Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Read chapter Read first chapter

2018 | OriginalPaper | Chapter

12. Epidemic Automaton and the Eden Model: Various Aspects of Robustness

Author : Lucas Gerin

Published in: Probabilistic Cellular Automata

Publisher: Springer International Publishing

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

The two-dimensional probabilistic cellular automaton Epidemic models the spread of an epidemic without recovering on graph. We discuss some well-known and less well-known properties of Epidemic on a finite grid and its analogous on the infinite square lattice: the Eden model. This survey is intended for non-probabilists and gives a detailed study of the robustness of a cellular automaton with respect to several sources of randomness.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Statistical Equilibrium in Deterministic Cellular Automata
next chapter Convergence Time of Probabilistic Cellular Automata on the Torus
Footnotes
1
We write \(f_n=\varTheta (g_n)\) when there exist two positive numbers \(c_1,c_2\) such that, for n large enough, \(c_1g_n\le f_n\le c_2g_n\).
 
Literature
1.
Basdevant, A.L., Enriquez, N., Gerin, L., Gouéré, J.B.: The shape of large balls in highly supercritical percolation. Electron. J. Probab. 19, 1–14 (2014)MathSciNetCrossRefMATH
2.
3.
Couronné, O., Enriquez, N., Gerin, L.: Construction of a short path in high dimensional first-passage percolation. Electron. Commun. Probab. 16, 22–28 (2011)MathSciNetCrossRefMATH
4.
Cox, J., Durrett, R.: Some limit theorems for percolation processes with necessary and sufficient conditions. Ann. Probab. 9(4), 583–603 (1981)MathSciNetCrossRefMATH
5.
6.
7.
Eden, M.: A two-dimensional growth process. In: Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, vol. IV, pp. 223–239. University of California Press, Berkeley, Calif (1961)
8.
Fatès, N., Gerin, L.: Examples of fast and slow convergence of 2d asynchronous cellular systems. J. Cell. Autom 4, 323–337 (2009)MathSciNetMATH
9.
Fatès, N., Morvan, M., Schabanel, N., Thierry, E.: Fully asynchronous behavior of double-quiescent elementary cellular automata. Theoretical Comput. Sci. 362, 1–16 (2006)MathSciNetCrossRefMATH
10.
Garet, O., Marchand, R.: Asymptotic shape for the chemical distance and first-passage percolation on the infinite Bernoulli cluster. ESAIM Probab. Stat. 8, 169–199 (2004) (electronic)
11.
Gerin, L.: Aspects probabilistes des automates cellulaires, et d’autres problèmes en informatique théorique. (2008). Thèse de l’Université Nancy 1
12.
Grimmett, G.: Percolation, vol. 321, 2nd edn. Springer, Berlin (1999)
13.
Hammersley, J.M., Welsh, D.J.A.: First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory. In: Proceedings of the International Research Seminar Statistical Laboratory, University of California, Berkeley, Calif pp. 61–110. Springer, New York (1965)
14.
Janson, S., Łuczak, T., Rucinski, A.: RanDom Graphs. In: Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley-Interscience, New York (2000)
15.
Kesten, K.: Aspects of first passage percolation. In: École d’été de probabilités de Saint-Flour, XIV—1984, Lecture Notes in Mathematics, vol. 1180, pp. 125–264. Springer, Berlin (1986)
16.
17.
Richardson, D.: Random growth in a tessellation. In: Proceedings of the Cambridge Philosophical Society 74, 515–528 (1973)
18.
Metadata
Title
Epidemic Automaton and the Eden Model: Various Aspects of Robustness
Author
Lucas Gerin
Copyright Year
2018
Book
Probabilistic Cellular Automata

Print ISBN: 978-3-319-65556-7

Electronic ISBN: 978-3-319-65558-1

Copyright Year: 2018

DOI
https://doi.org/10.1007/978-3-319-65558-1_12

Premium Partner

    Image Credits