Optimal local interventions in the two-dimensional Abelian sandpile model

TL;DR

This paper rigorously analyzes how targeted interventions in the Abelian sandpile model can effectively reduce the size of avalanches, balancing mitigation of large events with the frequency of smaller ones.

Contribution

It introduces a formal method to evaluate and optimize interventions in the sandpile model, providing new insights into controlling self-organized criticality.

Findings

Optimal intervention locations balance reducing large avalanches and increasing smaller ones.

Explicit analysis of interventions on square critical components.

Development of a method to compute expected avalanche sizes from critical vertices.

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.