Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimal local interventions in the two-dimensional Abelian sandpile model

Maike C. de Jongh, Richard J. Boucherie, M. N. M. van Lieshout

Abstract

The Abelian sandpile model serves as a canonical example of self-organized criticality. This critical behavior manifests itself through large cascading events triggered by small perturbations. Such large-scale events, known as avalanches, are often regarded as stylized representations of catastrophic phenomena, such as earthquakes or forest fires. Motivated by this perspective, we study strategies to reduce avalanche sizes. We provide a first rigorous analysis of the impact of interventions in the Abelian sandpile model, considering a setting in which an external controller can perturb a configuration by removing sand grains at selected locations. We first develop and formalize an extended method to compute the expected size of an avalanche originating from a connected component of critical vertices, i.e., vertices at maximum height. Using this method, we characterize the structure of avalanches starting from square components and explicitly analyze the effect of interventions in such components. Our results show that the optimal intervention locations strike an interesting balance between reduction of largest avalanche sizes and increasing the number of mitigated avalanches.

Optimal local interventions in the two-dimensional Abelian sandpile model

Abstract

The Abelian sandpile model serves as a canonical example of self-organized criticality. This critical behavior manifests itself through large cascading events triggered by small perturbations. Such large-scale events, known as avalanches, are often regarded as stylized representations of catastrophic phenomena, such as earthquakes or forest fires. Motivated by this perspective, we study strategies to reduce avalanche sizes. We provide a first rigorous analysis of the impact of interventions in the Abelian sandpile model, considering a setting in which an external controller can perturb a configuration by removing sand grains at selected locations. We first develop and formalize an extended method to compute the expected size of an avalanche originating from a connected component of critical vertices, i.e., vertices at maximum height. Using this method, we characterize the structure of avalanches starting from square components and explicitly analyze the effect of interventions in such components. Our results show that the optimal intervention locations strike an interesting balance between reduction of largest avalanche sizes and increasing the number of mitigated avalanches.

Paper Structure

This paper contains 6 sections, 10 theorems, 60 equations, 14 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Definition of the Abelian sandpile model
  3. Avalanche development
  4. Reducing avalanche sizes through strategic interventions
  5. Conclusion
  6. Remaining proofs

Key Result

Lemma 3.1

Given a stable sandpile configuration $\eta \in \Omega$ and a generator $A = \{v_{i_1}, v_{i_2}, \ldots, v_{i_k}\} \subseteq V$, we have $W_{i_{\ell}1}(\eta) = W_{i_m1}(\eta)$ for all $\ell, m = 1, \ldots, k$.

Figures (14)

  • Figure 1: Example to which the method of Dorso does not apply. The generator outlined in bold splits into two components after the first wave, each producing second waves of different sizes.
  • Figure 2: Directed graph constructed in Algorithm 1 applied to a square generator (left) with $N = 2$ and sandpile configuration $\eta^{A^{(2)}}$ (right).
  • Figure 3: Directed graph constructed in Algorithm 1 applied to a square generator (left) and sandpile configuration $\eta^{A^{(k)}}$ (right).
  • Figure 4: Directed graph constructed in Algorithm 1 applied to generator $\tilde{A}_R^{2,1}$ in configuration $\gamma_i \eta$ (left) and sandpile configuration $(\gamma_i \eta)^{\tilde{A}_R^{2,1}}$ (right).
  • Figure 5: Directed graph constructed in Algorithm 1 applied to generator $\tilde{A}_R^{3,1}$ in configuration $\gamma_i \eta$ (left) and sandpile configuration $(\gamma_i\eta)^{\tilde{A}_R^{3,1}}$ (right).
  • ...and 9 more figures

Theorems & Definitions (21)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 4.1
  • proof
  • Remark 1: Local confinement of avalanches and upper bound
  • Theorem 4.2
  • ...and 11 more