# Entropy Analysis of a Railway Network’s Complexity

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## Abstract

**:**

## 1. Introduction

## 2. The Portuguese Railway System

## 3. Fractal Dimension of the Portuguese Railway Network

- Repeat:
- -
- cover the fractal object S with a grid consisting of squares (the boxes) with size $\u03f5>0$,
- -
- find the number of boxes that include part of the fractal ${N}_{\u03f5}(S)\in \mathbb{N}$,
- -
- decrease ϵ.

- The fractal dimension $b\in \mathbb{R}$ is the slope of the log-log plot of ${N}_{\u03f5}(S)$ vs. ϵ, i.e.,$$b(S)=-\underset{\u03f5\to {0}^{+}}{lim}\frac{log{N}_{\u03f5}(S)}{log\u03f5}.$$

## 4. Entropy and SSP Analysis of the Portuguese Railway Network

#### 4.1. Entropy of the Railway Network

#### 4.2. The SSP of the Railway Network Entropy

## 5. Discussion

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Timeline of Portuguese history over the last 165 years; see [15] for more details.

**Figure 2.**Snapshots of the Portuguese railway network in years 1856, 1864, 1882, 1945, 1987 and 2012.

**Figure 5.**Tsallis, Rényi and fractional entropies, versus year and parameters $\{q,r,\alpha \}\in [0.1,0.9]$, respectively: (

**a**) ${S}_{q}$; (

**b**) ${S}_{r}$; and (

**c**) ${S}_{\alpha}$.

**Figure 6.**Time evolution of the Boltzmann–Gibbs–Shannon entropy S and of the fractional entropy ${S}_{\alpha}$ for $\alpha =0.76$, in the 1856–2012 period.

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Valério, D.; Lopes, A.M.; Tenreiro Machado, J.A.
Entropy Analysis of a Railway Network’s Complexity. *Entropy* **2016**, *18*, 388.
https://doi.org/10.3390/e18110388

**AMA Style**

Valério D, Lopes AM, Tenreiro Machado JA.
Entropy Analysis of a Railway Network’s Complexity. *Entropy*. 2016; 18(11):388.
https://doi.org/10.3390/e18110388

**Chicago/Turabian Style**

Valério, Duarte, António M. Lopes, and José A. Tenreiro Machado.
2016. "Entropy Analysis of a Railway Network’s Complexity" *Entropy* 18, no. 11: 388.
https://doi.org/10.3390/e18110388