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Determination of fractal dimensions for geometrical multifractals

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  • Tél, Tamás
  • Fülöp, Ágnes
  • Vicsek, Tamás

Abstract

Two independent approaches, the box counting and the sand box methods are used for the determination of the generalized dimensions (Dq) associated with the geometrical structure of growing deterministic fractals. We find that the multifractal nature of the geometry results in an unusually slow convergence of the numerically calculated Dq's to their true values. Our study demonstrates that the above-mentioned two methods are equivalent only if the sand box method is applied with an averaging over randomly selected centres. In this case the latter approach provides better estimates of the generalized dimensions.

Suggested Citation

  • Tél, Tamás & Fülöp, Ágnes & Vicsek, Tamás, 1989. "Determination of fractal dimensions for geometrical multifractals," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 159(2), pages 155-166.
    • Handle: RePEc:eee:phsmap:v:159:y:1989:i:2:p:155-166
    DOI: 10.1016/0378-4371(89)90563-3
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    Cited by:

    1. Yu, Z.G. & Anh, V.V. & Wanliss, J.A. & Watson, S.M., 2007. "Chaos game representation of the Dst index and prediction of geomagnetic storm events," Chaos, Solitons & Fractals, Elsevier, vol. 31(3), pages 736-746.
    2. Kanevski, Mikhail & Pereira, Mário G., 2017. "Local fractality: The case of forest fires in Portugal," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 479(C), pages 400-410.
    3. Alison K. Cheeseman & Edward R. Vrscay, 2022. "Estimating the Fractal Dimensions of Vascular Networks and Other Branching Structures: Some Words of Caution," Mathematics, MDPI, vol. 10(5), pages 1-21, March.
    4. Yuxin Zhao & Shuai Chang & Chang Liu, 2015. "Multifractal theory with its applications in data management," Annals of Operations Research, Springer, vol. 234(1), pages 133-150, November.
    5. Pavón-Domínguez, Pablo & Moreno-Pulido, Soledad, 2022. "Sandbox fixed-mass algorithm for multifractal unweighted complex networks," Chaos, Solitons & Fractals, Elsevier, vol. 156(C).
    6. Huang, Da-Wen & Yu, Zu-Guo & Anh, Vo, 2017. "Multifractal analysis and topological properties of a new family of weighted Koch networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 469(C), pages 695-705.
    7. Retière, N. & Sidqi, Y. & Frankhauser, P., 2022. "A steady-state analysis of distribution networks by diffusion-limited-aggregation and multifractal geometry," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 600(C).
    8. Pavón-Domínguez, Pablo & Moreno-Pulido, Soledad, 2020. "A Fixed-Mass multifractal approach for unweighted complex networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 541(C).
    9. Golay, Jean & Kanevski, Mikhail & Vega Orozco, Carmen D. & Leuenberger, Michael, 2014. "The multipoint Morisita index for the analysis of spatial patterns," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 406(C), pages 191-202.
    10. Pavón-Domínguez, P. & Rincón-Casado, A. & Ruiz, P. & Camacho-Magriñán, P., 2018. "Multifractal approach for comparing road transport network geometry: The case of Spain," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 510(C), pages 678-690.
    11. Dailyudenko, Victor F., 2008. "Topological considerations of an attractor based on temporal locality along its phase trajectories," Chaos, Solitons & Fractals, Elsevier, vol. 37(3), pages 876-893.
    12. François Sémécurbe & Cécile Tannier & Stéphane G. Roux, 2019. "Applying two fractal methods to characterise the local and global deviations from scale invariance of built patterns throughout mainland France," Journal of Geographical Systems, Springer, vol. 21(2), pages 271-293, June.

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