Statistical Investigation of Avalanches of Three Dimensional Small-World Networks and their Boundary and Bulk Cross-Sections

TL;DR

This paper investigates avalanche behavior in a 3D small-world sandpile model, revealing how avalanche statistics transition from regular BTW behavior to dissipative regimes as long-range link concentration varies, with implications for energy propagation in complex networks.

Contribution

It introduces a 3D small-world sandpile model with boundary dissipation, analyzing how avalanche statistics evolve with long-range link density and establishing scaling relations.

Findings

Avalanche statistics follow a power-law trend with increasing long-range links.

Different regimes of behavior are identified for small and intermediate link densities.

Spatial extent of BTW-like behavior scales with the link concentration, akin to dissipative BTW models.

Abstract

In many situations we are interested in the propagation of energy in some portions of a three dimensional system with dilute long-range links. In this paper sandpile model is defined on the three-dimensional small world network with real dissipative boundaries and the energy propagation is studied in three-dimensions as well as the two-dimensional cross sections. Two types of cross section are defined in the system, one in the bulk and another in the system boundary. The motivation of this is to make clear how the statistics of the avalanches in the bulk cross section tend to the statistics of the dissipative avalanches, defined in the boundaries as the concentration of long range links ($\alpha$) increases. This trend is numerically shown to be power law in a manner described in the paper. Two regimes of $\alpha$ are considered in this work. For sufficiently small $\alpha$s the dominant behavior of the system is just like that of the regular BTW, whereas for the intermediate values the behavior is non-trivial with some exponents that are reported in the paper. It is shown that the spatial extent up to which the statistics is similar to the regular BTW model scales with $\alpha$ just like the dissipative BTW model with the dissipation factor (mass in the corresponding ghost model) $m^2\sim \alpha$ for the three-dimensional system as well as its two-dimensional cross-sections.