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A review of fractality and self-similarity in complex networks

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  • Gallos, Lazaros K.
  • Song, Chaoming
  • Makse, Hernán A.

Abstract

We review recent findings of self-similarity in complex networks. Using the box-covering technique, it was shown that many networks present a fractal behavior, which is seemingly in contrast to their small-world property. Moreover, even non-fractal networks have been shown to present a self-similar picture under renormalization of the length scale. These results have an important effect in our understanding of the evolution and behavior of such systems. A large number of network properties can now be described through a set of simple scaling exponents, in analogy with traditional fractal theory.

Suggested Citation

  • Gallos, Lazaros K. & Song, Chaoming & Makse, Hernán A., 2007. "A review of fractality and self-similarity in complex networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 386(2), pages 686-691.
    • Handle: RePEc:eee:phsmap:v:386:y:2007:i:2:p:686-691
    DOI: 10.1016/j.physa.2007.07.069
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    Cited by:

    1. Xie, Wen-Jie & Zhou, Wei-Xing, 2011. "Horizontal visibility graphs transformed from fractional Brownian motions: Topological properties versus the Hurst index," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(20), pages 3592-3601.
    2. Yu-Hsiang Fu & Chung-Yuan Huang & Chuen-Tsai Sun, 2017. "A community detection algorithm using network topologies and rule-based hierarchical arc-merging strategies," PLOS ONE, Public Library of Science, vol. 12(11), pages 1-30, November.
    3. Ikeda, Nobutoshi, 2021. "Stratified structure of fractal scale-free networks generated by local rules," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 583(C).
    4. Ma, Jing & Li, Dandan & Tian, Zihao, 2016. "Rumor spreading in online social networks by considering the bipolar social reinforcement," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 447(C), pages 108-115.
    5. Ikeda, Nobutoshi, 2019. "Growth model for fractal scale-free networks generated by a random walk," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 521(C), pages 424-434.
    6. Liu, Ying & Tang, Ming & Zhou, Tao & Do, Younghae, 2016. "Identify influential spreaders in complex networks, the role of neighborhood," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 452(C), pages 289-298.
    7. Chen, Jin & Le, Anbo & Wang, Qin & Xi, Lifeng, 2016. "A small-world and scale-free network generated by Sierpinski Pentagon," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 449(C), pages 126-135.
    8. Feng, Shiyuan & Weng, Tongfeng & Chen, Xiaolu & Ren, Zhuoming & Su, Chang & Li, Chunzi, 2024. "Scaling law of diffusion processes on fractal networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 640(C).
    9. Rosenberg, Eric, 2018. "Generalized Hausdorff dimensions of a complex network," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 511(C), pages 1-17.
    10. Li, Dongyan & Wang, Xingyuan & Huang, Penghe, 2017. "A fractal growth model: Exploring the connection pattern of hubs in complex networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 471(C), pages 200-211.
    11. Fu, Yu-Hsiang & Huang, Chung-Yuan & Sun, Chuen-Tsai, 2016. "Using a two-phase evolutionary framework to select multiple network spreaders based on community structure," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 461(C), pages 840-853.
    12. Xi, Lifeng & Wang, Lihong & Wang, Songjing & Yu, Zhouyu & Wang, Qin, 2017. "Fractality and scale-free effect of a class of self-similar networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 478(C), pages 31-40.
    13. Blagus, Neli & Šubelj, Lovro & Bajec, Marko, 2012. "Self-similar scaling of density in complex real-world networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(8), pages 2794-2802.
    14. Rasul Kochkarov & Azret Kochkarov, 2022. "Introduction to the Class of Prefractal Graphs," Mathematics, MDPI, vol. 10(14), pages 1-17, July.
    15. Maiorino, Enrico & Livi, Lorenzo & Giuliani, Alessandro & Sadeghian, Alireza & Rizzi, Antonello, 2015. "Multifractal characterization of protein contact networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 428(C), pages 302-313.
    16. Blagus, Neli & Šubelj, Lovro & Bajec, Marko, 2014. "Assessing the effectiveness of real-world network simplification," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 413(C), pages 134-146.
    17. Jiang, Jincheng & Xu, Zhihua & Zhang, Zhenxin & Zhang, Jie & Liu, Kang & Kong, Hui, 2023. "Revealing the fractal and self-similarity of realistic collective human mobility," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 630(C).
    18. Karimui, Reza Yaghoobi, 2021. "A new approach to measure the fractal dimension of a trajectory in the high-dimensional phase space," Chaos, Solitons & Fractals, Elsevier, vol. 151(C).
    19. Craig, Adam & Yücel, Mesut & Muchnik, Lev & Hershberg, Uri, 2022. "Impact of finite size effect on applicability of generalized fractal and spectral dimensions to biological networks," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).
    20. Liudvikas Kaklauskas & Leonidas Sakalauskas, 2013. "Study of on-line measurement of traffic self-similarity," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 21(1), pages 63-84, January.
    21. Xu, Yan & Gurfinkel, Aleks Jacob & Rikvold, Per Arne, 2014. "Architecture of the Florida power grid as a complex network," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 401(C), pages 130-140.
    22. Dai, Meifeng & Shao, Shuxiang & Su, Weiyi & Xi, Lifeng & Sun, Yanqiu, 2017. "The modified box dimension and average weighted receiving time of the weighted hierarchical graph," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 475(C), pages 46-58.
    23. Ikeda, Nobutoshi, 2020. "Fractal networks induced by movements of random walkers on a tree graph," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 537(C).

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