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Avalanches in a short-memory excitable network

Part of: Markov processes Special processes Operations research and management science

Published online by Cambridge University Press:  08 October 2021

Reza Rastegar  and
Alexander Roitershtein
Reza Rastegar*
Affiliation:
Occidental Petroleum Corporation and University of Tulsa
Alexander Roitershtein*
Affiliation:
Texas A&M University
*
*Postal address: Occidental Petroleum Corporation, Houston, TX 77046. Email address: reza_rastegar2@oxy.com
**Postal address: Department of Statistics, Texas A&M University, College Station, TX 77843. Email address: alexander@stat.tamu.edu
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Abstract

We study propagation of avalanches in a certain excitable network. The model is a particular case of the one introduced by Larremore et al. (Phys. Rev. E, 2012) and is mathematically equivalent to an endemic variation of the Reed–Frost epidemic model introduced by Longini (Math. Biosci., 1980). 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.

Keywords

Complex networkscascading failurescriticalitybranching processesdynamic graphs

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60J85: Applications of branching processes
Secondary: 92D25: Population dynamics (general) 90B15: Network models, stochastic 60K40: Other physical applications of random processes
Type
Original Article
Information
Advances in Applied Probability , Volume 53 , Issue 3 , September 2021 , pp. 609 - 648
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Copyright
© The Author(s) 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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