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Avalanches in a short-memory excitable network

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

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

Information

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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References

Agresti, A. (1974). Bounds on the extinction time distribution of a branching process. Adv. Appl. Prob. 6, 322–335.10.2307/1426296CrossRefGoogle Scholar
Andersson, H. (1998). Limit theorems for a random graph epidemic model. Ann. Appl. Prob. 8, 1331–1349.10.1214/aoap/1028903384CrossRefGoogle Scholar
Athreya, K. B. (1988). On the maximum sequence in a critical branching processes. Ann. Prob. 16, 502–507.10.1214/aop/1176991770CrossRefGoogle Scholar
Athreya, K. B. (1994). Large deviation rates for branching processes—I. Single type case. Ann. Appl. Prob. 4, 779–790.Google Scholar
Athreya, K. B. and Ney, P. E. (2004). Branching Processes. Dover, Mineola, NY.Google Scholar
Aydogmus, O. (2016). On extinction time of a generalized endemic chain-binomial model. Math. Biosci. 279, 38–42.10.1016/j.mbs.2016.06.010CrossRefGoogle ScholarPubMed
Bailey, N. T. J. (1975). The Mathematical Theory of Infectious Diseases and Its Applications, 2nd edn. Hafner Press/Macmillan, New York.Google Scholar
Barbour, A. D. (2014). Couplings for locally branching epidemic processes. J. Appl. Prob. 51, 43–56.10.1239/jap/1417528465CrossRefGoogle Scholar
Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximation. The Clarendon Press, Oxford University Press, New York.Google Scholar
Bollobás, B. (2001). Random Graphs, 2nd edn. Cambridge University Press.10.1017/CBO9780511814068CrossRefGoogle Scholar
Buckley, F. M. and Pollett, P. K. (2010). Limit theorems for discrete-time metapopulation models. Prob. Surveys 7, 53–83.10.1214/10-PS158CrossRefGoogle Scholar
Chatterjee, S., Diaconis, P. and Meckes, E. (2005). Exchangeable pairs and Poisson approximation. Prob. Surveys 2, 64–106.10.1214/154957805100000096CrossRefGoogle Scholar
Chumley, T., Aydogmus, O., Matzavinos, A. and Roitershtein, A. (2018). Moran-type bounds for the fixation probability in a frequency-dependent Wright–Fisher model. J. Math. Biol. 76, 1–35.10.1007/s00285-017-1137-2CrossRefGoogle Scholar
Cocchi, L., Gollo, L. L., Zalesky, A. and Breakspear, M. (2017). Criticality in the brain: a synthesis of neurobiology, models and cognition. Prog. Neurobiol. 158, 132–152.10.1016/j.pneurobio.2017.07.002CrossRefGoogle ScholarPubMed
Cooke, K. L., Calef, D. F. and Level, E. V. (1977). Stability or chaos in discrete epidemic models. In Nonlinear Systems and Applications, ed. Lakshmikantham, V., Press, Academic, York, New, pp. 73–93.10.1016/B978-0-12-434150-0.50013-8CrossRefGoogle Scholar
Durrett, R. (2010). Probability Models for DNA Sequence Evolution, 2nd edn. Springer, New York.Google Scholar
Dwass, M. (1969). The total progeny in a branching process and a related random walk. J. Appl. Prob. 6, 682–686.10.2307/3212112CrossRefGoogle Scholar
Gleeson, J. P. (2008). Mean size of avalanches on directed random networks with arbitrary degree distributions. Phys. Rev. E 77, 057101.10.1103/PhysRevE.77.057101CrossRefGoogle ScholarPubMed
Gleeson, J. P. and Durrett, R. (2017). Temporal profiles of avalanches on networks. Nature Commun. 8, 1227.10.1038/s41467-017-01212-0CrossRefGoogle ScholarPubMed
