[Paper Review] Self-organized (quasi-)criticality: the extremal Feder and Feder model

Osame Kinouchi|arXiv (Cornell University)|Feb 27, 1998

Markov Chains and Monte Carlo Methods1 references3 citations

TL;DR

This paper introduces a novel self-organized criticality (SOC) model combining features of the Bak-Sneppen and slip-stick models, using a random-neighbor extremal dynamics approach. It analytically derives the stationary state distribution and exactly calculates the mean avalanche size as a function of coupling strength in the thermodynamic limit, revealing a (quasi-)critical regime with finite but diverging mean avalanche sizes.

ABSTRACT

A simple random-neighbor SOC model that combines properties of the Bak-Sneppen and the relaxation oscillators (slip-stick) models is introduced. The analysis in terms of branching processes is transparent and gives insight about the development of large but finite mean avalanche sizes in dissipative models. In the thermodynamic limit, the distribution of states has a simple analytical form and the mean avalanche size, as a function of the coupling parameter strength, is exactly calculable.

Motivation & Objective

• To develop a simple, analytically tractable SOC model that unifies key features of the Bak-Sneppen and relaxation oscillator (slip-stick) models.
• To understand the emergence of large but finite mean avalanche sizes in dissipative SOC systems through a branching process framework.
• To derive the exact stationary state distribution and mean avalanche size in the thermodynamic limit.
• To explore how coupling strength controls the transition to a (quasi-)critical state with scale-invariant behavior.

Proposed method

• The model uses extremal dynamics where the least stable element is updated at each time step, combining random neighbor selection with a threshold-based failure mechanism.
• The system evolves via a branching process, where failures trigger cascades (avalanches) that propagate stochastically through the network.
• The stationary state distribution is derived analytically using mean-field approximation and branching process theory.
• The mean avalanche size is calculated exactly as a function of the coupling parameter, using generating functions and fixed-point analysis.
• The model is studied in the thermodynamic limit, where the system reaches a stationary state with power-law-like behavior.
• The analysis reveals a critical-like regime where the mean avalanche size diverges in the limit of infinite system size, though finite in practice.

Experimental results

Research questions

• RQ1 How does a random-neighbor SOC model with extremal dynamics exhibit (quasi-)critical behavior without fine-tuning?
• RQ2 What is the exact form of the stationary state distribution in the thermodynamic limit of this model?
• RQ3 How does the mean avalanche size depend on the coupling parameter in a dissipative SOC system?
• RQ4 Can the branching process framework accurately describe the avalanche dynamics in this model?
• RQ5 What conditions lead to the emergence of large but finite avalanche sizes in such models?

Key findings

• The stationary state distribution of the model has a simple, analytically tractable form in the thermodynamic limit.
• The mean avalanche size is exactly calculable as a function of the coupling parameter, showing a non-trivial dependence.
• The system exhibits (quasi-)critical behavior with a diverging mean avalanche size in the thermodynamic limit, indicating scale-invariant dynamics.
• The model displays finite but large avalanche sizes due to the interplay between local failure and global relaxation.
• The branching process analysis provides a transparent and intuitive explanation for the emergence of large-scale avalanches.
• The model successfully unifies features of the Bak-Sneppen and slip-stick models while remaining analytically solvable.