Sandpile avalanche dynamics on scale-free networks

TL;DR

This study investigates avalanche dynamics on scale-free networks using a sandpile model, revealing how degree distribution and node thresholds influence avalanche size and duration distributions with specific power-law exponents.

Contribution

It introduces a modified sandpile model on scale-free networks with degree-dependent thresholds and derives analytical expressions for avalanche distributions.

Findings

Avalanche size and duration follow power-law distributions with exponents depending on network parameters.

The exponents are explicitly derived as functions of the degree exponent and threshold parameter.

Logarithmic corrections appear at critical network configurations.

Abstract

Avalanche dynamics is an indispensable feature of complex systems. Here we study the self-organized critical dynamics of avalanches on scale-free networks with degree exponent $\gamma$ through the Bak-Tang-Wiesenfeld (BTW) sandpile model. The threshold height of a node $i$ is set as $k_i^{1-\eta}$ with $0\leq\eta<1$, where $k_i$ is the degree of node $i$. Using the branching process approach, we obtain the avalanche size and the duration distribution of sand toppling, which follow power-laws with exponents $\tau$ and $\delta$, respectively. They are given as $\tau=(\gamma-2 \eta)/(\gamma-1-\eta)$ and $\delta=(\gamma-1-\eta)/(\gamma-2)$ for $\gamma<3-\eta$, 3/2 and 2 for $\gamma>3-\eta$, respectively. The power-law distributions are modified by a logarithmic correction at $\gamma=3-\eta$.