Karr, A. F. (1975). Weak convergence of a sequence of Markov chains. Z. Wahrscheinlichkeitsth. 33, 41–48.10.1007/BF00539859CrossRefGoogle Scholar
Kello, C. T. (2013). Critical branching neural networks. Psych. Rev. 120, 230–254.10.1037/a0030970CrossRefGoogle ScholarPubMed
Kinouchi, O. and Copelli, M. (2006). Optimal dynamical range of excitable networks at criticality. Nature Phys. 2, 348–351.10.1038/nphys289CrossRefGoogle Scholar
Klebaner, F. C. and Nerman, O. (1994). Autoregressive approximation in branching processes with a threshold. Stoch. Process. Appl. 51, 1–7.10.1016/0304-4149(93)00000-6CrossRefGoogle Scholar
Larremore, D. B., Carpenter, M. Y., Ott, E. and Restrepo, J. G. (2012). Statistical properties of avalanches in networks. Phys. Rev. E 85, 066131.10.1103/PhysRevE.85.066131CrossRefGoogle ScholarPubMed
Larremore, D. B., Shew, W. L. and Restrepo, J. G. (2011). Predicting criticality and dynamic range in complex networks: effects of topology. Phys. Rev. Lett. 106, 058101.10.1103/PhysRevLett.106.058101CrossRefGoogle ScholarPubMed
Larremore, D. B., Shew, W. L., Ott, E. and Restrepo, J. G. (2011). Effects of network topology, transmission delays, and refractoriness on the response of coupled excitable systems to a stochastic stimulus. Chaos 21, 025117.10.1063/1.3600760CrossRefGoogle ScholarPubMed
Lindvall, T. (1976). On the maximum of a branching process. Scand. J. Statist. 3, 209–214.Google Scholar
Longini, I. M., Jr. (1980). A chain binomial model of endemicity. Math. Biosci. 50, 85–93.10.1016/0025-5564(80)90123-6CrossRefGoogle Scholar
Lyons, R., Pemantle, R. and Peres, Y. (1995). Conceptual proofs of $L \log L$ criteria for mean behavior of branching processes. Ann. Prob. 23, 1125–1138.Google Scholar
Moosavi, S. A., Montakhab, A. and Valizadeh, A. (2017). Refractory period in network models of excitable nodes: self-sustaining stable dynamics, extended scaling region and oscillatory behavior. Sci. Reports 7, 7107.Google ScholarPubMed
Muñoz, M. A. (2018). Colloquium: criticality and dynamical scaling in living systems. Rev. Mod. Phys 90, 031001.10.1103/RevModPhys.90.031001CrossRefGoogle Scholar
Nerman, O. (1977). On the maximal generation size of a non-critical Galton–Watson process. Scand. J. Statist. 4, 131–135.Google Scholar
Rämö, P., Kesseli, J. and Yli-Harja, O. (2006). Perturbation avalanches and criticality in gene regulatory networks. J. Theoret. Biol. 242, 164–170.10.1016/j.jtbi.2006.02.011CrossRefGoogle ScholarPubMed
Rämö, P., Kauffman, S., Kesseli, J. and Yli-Harja, O. (2007). Measures for information propagation in Boolean networks. Physica D 227, 100–104.10.1016/j.physd.2006.12.005CrossRefGoogle Scholar
Rastegar, R. and Roitershtein, A. Duration of avalanches in an excitable network. In preparation.Google Scholar
Rastegar, R. and Roitershtein, A. Approximation schemes for avalanches in a complex network with community structure. In preparation.Google Scholar
Thorisson, H. (2000). Coupling, Stationarity, and Regeneration. Springer, New York.10.1007/978-1-4612-1236-2CrossRefGoogle Scholar
Weiß, C. H. and Pollett, P. K. (2012). Chain binomial models and binomial autoregressive processes. Biometrics 68, 815–824.10.1111/j.1541-0420.2011.01716.xCrossRefGoogle Scholar
Zapperi, S., Lauritsen, K. B. and Stanley, H. E. (1995). Self-organized branching processes: mean-field theory for avalanches. Phys. Rev. Lett. 75, 4071.10.1103/PhysRevLett.75.4071CrossRefGoogle Scholar