Number theoretic example of scale-free topology inducing self-organized criticality

TL;DR

This paper introduces a simple number-theoretic model demonstrating how scale-free network topologies can lead to self-organized criticality, linking statistical physics and number theory.

Contribution

It presents the simplest analytically tractable self-organized critical model based on number theory, illustrating the role of network topology in criticality emergence.

Findings

Model exhibits scale-invariance and critical scaling relations

Analytical characterization of the critical state

Connections established between physics and number theory

Abstract

In this work we present a general mechanism by which simple dynamics running on networks become self-organized critical for scale free topologies. We illustrate this mechanism with a simple arithmetic model of division between integers, the division model. This is the simplest self-organized critical model advanced so far, and in this sense it may help to elucidate the mechanism of self-organization to criticality. Its simplicity allows analytical tractability, characterizing several scaling relations. Furthermore, its mathematical nature brings about interesting connections between statistical physics and number theoretical concepts. We show how this model can be understood as a self-organized stochastic process embedded on a network, where the onset of criticality is induced by the topology.