#
A Fractional (q,q^{′}) Non-Extensive Information Dimension for Complex Networks

^{1}

^{2}

^{*}

## Abstract

**:**

## Simple Summary

## Abstract

## 1. Introduction

## 2. Preliminaries

#### 2.1. Fractional $(q,{q}^{\prime})-$Entropy

#### 2.2. Information Dimension of Networks

## 3. Fractional $\mathbf{(}\mathit{q}\mathbf{,}{\mathit{q}}^{\mathbf{\prime}}\mathbf{)}$ Information Dimension of Complex

#### Computation of $q,{q}^{\prime}$

## 4. Results

#### 4.1. Real-World Networks

#### 4.2. Synthetic Networks

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ACC | Accuracy |

MCC | Matthew’s Correlation Coefficient |

MDPI | Multidisciplinary Digital Publishing Institute |

SBICR | Bayesian Information Criterion with Bonuses |

BA | Barabasi–Albert |

SHM | Song, Havlin, and Makse |

WS | Watts and Strogatz |

## References

- Clausius, R. The Mechanical Theory of Heat: With Its Applications to the Steam-Engine and to the Physical Properties of Bodies; Creative Media Partners: Sacramento, CA, USA, 1867. [Google Scholar]
- Shannon, C.E. A mathematical theory of communication. Bell Syst. Tech. J.
**1948**, 27, 379–423. [Google Scholar] [CrossRef] - Beck, C. Generalized information and entropy measures in physics. Contemp. Phys.
**2009**, 50, 495–510. [Google Scholar] [CrossRef] - Esteban, M.D. A general class of entropy statistics. Appl. Math.
**1997**, 42, 161–169. [Google Scholar] [CrossRef] - Esteban, M.D.; Morales, D. A summary on entropy statistics. Kybernetika
**1995**, 31, 337–346. [Google Scholar] - Ribeiro, M.; Henriques, T.; Castro, L.; Souto, A.; Antunes, L.; Costa-Santos, C.; Teixeira, A. The entropy universe. Entropy
**2021**, 23, 222. [Google Scholar] [CrossRef] [PubMed] - Lopes, A.M.; Tenreiro Machado, J.A. A review of fractional order entropies. Entropy
**2020**, 22, 1374. [Google Scholar] [CrossRef] [PubMed] - Amigó, J.M.; Balogh, S.G.; Hernández, S. A brief review of generalized entropies. Entropy
**2018**, 20, 813. [Google Scholar] [CrossRef] - Abe, S. A note on the q-deformation-theoretic aspect of the generalized entropies in nonextensive physics. Phys. Lett. A
**1997**, 224, 326–330. [Google Scholar] [CrossRef] - Tsallis, C. Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys.
**1988**, 52, 479–487. [Google Scholar] [CrossRef] - Tsallis, C.; Tirnakli, U. Non-additive entropy and nonextensive statistical mechanics—Some central concepts and recent applications. J. Phys. Conf. Ser.
**2010**, 201, 012001. [Google Scholar] [CrossRef] - Tsallis, C. Introduction to non-extensive Statistical Mechanics: Approaching a Complex World. In Introduction to Non-Extensive Statistical Mechanics: Approaching a Complex World; Chapter Thermodynamical and Nonthermodynamical Applications; Springer: New York, NY, USA, 2009; pp. 221–301. [Google Scholar]
- Jackson, D.O.; Fukuda, T.; Dunn, O.; Majors, E. On q-definite integrals. Quart. J. Pure Appl. Math
**1910**, 41, 193–203. [Google Scholar] - Johal, R.S. q calculus and entropy in nonextensive statistical physics. Phys. Rev. E
**1998**, 58, 4147. [Google Scholar] [CrossRef] - Lavagno, A.; Swamy, P.N. q-Deformed structures and nonextensive-statistics: A comparative study. Phys. A Stat. Mech. Appl.
**2002**, 305, 310–315. [Google Scholar] [CrossRef] - Shafee, F. Lambert function and a new non-extensive form of entropy. IMA J. Appl. Math.
