Avalanches in an excitable network

TL;DR

This paper rigorously analyzes avalanche propagation in an excitable network, establishing precise relations and bounds between the stochastic model and its deterministic and branching process approximations.

Contribution

It provides mathematical proofs connecting the avalanche model to its limiting approximations, including convergence rates and bounds.

Findings

Established convergence rates of the avalanche process to its deterministic limit

Derived bounds for avalanche size and duration

Clarified the relation between stochastic avalanches and branching process approximations

Abstract

We study propagation of avalanches in a certain excitable network. The model is a particular case of the one introduced in [23], and is mathematically equivalent to an endemic variation of the Reed-Frost epidemic model introduced in [27]. Two types of heuristic approximation are frequently used for models of this type in applications, a branching process for avalanches of a small size at the beginning of the process and a deterministic dynamical system once the avalanche spreads to a significant fraction of a large network. In this paper we prove several results concerning the exact relation between the avalanche model and these limits, including rates of convergence and rigorous bounds for common characteristics of the model.