 | 95.39 |

12 | Females | 512.9 | 67.19 | 81.15 | 21.24 | 238.20 | 80.92 |

12 | Males | 471.3 | 54.36 | 75.83 | 25.72 | 238.04 | 306.2 |

Overall | Females | 629.5 | 78.30 | 111.50 | 27.86 | 322.87 | 102.6 |

Overall | Males | 572.7 | 96.78 | 113.48 | 49.22 | 252.15 | 142.1 |

Participant | Age | Sex | Mean Reaction Time | # Hits | # Omission Errors | # Commission Errors | Shannon Entropy |
---|---|---|---|---|---|---|---|

89 | 12 | M | 1338.7* | 91 | 27 | 12 | 3.44 |

120 | 8 | F | 974.5 | 14 | 7 | 109* | 3.92 |

84 | 12 | M | 492.7* | 124 | 0 | 6 | 3.12 |

Participant | $\mathit{\alpha}$ | $\mathit{\mu}$ | $\mathit{\sigma}$ | $\mathit{\tau}$ |
---|---|---|---|---|

89 | 5.8 | 481.9 | 46.9 | 856.9 |

120 | 2.2 | 580.5 | 136.8 | 394.0 |

84 | 3.2 | 437.8 | 54.1 | 54.85 |

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**MDPI and ACS Style**

Mira-Iglesias, A.; Navarro-Pardo, E.; Conejero, J.A.
Power-Law Distribution of Natural Visibility Graphs from Reaction Times Series. *Symmetry* **2019**, *11*, 563.
https://doi.org/10.3390/sym11040563

**AMA Style**

Mira-Iglesias A, Navarro-Pardo E, Conejero JA.
Power-Law Distribution of Natural Visibility Graphs from Reaction Times Series. *Symmetry*. 2019; 11(4):563.
https://doi.org/10.3390/sym11040563

**Chicago/Turabian Style**

Mira-Iglesias, Ainara, Esperanza Navarro-Pardo, and J. Alberto Conejero.
2019. "Power-Law Distribution of Natural Visibility Graphs from Reaction Times Series" *Symmetry* 11, no. 4: 563.
https://doi.org/10.3390/sym11040563