**2007**, 72, 785–800. [Google Scholar] [CrossRef] - Ubriaco, M.R. Entropies based on fractional calculus. Phys. Lett. A
**2009**, 373, 2516–2519. [Google Scholar] [CrossRef] - Ubriaco, M.R. A simple mathematical model for anomalous diffusion via Fisher’s information theory. Phys. Lett. A
**2009**, 373, 4017–4021. [Google Scholar] [CrossRef] - Karci, A. Fractional order entropy: New perspectives. Optik
**2016**, 127, 9172–9177. [Google Scholar] [CrossRef] - Karci, A. Notes on the published article “Fractional order entropy: New perspectives” by Ali KARCI, Optik-International Journal for Light and Electron Optics, Volume 127, Issue 20, October 2016, Pages 9172–9177. Optik
**2018**, 171, 107–108. [Google Scholar] [CrossRef] - Radhakrishnan, C.; Chinnarasu, R.; Jambulingam, S. A Fractional Entropy in Fractal Phase Space: Properties and Characterization. Int. J. Stat. Mech.
**2014**, 2014, 460364. [Google Scholar] [CrossRef] - Ferreira, R.A.C.; Tenreiro Machado, J. An Entropy Formulation Based on the Generalized Liouville Fractional Derivative. Entropy
**2019**, 21, 638. [Google Scholar] [CrossRef] - Ferreira, R.A.C. An entropy based on a fractional difference operator. J. Differ. Equations Appl.
**2021**, 27, 218–222. [Google Scholar] [CrossRef] - Lad, F.; Sanfilippo, G.; Agrò, G. Extropy: Complementary dual of entropy. Statist. Sci.
**2015**, 30, 40–58. [Google Scholar] [CrossRef] - Xue, Y.; Deng, Y. Tsallis eXtropy. Comm. Statist. Theory Methods
**2023**, 52, 751–762. [Google Scholar] [CrossRef] - Liu, J.; Xiao, F. Renyi extropy. Comm. Statist. Theory Methods
**2023**, 52, 5836–5847. [Google Scholar] [CrossRef] - Buono, F.; Longobardi, M. A dual measure of uncertainty: The Deng extropy. Entropy
**2020**, 22, 582. [Google Scholar] [CrossRef] - González, S.H.; De La Mota, I.F. Applying complex network theory to the analysis of Mexico city metro network (1969–2018). Case Stud. Transp. Policy
**2021**, 9, 1344–1357. [Google Scholar] [CrossRef] - Eguíluz, V.M.; Hernández-García, E.; Piro, O.; Klemm, K. Effective dimensions and percolation in hierarchically structured scale-free networks. Phys. Rev. E
**2003**, 68, 055102. [Google Scholar] [CrossRef] - Ortiz-Vilchis, P.; Ramirez-Arellano, A. Learning Pathways and Students Performance: A Dynamic Complex System. Entropy
**2023**, 25, 291. [Google Scholar] [CrossRef] - Ramirez-Arellano, A.; Ortiz-Vilchis, P.; Bory-Reyes, J. The role of D-summable information dimension in differentiating covid-19 disease. Fractals
**2021**, 29, 2150255. [Google Scholar] [CrossRef] - Ramirez-Arellano, A. Students learning pathways in higher blended education: An analysis of complex networks perspective. Comput. Educ.
**2019**, 141, 103634. [Google Scholar] [CrossRef] - Ortiz-Vilchis, P.; De-la Cruz-García, J.S.; Ramirez-Arellano, A. Identification of Relevant Protein Interactions with Partial Knowledge: A Complex Network and Deep Learning Approach. Biology
**2023**, 12, 140. [Google Scholar] [CrossRef] - RamirezArellano, A. Classification of Literary Works: Fractality and Complexity of the Narrative, Essay, and Research Article. Entropy
**2020**, 22, 904. [Google Scholar] [CrossRef] [PubMed] - Ortiz-Vilchis, P.; Ramirez-Arellano, A. An Entropy-Based Measure of Complexity: An Application in Lung-Damage. Entropy
**2022**, 24, 1119. [Google Scholar] [CrossRef] [PubMed] - De la Cruz-García, J.S.; Bory-Reyes, J.; Ramirez-Arellano, A. A Two-Parameter Fractional Tsallis Decision Tree. Entropy
**2022**, 24, 572. [Google Scholar] [CrossRef] - Boccaletti, S.; Latora, V.; Moreno, Y.; Chavez, M.; Hwang, D.U. Complex networks: Structure and dynamics. Phys. Rep.
**2006**, 424, 175–308. [Google Scholar] [CrossRef] - Estrada, E. The Structure of Complex Networks: Theory and Applications; Oxford University Press, Inc.: New York, NY, USA, 2011. [Google Scholar]
- Rosenberg, E. A Survey of Fractal Dimensions of Networks; Springer Briefs in Computer Science; Springer: Cham, Switzerland, 2018. [Google Scholar]
