Bottom-up model of self-organized criticality on networks

Pierre-André Noël, Charles D. Brummitt, and Raissa M. D'Souza

Phys. Rev. E 89, 012807 – Published 15 January 2014

Abstract

The Bak-Tang-Wiesenfeld (BTW) sandpile process is an archetypal, stylized model of complex systems with a critical point as an attractor of their dynamics. This phenomenon, called self-organized criticality, appears to occur ubiquitously in both nature and technology. Initially introduced on the two-dimensional lattice, the BTW process has been studied on network structures with great analytical successes in the estimation of macroscopic quantities, such as the exponents of asymptotically power-law distributions. In this article, we take a microscopic perspective and study the inner workings of the process through both numerical and rigorous analysis. Our simulations reveal fundamental flaws in the assumptions of past phenomenological models, the same models that allowed accurate macroscopic predictions; we mathematically justify why universality may explain these past successes. Next, starting from scratch, we obtain microscopic understanding that enables mechanistic models; such models can, for example, distinguish a cascade's area from its size. In the special case of a 3-regular network, we use self-consistency arguments to obtain a zero-parameter mechanistic (bottom-up) approximation that reproduces nontrivial correlations observed in simulations and that allows the study of the BTW process on networks in regimes otherwise prohibitively costly to investigate. We then generalize some of these results to configuration model networks and explain how one could continue the generalization. The numerous tools and methods presented herein are known to enable studying the effects of controlling the BTW process and other self-organizing systems. More broadly, our use of multitype branching processes to capture information bouncing back and forth in a network could inspire analogous models of systems in which consequences spread in a bidirectional fashion.

Received 20 October 2013

DOI:https://doi.org/10.1103/PhysRevE.89.012807

Authors & Affiliations

Pierre-André Noël^1,2,*, Charles D. Brummitt^1,3, and Raissa M. D'Souza^1,2,4,5

¹Complexity Sciences Center, University of California, Davis, California 95616, USA
²Department of Computer Science, University of California, Davis, California 95616, USA
³Department of Mathematics, University of California, Davis, California 95616, USA
⁴Department of Mechanical and Aerospace Engineering, University of California, Davis, California 95616, USA
⁵Santa Fe Institute, Santa Fe, New Mexico 87501, USA

^*noel.pierre.andre@gmail.com

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Vol. 89, Iss. 1 — January 2014

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Figure 1
Cascade size distribution for a scale-free graph generated by the static model [32] with $N = 10^{7}$ nodes, power-law degree exponent $γ = 2.5$ , average degree $6.3$ , and dissipation rate $ε = 10^{- 3}$ . We use only the largest component (containing $9 503 592$ nodes), and we collect statistics on the last $10^{8}$ cascades out of $1.2 \times 10^{8}$ cascades. We logarithmically bin the data for all cascades (open circles) and for the bulk cascades in which no sand dissipates (pluses). Past work based on the $1 / k$ assumption [21] successfully predicts that the cascade size distribution should follow a power law with exponent $τ = γ / (γ - 1) = 5 / 3$ (line).
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Figure 2
Finite treelike cascades as an assemblage of patterns. (a) The graph before the cascade. Dark circles represent nodes initially at capacity, while light circles represent nodes initially not at capacity. (b) Starting at the root (indicated by an arrow), the cascade is assembled from patterns that encode how the grains of sand are shed during the process. Note that this assemblage may form a tree even though the original graph contained cycles. (c) The only possible patterns and rules for assembling them. The roman numerals correspond to the cases in Corollary 1. The convex shapes i and v represent root patterns, while the concave shapes represent nonroot patterns. The large arrow represents the first grain dropped on the root, a curved arrow indicates dissipation, and the other arrows indicate exchanges of sand between neighboring nodes. The signature of a nonroot pattern may be inferred by the arrows on its left. A large $\times$ marks a forbidden special case for patterns v and vi.
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Figure 3
Example of a cascade forming a finite tree $G^{†}$ from the pattern perspective. Refer to the caption of Fig. 2 for notation. There are 16 nodes in $G^{†}$ , so 16 patterns are required. All 8 nodes initially not at capacity do not topple [all pattern (ii)]. A single node at capacity does not topple (top left) because the grain sent towards it dissipates [pattern (iv)]. The 7 remaining nodes are all initially at capacity: 2 nodes each topple twice [the root, center left, has pattern (v) and the node immediately to the right of the root has pattern (vii)] and the 5 others topple once [4 have pattern (vi) and 1 has pattern (vii)]. The resulting cascade area is 7 and the cascade size is 9.
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Figure 4
Cascade size distribution for a 3-regular network with $N = 10^{7}$ nodes and dissipation rate $ε = 10^{- 3}$ . The data for both Monte Carlo simulations (open circles) and theory (closed circles) have been logarithmically binned. The theory is obtained for $ψ_{i}$ and $ϕ_{i j}$ shown in Eq. (5d), i.e., those satisfying the constraints [Eq. (5d)] of the self-organizing model.
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Figure 5
Distribution of the difference between cascade size and cascade area for a 3-regular network in the same conditions as in Fig. 4. We consider the difference between the two quantities to facilitate the comparison. Note that the size of a cascade is always greater than or equal to its area.
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