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Directed self-organized critical patterns emerging from fractional Brownian paths

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  • Carbone, Anna
  • Stanley, H.Eugene

Abstract

We discuss a family of clusters C corresponding to the region whose boundary is formed by a fractional Brownian path y(i) and by the moving average function yn(i)≡1n∑k=0n−1y(i−k). Our model generates fractal directed patterns showing spatio-temporal complexity, and we demonstrate that the cluster area, length and duration exhibit the characteristic scaling behavior of SOC clusters. The function Cn(i) acts as a magnifying lens, zooming in (or out) the ‘avalanches’ formed by the cluster construction rule, where the magnifying power of the zoom is set by the value of the amplitude window n. On the basis of the construction rule of the clusters Cn(i)≡y(i)−yn(i) and of the relationship among the exponents, we hypothesize that our model might be considered to be a generalized stochastic directed model, including the Dhar–Ramaswamy (DR) model and the stochastic models as particular cases. As in the DR model, the growth and annihilation of our clusters are obtained from the set of intersections of two random walk paths, and we argue that our model is a variant of the directed self-organized criticality scheme of the DR model.

Suggested Citation

  • Carbone, Anna & Stanley, H.Eugene, 2004. "Directed self-organized critical patterns emerging from fractional Brownian paths," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 340(4), pages 544-551.
    • Handle: RePEc:eee:phsmap:v:340:y:2004:i:4:p:544-551
    DOI: 10.1016/j.physa.2004.05.004
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    Citations

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    Cited by:

    1. Stanley, H.E. & Gabaix, Xavier & Gopikrishnan, Parameswaran & Plerou, Vasiliki, 2007. "Economic fluctuations and statistical physics: Quantifying extremely rare and less rare events in finance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 382(1), pages 286-301.
    2. Jiang, Zhi-Qiang & Xie, Wen-Jie & Zhou, Wei-Xing, 2014. "Testing the weak-form efficiency of the WTI crude oil futures market," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 405(C), pages 235-244.
    3. Dumansky, Y.V. & Lyakh, Y.E. & Gorshkov, O.G. & Gurianov, V.G. & Prihodchenko, V.V., 2012. "Fractal dimensionality analysis of normal and cancerous mammary gland thermograms," Chaos, Solitons & Fractals, Elsevier, vol. 45(12), pages 1494-1500.
    4. Stanley, H. Eugene & Plerou, Vasiliki & Gabaix, Xavier, 2008. "A statistical physics view of financial fluctuations: Evidence for scaling and universality," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(15), pages 3967-3981.
    5. He, Ling-Yun & Chen, Shu-Peng, 2010. "Are developed and emerging agricultural futures markets multifractal? A comparative perspective," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(18), pages 3828-3836.
    6. Xie, Wen-Jie & Jiang, Zhi-Qiang & Zhou, Wei-Xing, 2014. "Extreme value statistics and recurrence intervals of NYMEX energy futures volatility," Economic Modelling, Elsevier, vol. 36(C), pages 8-17.

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