- Rosenberg, E. Fractal Dimensions of Networks; Springer: Cham, Switzerland, 2020. [Google Scholar]
- Wen, T.; Cheong, K.H. The fractal dimension of complex networks: A review. Inf. Fusion
**2021**, 73, 87–102. [Google Scholar] [CrossRef] - Xu, T.; Moore, I.D.; Gallant, J.C. Fractals, fractal dimensions and landscapes—A review. Geomorphology
**1993**, 8, 245–262. [Google Scholar] [CrossRef] - Camacho, D.; Panizo-LLedot, Á.; Bello-Orgaz, G.; Gonzalez-Pardo, A.; Cambria, E. The four dimensions of social network analysis: An overview of research methods, applications, and software tools. Inf. Fusion
**2020**, 63, 88–120. [Google Scholar] [CrossRef] - Peach, R.; Arnaudon, A.; Barahona, M. Relative, local and global dimension in complex networks. Nat. Commun.
**2022**, 13, 3088. [Google Scholar] [CrossRef] - Ramirez-Arellano, A.; Sigarreta-Almira, J.M.; Bory-Reyes, J. Fractional information dimensions of complex networks. Chaos Interdiscip. J. Nonlinear Sci.
**2020**, 30, 093125. [Google Scholar] [CrossRef] - Ramirez-Arellano, A.; Hernández-Simón, L.M.; Bory-Reyes, J. Two-parameter fractional Tsallis information dimensions of complex networks. Chaos Solitons Fractals
**2021**, 150, 111113. [Google Scholar] [CrossRef] - Borges, E.P.; Roditi, I. A family of nonextensive entropies. Phys. Lett. A
**1998**, 246, 399–402. [Google Scholar] [CrossRef] - Chakrabarti, R.; Jagannathan, R. A (p, q)-oscillator realization of two-parameter quantum algebras. J. Phys. Math. Gen.
**1991**, 24, L711. [Google Scholar] [CrossRef] - Wei, D.; Wei, B.; Hu, Y.; Zhang, H.; Deng, Y. A new information dimension of complex networks. Phys. Lett. A
**2014**, 378, 1091–1094. [Google Scholar] [CrossRef] - Song, C.; Havlin, S.; Makse, H.A. Self-similarity of complex networks. Nature
**2005**, 433, 392. [Google Scholar] [CrossRef] [PubMed] - Song, C.; Gallos, L.K.; Havlin, S.; Makse, H.A. How to calculate the fractal dimension of a complex network: The box covering algorithm. J. Stat. Mech. Theory Exp.
**2007**, 2007, P03006. [Google Scholar] [CrossRef] - Rosenberg, E. Maximal entropy coverings and the information dimension of a complex network. Phys. Lett. A
**2017**, 381, 574–580. [Google Scholar] [CrossRef] - Ramirez-Arellano, A.; Hernández-Simón, L.M.; Bory-Reyes, J. A box-covering Tsallis information dimension and non-extensive property of complex networks. Chaos Solitons Fractals
**2020**, 132, 109590. [Google Scholar] [CrossRef] - Ramirez-Arellano, A.; Bermúdez-Gómez, S.; Hernández-Simón, L.M.; Bory-Reyes, J. d-summable fractal dimensions of complex networks. Chaos Solitons Fractals
**2019**, 119, 210–214. [Google Scholar] [CrossRef] - Seber, G.; Wild, C. Nonlinear Regression; Wiley Series in Probability and Statistics; Wiley: Hoboken, NJ, USA, 2003. [Google Scholar]
- Rossi, R.A.; Ahmed, N.K. The Network Data Repository with Interactive Graph Analytics and Visualization. In Proceedings of the AAAI, Austin, TX, USA, 25–30 January 2015. [Google Scholar]
- Dudley, R.M.; Haughton, D. Information criteria for multiple data sets and restricted parameters. Stat. Sin.
**1997**, 7, 265–284. [Google Scholar] - Barabási, A.L.; Albert, R. Emergence of scaling in random networks. Science
**1999**, 286, 509–512. [Google Scholar] [CrossRef] [PubMed] - Song, C.; Havlin, S.; Makse, H.A. Origins of fractality in the growth of complex networks. Nat. Phys.
**2006**, 2, 275. [Google Scholar] [CrossRef] - Watts, D.J.; Strogatz, S.H. Collective dynamics of ‘small-world’ networks. Nature
**1998**, 393, 440. [Google Scholar] [CrossRef] - Hall, M.; Frank, E.; Holmes, G.; Pfahringer, B.; Reutemann, P.; Witten, I.H. The WEKA Data Mining Software: An Update. SIGKDD Explor. Newsl.
**2009**, 11, 10–18. [Google Scholar] [CrossRef]

