Sandpile models and random walkers on finite lattices

Shilo, Yehiel; Biham, Ofer

doi:10.1103/physreve.67.066102

Cited by 15 publications

(17 citation statements)

References 51 publications

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“…Therefore a crossover from diffusive to super diffusive sand transport occurs as RL is evolved to RN. It can be seen that not only the scaling of n φ and s φ are same but also the magnitude of n φ is just twice of s φ for both φ = 0 and φ = 1 on a given L. On RL it was already known that n φ = 2 s φ [35,40]. Such a relationship is then also valid on RN.…”

Section: Diffusive To Super Diffusive Sand Transportmentioning

confidence: 72%

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Critical properties of a dissipative sandpile model on small-world networks

Bhaumik

Santra

2013

Phys. Rev. E

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A dissipative sandpile model (DSM) is constructed and studied on small world networks (SWN). SWNs are generated adding extra links between two arbitrary sites of a two dimensional square lattice with different shortcut densities φ. Three different regimes are identified as regular lattice (RL) for φ 2 −12 , SWN for 2 −12 < φ < 0.1 and random network (RN) for φ ≥ 0.1. In the RL regime, the sandpile dynamics is characterized by usual Bak, Tang, Weisenfeld (BTW) type correlated scaling whereas in the RN regime it is characterized by the mean field (MF) scaling. On SWN, both the scaling behaviors are found to coexist. Small compact avalanches below certain characteristic size sc are found to belong to the BTW universality class whereas large, sparse avalanches above sc are found to belong to the MF universality class. A scaling theory for the coexistence of two scaling forms on SWN is developed and numerically verified. Though finite size scaling (FSS) is not valid for DSM on RL as well as on SWN, it is found to be valid on RN for the same model. FSS on RN is appeared to be an outcome of super diffusive sand transport and uncorrelated toppling waves.

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Section: Diffusive To Super Diffusive Sand Transportmentioning

confidence: 72%

“…It should be noted here that on a two dimensional regular square lattice n φ ≈ aL + bL 2 , where a = 0.56 and b = 0.14 for small L [40]. However in the limit L → ∞, such a scaling can be approximated as n φ (L) ≈ 0.14L 2 .…”

Section: Diffusive To Super Diffusive Sand Transportmentioning

confidence: 83%

Critical properties of a dissipative sandpile model on small-world networks

Bhaumik

Santra

2013

Phys. Rev. E

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“…These phenomena are frequently modelled by means of a class of cellular automata known as sandpile models [18] that can mimic financial distress propagation [24]; financial crisis fluctuations are described, at least qualitatively, by the avalanches of a sandpile model [25]. Conservation laws provide the connection between sandpile models and random-walk (diffusive) models [26].…”

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confidence: 99%

Robustness and assortativity for diffusion-like processes in scale-free networks

D’Agostino¹,

Scala²,

Zlatić³

et al. 2012

EPL

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-By analysing the diffusive dynamics of epidemics and of distress in complex networks, we study the effect of the assortativity on the robustness of the networks. We first determine by spectral analysis the thresholds above which epidemics/failures can spread; we then calculate the slowest diffusional times. Our results shows that disassortative networks exhibit a higher epidemiological threshold and are therefore easier to immunize, while in assortative networks there is a longer time for intervention before epidemic/failure spreads. Moreover, we study by computer simulations the sandpile cascade model, a diffusive model of distress propagation (financial contagion). We show that, while assortative networks are more prone to the propagation of epidemic/failures, degree-targeted immunization policies increases their resilience to systemic risk.

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“…It can be seen that all the α x (q)s do not converge for the BTW model upto q = 4 whereas in the case of SSM, there is a clear convergence of α x (q), which confirms that though the spatial structure is random fractal the BTW retains its multifractal behavior whereas the SSM obeys FSS. For lattices with integer dimension it is already known that the average toppling size s is equivalent to the average number of steps of a random walker on a given lattice before it reaches the boundary starting from an arbitrary lattice point [36,37]. Thus one could get a relation…”

Section: Probability Distribution Functionmentioning

confidence: 99%

Critical properties of deterministic and stochastic sandpile models on two-dimensional percolation backbone

Bhaumik

Santra

2020

Physica A: Statistical Mechanics and its Applications

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Both the deterministic and stochastic sandpile models are studied on the percolation backbone, a random fractal, generated on a square lattice in 2-dimensions. In spite of the underline random structure of the backbone, the deterministic Bak Tang Wiesenfeld (BTW) model preserves its positive time auto-correlation and multifractal behaviour due to its complete toppling balance, whereas the critical properties of the stochastic sandpile model (SSM) still exhibits finite size scaling (FSS) as it exhibits on the regular lattices. Analyzing the topography of the avalanches, various scaling relations are developed. While for the SSM, the extended set of critical exponents obtained is found to obey various the scaling relation in terms of the fractal dimension d B f of the backbone, whereas the deterministic BTW model, on the other hand, does not. As the critical exponents of the SSM defined on the backbone are related to d B f , the backbone fractal dimension, they are found to be entirely different from those of the SSM defined on the regular lattice as well as on other deterministic fractals. The SSM on the percolation backbone is found to obey FSS but belongs to a new stochastic universality class.

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Sandpile models and random walkers on finite lattices

Cited by 15 publications

References 51 publications

Critical properties of a dissipative sandpile model on small-world networks

Critical properties of a dissipative sandpile model on small-world networks

Robustness and assortativity for diffusion-like processes in scale-free networks

Critical properties of deterministic and stochastic sandpile models on two-dimensional percolation backbone

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