ArticleNo Access

HAUSDORFF DIMENSION OF A CLASS OF COLORED SUBSTITUTION NETWORKS

QINGCHENG ZENG

https://orcid.org/0009-0001-5196-0939

School of Mathematical Sciences, Beihang University, Beijing 100083, P. R. China

E-mail Address: qczeng@buaa.edu.cn

Search for more papers by this author

and

YUMEI XUE

https://orcid.org/0000-0002-2103-0052

School of Mathematical Sciences, Beihang University, Beijing 100083, P. R. China

E-mail Address: yxue@buaa.edu.cn

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S0218348X25500069Cited by:0 (Source: Crossref)

Abstract

In 2023, Xi et al. introduced the Hausdorff dimension of a family of networks which inspired by the potential theoretic methods in fractal geometry. In this paper, we will construct a class of colored substitution networks and obtain its Hausdorff dimension using the self-similarity.

Keywords:

References

1. D. J. Watts, S. H. Strogatz and H. Strogatz , Collective dynamics of ‘small-world’ networks, Nature 393(6684) (1998) 440–442. Crossref, Web of Science, Google Scholar
2. A. L. Barabási, R. Albert and R. Albert , Emergence of scaling in random networks, Science 286 (1999) 509–512. Crossref, Web of Science, Google Scholar
3. M. E. J. Barabási, R. Albert and M. E. J. Newman , Emergence of scaling in random networks, SIAM Rev. 45 (2003) 167–256. Google Scholar
4. B. Mandelbrot and B. Mandelbrot , How long is the coast of Britain? Statistical self-similarity and fractional dimension, Science 156 (1967) 636–638. Crossref, Web of Science, Google Scholar
5. J. Hutchinson and J. Hutchinson , Fractals and self similarity, Indiana Univ. Math. J. 30 (1981) 713–747. Crossref, Web of Science, Google Scholar
6. C. Song, S. Havlin, H. Makse and H. Makse , Self-similarity of complex networks, Nature 433 (2005) 392–395. Crossref, Web of Science, Google Scholar
7. A. Le, F. Gao, L. Xi, S. Yin and L. Xi , Complex networks modeled on the Sierpinski gasket, Phys. A 436 (2015) 646–657. Crossref, Web of Science, Google Scholar
8. S. Wang, L. Xi, H. Xu, L. Wang and L. Wang , Scale-free and small-world properties of Sierpinski networks, Phys. A 465 (2017) 690–700. Crossref, Web of Science, Google Scholar
9. Z. Zhang, S. Zhou, L. Fang, J. Guan, Y. Zhang and Y. Zhang , Maximal planar scale-free Sierpinski networks with small-world effect and power law strength-degree correlation, Europhys. Lett. 79 (2007) 38007. Crossref, Google Scholar
10. Z. Zhang, S. Zhou, Z. Su, T. Zou, J. Guan and J. Guan , Random Sierpinski network with scale-free small-world and modular structure, Eur. Phys. J. B 65 (2008) 141–147. Crossref, Web of Science, Google Scholar
11. M. Dai, D. Ye, J. Hou, X. Li and X. Li , Scaling of average weighted receiving time on double-weighted Koch networks, Fractals 23(2) (2015) 1550011. Link, Web of Science, Google Scholar
12. M. Dai, Q. Xie, L. Xi and L. Xi , Trapping on weighted tetrahedron Koch networks with small-world property, Fractals 22(01–02) (2014) 1450006. Link, Google Scholar
13. Z. Zhang, S. Zhou, L. Chen, M. Yin, J. Guan and J. Guan , The exact solution of the mean geodesic distance for Vicsek fractals, J. Phys. A: Math. Theor. 41 (2008) 485102. Crossref, Web of Science, Google Scholar
14. J. Deng, Q. Ye, Q. Wang and Q. Wang , Weighted average geodesic distance of Vicsek network, Phys. A 527 (2019) 121327. Crossref, Web of Science, Google Scholar
15. M. Dai, J. He, Y. Zong, T. Ju, Y. Sun, W. Su and W. Su , Laplacian spectrum and coherence analysis of weighted hypercube network, Chaos Solitons Fractals 115 (2018) 29–34. Crossref, Web of Science, Google Scholar
16. J. He, Y. Xue and Y. Xue , Scale-free and small-world properties of hollow cube networks, Chaos Solitons Fractals 113 (2018) 11–15. Crossref, Web of Science, Google Scholar
17. L. Xi, L. Wang, S. Wang, Z. Yu, Q. Wang and Q. Wang , Fractality and scale-free effect of a class of self-similar networks, Phys. A 478 (2017) 31–40. Crossref, Web of Science, Google Scholar
18. Z. Li, Z. Yu, L. Xi and L. Xi , Scale-free effect of substitution networks, Phys. A 492 (2018) 1449–1455. Crossref, Web of Science, Google Scholar
19. Z. Li, J. Yao, Q. Wan and Q. Wan , Fractality of multiple colored substitution networks, Phys. A 525 (2019) 402–408. Crossref, Web of Science, Google Scholar
20. Q. Ye, L. Xi and L. Xi , Average distance of substitution networks, Fractals 27(6) (2019) 1950097. Link, Web of Science, Google Scholar
21. K. Cui, C. Chen, Y. Ma, L. Xi and L. Xi , Eccentric Fermat distance sum on a class of substitution networks, Internat. J. Modern Phys. B 37(17) (2023) 22350165. Link, Web of Science, Google Scholar
22. Z. Li, T. Britz and T. Britz , On the scale-freeness of random colored substitution networks, Proc. Amer. Math. Soc. 152 (2024) 1377–1389. Web of Science, Google Scholar
23. H. Bao, L. Xi, B. Zhao and B. Zhao , Self-similar touching networks-part I, Fractals 29(6) (2021) 2150172. Link, Web of Science, Google Scholar
24. J. Fan, L. Xi, B. Zhao and B. Zhao , Self-similar touching networks-part II, Fractals 29(6) (2021) 2150173. Link, Web of Science, Google Scholar
25. Q. Zeng, L. Xi and L. Xi , Hausdorff dimension of a family of networks, Fractals 31(1) (2023) 2350016. Link, Web of Science, Google Scholar
26. R. Kaufman and R. Kaufman , On Hausdorff dimension of projections, Mathematika 15(2) (1968) 153–155. Crossref, Web of Science, Google Scholar
27. J. Fan, J. Xu, L. Xi and L. Xi , Shortest path distance and fractal dimension of Sierpinski carpet networks, Fractals 32(3) (2024) 2450056. Link, Web of Science, Google Scholar