**Figure 3.**(

**a**) ${d}_{I}$ and (

**b**) ${d}_{q.{q}^{\prime}}$ of BA, SHM, and WS networks. Mann–Whitney U test revealed a significant difference in ${d}_{q.{q}^{\prime}}$ for all types of networks. The * symbol indicates no statistical difference between them.

**Figure 4.**Accuracy (ACC) and Matthew’s correlation coefficient of decision trees built on data sets: (1) ${d}_{I}$, (2) ${d}_{q.{q}^{\prime}}$, and (3) ${d}_{q.{q}^{\prime}}$, q, and ${q}^{\prime}$.

**Table 1.**Diameter, number of nodes, source, ${d}_{I}$, and ${d}_{q,{q}^{\prime}}$ of real-world networks.

Network | Full Name | Source | Diameter | Nodes | Edges |
---|---|---|---|---|---|

ACF | American college football | [46] | 4 | 115 | 613 |

BCEPG | Bio-CE-PG | [46] | 8 | 1692 | 47,309 |

BGP | Bio-grid-plant | [46] | 26 | 1272 | 2726 |

BGW | Bio-grid-worm | [46] | 12 | 16,259 | 762,774 |

CEN | C. elegans neural network | [46] | 5 | 297 | 2148 |

CNC | Ca-netscience | [46] | 17 | 379 | 914 |

COL | SocfbColgate88 | [56] | 6 | 3482 | 155,044 |

DRO | Drosophilamedulla1 | [56] | 6 | 1770 | 33,635 |

DS | Dolphins social network | [46] | 8 | 62 | 159 |

ECC | E. coli cellular network | [46] | 18 | 2859 | 6890 |

EM | [46] | 8 | 1133 | 5451 | |

IOF | Infopenflights | [56] | 14 | 2905 | 30,442 |

JM | Jazz-musician | [46] | 6 | 198 | 2742 |

JUN | Jung2015 | [56] | 16 | 2989 | 31,548 |

LAS | Lada Adamic’s network | [46] | 8 | 350 | 3492 |

LDU | Labanderiadunne | [56] | 6 | 700 | 6444 |

MAR | Marvel | [56] | 11 | 19,365 | 96,616 |

MIT | SocfbMIT | [56] | 8 | 6402 | 251,230 |

PAIR | Pairdoc | [56] | 14 | 8914 | 25,514 |

PG | Power grid network | [46] | 46 | 4941 | 6594 |

PGP | Techpgp | [56] | 24 | 10,680 | 24,340 |

POW | Powerbcspwr10 | [56] | 49 | 5300 | 13,571 |

PRI | SocfbPrinceton12 | [56] | 9 | 6575 | 293,307 |

TC | Topology of communications | [46] | 7 | 174 | 557 |

USAA | USA airport network | [46] | 7 | 500 | 2980 |

WHO | TechWHOIS | [56] | 8 | 7476 | 56,943 |

YEAST | Protein interaction | [46] | 11 | 2223 | 7046 |

ZCK | Zachary’s karate club | [46] | 5 | 34 | 78 |

Network | ${\mathit{SBICR}}_{\mathit{I}}$ | ${\mathit{SBICR}}_{(\mathit{q},{\mathit{q}}^{\prime})}$ | ${\mathit{d}}_{\mathit{I}}$ | ${\mathit{d}}_{\mathit{q},{\mathit{q}}^{\prime}}$ | q | ${\mathit{q}}^{\prime}$ |
---|---|---|---|---|---|---|

