# Divisibility Networks of the Rational Numbers in the Unit Interval

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Matherials and Methods

#### 2.1. Networks $G\left({\mathcal{A}}_{n}^{1}\right)$

#### 2.2. Networks $G\left({\mathcal{A}}_{n}^{2}\right)$

#### 2.3. Networks $G\left({\mathcal{B}}_{n}^{1}\right)$

#### 2.4. Networks $G\left({\mathcal{B}}_{n}^{2}\right)$

## 3. Results

#### 3.1. Degree Distribution

#### 3.2. Density and Sparsity

#### 3.3. Local Clustering Coefficient

#### 3.4. Network Topologies

#### 3.4.1. Global Clustering Coefficient

#### 3.4.2. Assortative Coefficient

#### 3.4.3. Average Path Length

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

$G(V,L)$ | G is the network with V as set of nodes and L as set of links |

$G\left(A\right)$ | Netwotk with A as adjacency matrix |

${k}_{i}$ | Degree of the node i |

$\langle k\rangle $ | Average degree |

${\delta}_{ij}$ | Kronecker’s delta |

${d}_{i,j}$ | Distance given by the shortest path between nodes i and j |

$\langle d\rangle $ | Average path length |

$p\left(k\right)$ | Probability that a node has degree k |

$\rho \left(G\right)$ | Density of the network G |

${C}_{i}$ | Clustering coefficient of node i |

${C}_{\Delta}$ | Global clustering coefficient |

$\langle C\rangle $ | Average clustering coefficient |

r | Assortativity index |

## References

- Watts, D.J. Small Worlds: The Dynamics of Networks Between Order And Randomness; Princeton University Press: Princeton, NJ, USA, 2004; Volume 9. [Google Scholar]
- Pastor-Satorras, R.; Rubi, M.; Diaz-Guilera, A. Statistical Mechanics of Complex Networks; Springer Science & Business Media: Berlin, Germany, 2003; Volume 625. [Google Scholar]
- Newman, M. Networks; Oxford University Press: Oxford, UK, 2018. [Google Scholar]
- Barrat, A.; Barthelemy, M.; Vespignani, A. Dynamical Processes on Complex Networks; Cambridge University Press: Cambridge, UK, 2008. [Google Scholar]
- Barabási, A. Network Science; Cambridge University Press: Cambridge, UK, 2016. [Google Scholar]
- Brandes, U.; Freeman, L.; Wagner, D. Social Networks; CRC Press: Boca Raton, FL, USA, 2013. [Google Scholar]
- Estrada, E. The Structure of Complex Networks: Theory and Applications; Oxford University Press: Oxford, UK, 2012. [Google Scholar]
- Jian, F.; Dandan, S. Complex network theory and its application research on P2P networks. Appl. Math. Nonlinear Sci.
**2016**, 1, 45–52. [Google Scholar] [CrossRef] [Green Version] - Pérez-Benito, F.J.; García-Gómez, J.; Navarro-Pardo, E.; Conejero, J. Community detection-based deep neural network architectures: Afully automated framework based on Likert-scale data. Math. Methods Appl. Sci.
**2020**, 43, 8290–8301. [Google Scholar] [CrossRef] - Barabasi, A.; Oltvai, Z. Network biology: Understanding the cell’s functional organization. Nat. Rev. Gen.
**2004**, 5, 101–113. [Google Scholar] [CrossRef] [PubMed] - Borgatti, S.; Mehra, A.; Brass, D.; Labianca, G. Network analysis in the social sciences. Science
**2009**, 323, 892–895. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Shekatkar, S.M.; Bhagwat, C.; Ambika, G. Divisibility patterns of natural numbers on a complex network. Sci. Rep.
**2015**, 5, 14280. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Solares-Hernández, P.; Manzano, F.; Pérez-Benito, F.; Conejero, J. Divisibility patterns within Pascal divisibility networks. Mathematics
**2020**, 8, 254. [Google Scholar] [CrossRef] [Green Version] - Chandra, A.; Dasgupta, S. A small world network of prime numbers. Phys. A Stat. Mech. Appl.
**2005**, 357, 436–446. [Google Scholar] [CrossRef] [Green Version] - Yan, X.Y.; Wang, W.X.; Chen, G.R.; Shi, D.H. Multiplex congruence network of natural numbers. Sci. Rep.
**2016**, 6, 23714. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bunimovich, L.; Smith, D.; Webb, B. Finding hidden structures, hierarchies, and cores in networks via isospectral reduction. Appl. Math. Nonlinear Sci.
**2019**, 4, 225–248. [Google Scholar] [CrossRef] [Green Version] - Abiya, R.; Ambika, G. Patterns of primes and composites on divisibility graph. arXiv
**2020**, arXiv:2010.12153. [Google Scholar] - Rajans, A.; Ambika, G. Patterns of primes and composites from divisibility network of natural numbers. arXiv
**2020**, arXiv:2007.00769. [Google Scholar] - Mondreti, V. A Complex Networks Approach to Analysing the Erdös-Straus Conjecture and Related Problems. Preprint
**2019**. [Google Scholar] [CrossRef] - Jing-Yuan, Z.; Wei-Gang, S.; Li-Yan, T.; Chang-Pin, L. Topological properties of Fibonacci networks. Commun. Theor. Phys.
**2013**, 60, 375. [Google Scholar] - Vallin, R. The Elements of Cantor Sets: With Applications; Wiley Online Library: Hoboken, NJ, USA, 2013. [Google Scholar]
- Newman, M. Assortative mixing in networks. Phys. Rev.
**2002**, 89, 208701. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Newman, M. Mixing patterns in networks. Phys. Rev.
**2003**, 67, 026126. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Dorogovtsev, S.; Mendes, J. Evolution of networks. Adv. Phys.
**2002**, 51, 1079–1187. [Google Scholar] [CrossRef] [Green Version]

