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Investigating Scaling Behaviour of Avalanches in Sandpiles on Small-World Networks Using the Renormalization Approach

Moammar, Sina | 2025

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 58147 (04)
  4. University: Sharif University of Technology
  5. Department: Physics
  6. Advisor(s): Moghimi Araghi, Saman
  7. Abstract:
  8. The Manna sandpile model is a significant and widely-used model in the study of self-organized criticality. Various avalanche-related quantities, such as area, size, duration, and others, exhibit power-law distributions with finite-size effects. It has been demonstrated that this model exhibits simple monofractal behavior on both regular lattices and random networks, and the finite-size scaling (FSS) hypothesis holds for the avalanche distribution functions. On the other hand, it has been observed that a wide range of natural and human-made networks are small-world networks. Consequently, studying the Manna sandpile model on such networks and understanding its features and behavior can provide deeper insights into how self-organized criticality operates in the real world. Previous studies have reported that for such small-world networks, the FSS hypothesis does not hold, and data collapse across different network sizes is not observed. However, in this research, by reviewing the results of prior works and drawing inspiration from the sandpile model with bulk dissipation, we argue that the breakdown of FSS on these networks is not inevitable. The recovery of FSS depends on understanding the changes in the tail shapes of the distribution functions as a function of the network's dissipation strength. To address this, we introduce a generalized Manna model with arbitrary boundary dissipation and investigate the effects of network size and boundary dissipation probability on the shape of its distribution functions. Using simulation data and, ultimately, analytical and theoretical approaches, we demonstrate that the FSS hypothesis is valid for our generalized model. Additionally, we derive the necessary conditions among the model parameters to achieve data collapse. This relationship represents a renormalization group transformation in the model's parameter space. Accordingly, we calculate the beta function of this transformation, along with its fixed points and their properties. We find that the original Manna model and the Manna model with zero dissipation are, respectively, attractive and repulsive fixed points of this transformation.
  9. Keywords:
  10. Self Organized Criticality ; Manna Model ; Small World Network ; Finite Size Scaling ; Renormalization Group ; Sand Pile Model

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