ACF | −10.135 | −7.348 | 1.913 | 0.930 | 2.976 | 0.442 |

BCEPG | −35.380 | −20.988 | 1.828 | 1.004 | 6.109 | 1.814 |

BGP | −95.919 | −95.442 | 1.591 | 1.037 | 3.558 | 0.817 |

BGW | −64.525 | −42.927 | 1.949 | 1.006 | 2.437 | 1.219 |

CEN | −13.897 | −12.103 | 1.822 | 0.988 | 3.253 | 0.805 |

CNC | −58.166 | −49.747 | 1.835 | 1.014 | 3.606 | 0.733 |

COL | −25.187 | −18.343 | 2.679 | 1.001 | 3.655 | 1.364 |

DRO | −23.34 | −18.202 | 1.443 | 0.998 | 5.856 | 0.662 |

DS | −17.868 | −14.616 | 1.498 | 0.989 | 3.959 | 0.788 |

ECC | −87.426 | −66.706 | 1.625 | 1.029 | 5.044 | 1.442 |

EM | −34.459 | −31.15 | 1.547 | 1.001 | 4.663 | 0.915 |

IOF | −65.328 | −51.656 | 1.736 | 1.028 | 4.846 | 1.112 |

JM | −19.449 | −13.532 | 2.805 | 0.986 | 2.659 | 0.726 |

JUN | −63.643 | −58.511 | 2.485 | 1.006 | 5.95 | 1.105 |

LAS | −30.269 | −21.898 | 2.103 | 1.020 | 3.459 | 0.365 |

LDU | −20.750 | −20.091 | 1.850 | 0.998 | 2.879 | 0.864 |

MAR | −52.612 | −46.428 | 1.644 | 1.003 | 3.976 | 0.900 |

MIT | −39.208 | −25.351 | 2.295 | 1.006 | 4.353 | 1.165 |

PAIR | −68.947 | −57.065 | 1.582 | 1.004 | 2.726 | 0.670 |

PG | −186.322 | −199.049 | 1.463 | 0.999 | 5.221 | 1.324 |

PGP | −117.159 | −100.827 | 1.573 | 1.013 | 2.930 | 1.666 |

POW | −186.721 | −206.691 | 1.589 | 0.994 | 5.176 | 0.489 |

PRI | −45.866 | −27.325 | 2.533 | 1.016 | 6.338 | 1.234 |

TC | −22.759 | −22.025 | 1.57 | 0.991 | 5.494 | 1.143 |

USAA | −26.655 | −18.497 | 1.678 | 0.996 | 2.358 | 0.626 |

WHO | −36.125 | −29.873 | 1.675 | 1.002 | 4.751 | 1.280 |

YEAST | −48.576 | −43.622 | 1.533 | 1.006 | 6.500 | 0.594 |

ZCK | −9.77 | −3.672 | 1.500 | 0.950 | 5.094 | 0.696 |

Network Model | Nodes | Edges | ${\mathit{d}}_{\mathit{I}}$ | ${\mathit{d}}_{\mathit{q},{\mathit{q}}^{\prime}}$ | Model |
---|---|---|---|---|---|

