# A New Method of Quantifying the Complexity of Fractal Networks

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

^{H}y) is statistically similar to f(x, y) where H is known as the Hurst exponent [4]; we shall relate H to the fractal dimension in what follows. Some networks also have fractal properties, but others do not. Almost all models and real networks do not have fractal properties. If we convert to another space, however, we can estimate their statistical self-affine fractal dimension, i.e., we can determine their complexity.

## 2. Overview of Fractal Networks

^{D}

^{−y}

## 3. New Method for Fractal Network

^{−y}for an appropriate renormalization procedure. The fractality of a network (also called fractal scaling or topological fractality) infers a power-law relationship between the minimum number of boxes needed to cover the entire network and the size of the boxes, i.e., a network is fractal if the box dimension D exists. However, some networks are self-similar but not fractal. Typical examples of such networks are the Internet and hierarchical graph sequence models. Thus, self-similarity and fractality do not always imply each other for complex networks [23,24,25,26]. Figure 1, Figure 2 and Figure 3 demonstrate our approach and analysis.

_{i}, we see how many of the nodes are on the circle with radius i. We calculate n

_{i}, which represents the number of nodes in the polar coordinate system with the circle with radius i as (3):

_{i}(r

_{i}, n

_{i}) represent all points from the graph network in the polar coordinate system, which is a linear graph Γ(r, n). In addition, the linear graph Γ(r, n) presents points T

_{1}(r

_{1}, n

_{1}), T

_{2}(r

_{2}, n

_{2}) … T

_{k}(r

_{k}, n

_{k}).

## 4. Bus Transport System

^{2}. The municipality measures 236 km

^{2}. Novo mesto is the urban center of the Municipality of Novo mesto. It is also the administrative, educational, health, economic, and cultural center of the wider region of Southeastern Slovenia. With its industry, Novo mesto is the carrier of the fastest economic development in the region. A strong automotive, pharmaceutical, and cosmetic industry, and the insulation materials industry (Krka, Revoz, Adria Mobil, TPV) have developed, which also attracts labor from elsewhere. Maribor is the second-largest city in Slovenia and has a population of 95,000. Maribor has a developed public transport system. Celje is the third-largest city in Slovenia, with a population of 38,000, and has a developed public transport system. The city of Nova Gorica is the ninth-largest city of Slovenia, with a population of 13,000, and has a developed public transport system.

## 5. Data Mining

## 6. Results and Discussion

**MODEL OF GP**:

^{2}= 0.9474 or R

^{2}= 0.9277, depending on the specific ML methods used (GP and NN, respectively). The correspondence between the coefficients suggests that the two forecasting techniques are similarly effective. The coefficient of determination increases (more precisely, it almost always increases) as the number of regressors in the model increases. This leads to the fact that if, when assessing the quality of the model, one is guided by the usual coefficient R

^{2}, then equations with a large number of regressors will give a better result than with a smaller one. This can lead to the unjustified inclusion of a large number of insignificant regressors in the model. The inclusion of each additional regressor results in the loss of one degree of freedom. Figure 7 represents a graphical representation of data with the correlation of determination. It is the case of FD (Complexity) with coefficients.

_{GP}= 0.9997 and ρ

_{NN}= 0.7387.

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 7.**A graphical representation of data with the correlation of determination. It is the case of FD (Complexity) with coefficients.

City | # Passengers (PA) | # Population (PO) | # Stations (S) | # Line (L) | FD—Complexity |
---|---|---|---|---|---|

MB | 4,000,000 | 95,000 | 200 | 23 | 1.6936 |

NM | 300,000 | 24,000 | 94 | 7 | 1.6121 |

CE | 150,000 | 38,000 | 35 | 6 | 1.5697 |

NG | 500,000 | 13,000 | 45 | 5 | 1.4951 |

FD | GP | NN |
---|---|---|

1.6936 | 1.6973 | 1.722 |

1.6121 | 1.6126 | 1.6852 |

1.5697 | 1.5476 | 1.6231 |

1.4951 | 1.4956 | 1.5241 |

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**MDPI and ACS Style**

Babič, M.; Marinković, D.; Kovačič, M.; Šter, B.; Calì, M.
A New Method of Quantifying the Complexity of Fractal Networks. *Fractal Fract.* **2022**, *6*, 282.
https://doi.org/10.3390/fractalfract6060282

**AMA Style**

Babič M, Marinković D, Kovačič M, Šter B, Calì M.
A New Method of Quantifying the Complexity of Fractal Networks. *Fractal and Fractional*. 2022; 6(6):282.
https://doi.org/10.3390/fractalfract6060282

**Chicago/Turabian Style**

Babič, Matej, Dragan Marinković, Miha Kovačič, Branko Šter, and Michele Calì.
2022. "A New Method of Quantifying the Complexity of Fractal Networks" *Fractal and Fractional* 6, no. 6: 282.
https://doi.org/10.3390/fractalfract6060282