Scaling of spin avalanches in growing networks

TL;DR

This paper investigates how spin avalanches propagate in growing complex networks with antiferromagnetic interactions, revealing that avalanche statistics mirror network degree distributions and scale with network size.

Contribution

It introduces the damage spreading method to analyze spin avalanches, showing their statistics match degree distributions and identifying scaling laws for avalanche range.

Findings

Avalanche spectra follow the same statistics as degree distributions.

The mean avalanche range scales with size as Z/N^β = f(s/N^α).

Scaling exponents α=0.5 and β=0.33 are consistent across network types.

Abstract

Growing networks decorated with antiferromagnetically coupled spins are archetypal examples of complex systems due to the frustration and the multivalley character of their energy landscapes. Here we use the damage spreading method (DS) to investigate the cohesion of spin avalanches in the exponential networks and the scale-free networks. On the contrary to the conventional methods, the results obtained from DS suggest that the avalanche spectra are characterized by the same statistics as the degree distribution in their home networks. Further, the obtained mean range $Z$ of an avalanche, i.e. the maximal distance reached by an avalanche from the damaged site, scales with the avalanche size $s$ as $Z/N^\beta =f(s/N^{\alpha})$, where $\alpha=0.5$ and $\beta=0.33$. These values are true for both kinds of networks for the number $M$ of nodes to which new nodes are attached between 4 and 10; a check for M=25 confirms these values as well.