BA | 2000–4500 | 2685–40,455 | 3.930–7.412 | 0.864–1.729 | $q,{q}^{\prime}$ (100%) |

SHM | 10–36,480 | 9–880,475 | 0.831–12.689 | 0.005–2.432 | $q,{q}^{\prime}$ (70.83%) |

WS | 2000–4000 | 4000–40,000 | 0.955–7.363 | 0.015–1.540 | $q,{q}^{\prime}$ (100%) |

**Table 4.**The parameters of the SHM model that produced networks for which the information model fit better.

G | M | $\mathit{IB}$ | $\mathit{BB}$ | $\mathit{MODE}$ |
---|---|---|---|---|

2 | 2 | 0 | 0.400 | 1 |

2 | 2 | 0.400 | 0.400 | 1 |

2 | 3 | 0 | 0.400 | 1 |

2 | 3 | 0.400 | $[0,1]$ | 1 |

2 | 4 | 0 | 0.800 | 1 |

3 | 2 | 0 | $[0,1]$ | 1 |

3 | 2 | 0.400 | $[0.200,0.800]$ | 1 |

3 | 3 | $[0,0.400]$ | ≤0.800 | 1 |

3 | 4 | 0 | 1 | 1 |

4 | $[2,3]$ | 0 | 0.400 | 1 |

2 | 2 | 0 | $[0,0.200,0.800]$ | 2 |

2 | 2 | 0.400 | $[0,1]$ | 2 |

2 | 2 | 1 | ≤0.800 | 2 |

2 | 3 | 0 | ≤0.400 | 2 |

2 | 3 | 0.400 | 0.200 | 2 |

3 | 2 | 0 | $[2,0.600,1]$ | 2 |

3 | 2 | 0.400 | 0.800 | 2 |

3 | 4 | 0 | 0.400 | 2 |

4 | 2 | 0 | ≤0.200 | 2 |

4 | 2 | 0.400 | $[0.400,0.200]$ | 2 |

4 | 3 | 0 | 0.800 | 2 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ramirez-Arellano, A.; De-la-Cruz-Garcia, J.-S.; Bory-Reyes, J.
A Fractional (*q*,*q*^{′}) Non-Extensive Information Dimension for Complex Networks. *Fractal Fract.* **2023**, *7*, 702.
https://doi.org/10.3390/fractalfract7100702

**AMA Style**

Ramirez-Arellano A, De-la-Cruz-Garcia J-S, Bory-Reyes J.
A Fractional (*q*,*q*^{′}) Non-Extensive Information Dimension for Complex Networks. *Fractal and Fractional*. 2023; 7(10):702.
https://doi.org/10.3390/fractalfract7100702

**Chicago/Turabian Style**

Ramirez-Arellano, Aldo, Jazmin-Susana De-la-Cruz-Garcia, and Juan Bory-Reyes.
2023. "A Fractional (*q*,*q*^{′}) Non-Extensive Information Dimension for Complex Networks" *Fractal and Fractional* 7, no. 10: 702.
https://doi.org/10.3390/fractalfract7100702