**Figure 2.**Divisibility networks $G\left({\mathcal{A}}_{6}^{1}\right)$ and $G\left({\mathcal{A}}_{6}^{2}\right)$. (

**a**) The first one has 17 nodes and 71 links. (

**b**) The second one has 9 nodes and 15 links.

**Figure 3.**Divisibility networks $G\left({\mathcal{B}}_{6}^{1}\right)$ and $G\left({\mathcal{B}}_{6}^{2}\right)$. The network in (

**a**) has 12 nodes and 22 links and the network in (

**b**) has 6 nodes and 10 links.

**Figure 4.**Degree distribution with log-binning for the networks: (

**a**) $G\left({\mathcal{A}}_{n}^{1}\right)$, (

**b**) $G\left({\mathcal{A}}_{n}^{2}\right)$, (

**c**) $G\left({\mathcal{B}}_{n}^{1}\right)$, (

**d**) $G\left({\mathcal{B}}_{n}^{2}\right)$ for a fixed network size of ${2}^{16}$ nodes.

**Figure 5.**Evolution of the (

**a**) k-cumulative and (

**b**) ${\langle k\rangle}_{n}$ for the networks $G\left({\mathcal{A}}_{n}^{1}\right)$, $G\left({\mathcal{A}}_{n}^{2}\right)$, $G\left({\mathcal{B}}_{n}^{1}\right)$, and $G\left({\mathcal{B}}_{n}^{2}\right)$ for different network sizes up to ${2}^{16}$ nodes.

**Figure 6.**Evolution of the density of networks $G\left({\mathcal{A}}_{n}^{1}\right)$, $G\left({\mathcal{A}}_{n}^{2}\right)$, $G\left({\mathcal{B}}_{n}^{1}\right)$ and $G\left({\mathcal{B}}_{n}^{2}\right)$ for different network sizes up to ${2}^{16}$ nodes.

**Figure 7.**Local clustering coefficient of the networks (

**a**,

**b**) $G\left({\mathcal{A}}_{n}^{1}\right)$, (2nd row) $G\left({\mathcal{A}}_{n}^{2}\right)$, (

**c**,

**d**) $G\left({\mathcal{B}}_{n}^{1}\right)$, and (

**e**,

**f**) $G\left({\mathcal{B}}_{n}^{2}\right)$ (

**g**,

**h**). On the left (right), we separate the values taking into account if the values of the numerator (denominator) are prime or not. Values are colored according to the order of the number following the diagonal argument on the corresponding matrix.

**Figure 8.**Evolution of the global clustering coeffcient, ${C}_{\Delta}$, and the average clustering coefficient $\langle C\rangle $, for different network sizes up to $n={2}^{16}$.

**Figure 9.**Evolution of the assortativity coefficient, r, and the average path length, $\langle d\rangle $, for different network sizes.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Solares-Hernández, P.A.; García-March, M.A.; Conejero, J.A.
Divisibility Networks of the Rational Numbers in the Unit Interval. *Symmetry* **2020**, *12*, 1879.
https://doi.org/10.3390/sym12111879

**AMA Style**

Solares-Hernández PA, García-March MA, Conejero JA.
Divisibility Networks of the Rational Numbers in the Unit Interval. *Symmetry*. 2020; 12(11):1879.
https://doi.org/10.3390/sym12111879

**Chicago/Turabian Style**

Solares-Hernández, Pedro A., Miguel A. García-March, and J. Alberto Conejero.
2020. "Divisibility Networks of the Rational Numbers in the Unit Interval" *Symmetry* 12, no. 11: 1879.
https://doi.org/10.3390/sym12111879