Skip to main content
SpringerLink
Log in

Steady-state dynamics of the forest fire model on complex networks

  • Interdisciplinary Physics
  • Published:
The European Physical Journal B Aims and scope Submit manuscript

Abstract

Many sociological networks, as well as biological and technological ones, can be represented in terms of complex networks with a heterogeneous connectivity pattern. Dynamical processes taking place on top of them can be very much influenced by this topological fact. In this paper we consider a paradigmatic model of non-equilibrium dynamics, namely the forest fire model, whose relevance lies in its capacity to represent several epidemic processes in a general parametrization. We study the behavior of this model in complex networks by developing the corresponding heterogeneous mean-field theory and solving it in its steady state. We provide exact and approximate expressions for homogeneous networks and several instances of heterogeneous networks. A comparison of our analytical results with extensive numerical simulations allows to draw the region of the parameter space in which heterogeneous mean-field theory provides an accurate description of the dynamics, and enlights the limits of validity of the mean-field theory in situations where dynamical correlations become important.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Barrat, M. Barthélemy, A. Vespignani, Dynamical Processes on Complex Networks (Cambridge University Press, Cambridge, 2008)

  2. S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, Rev. Mod. Phys. 80, 1275 (2008)

    Article  ADS  Google Scholar 

  3. A.L. Barabási, R. Albert, Science 286, 509 (1999)

    Article  MathSciNet  Google Scholar 

  4. G. Caldarelli, Scale-Free Networks: Complex Webs in Nature and Technology (Oxford University Press, Oxford, 2007)

  5. D.S. Callaway, M.E.J. Newman, S.H. Strogatz, D.J. Watts, Phys. Rev. Lett. 85, 5468 (2000)

    Article  ADS  Google Scholar 

  6. R. Cohen, K. Erez, D. Ben Avraham, S. Havlin, Phys. Rev. Lett. 86, 3682 (2001)

    Article  ADS  Google Scholar 

  7. C. Castellano, R. Pastor-Satorras, Phys. Rev. Lett. 96, 038701 (2006)

    Article  ADS  Google Scholar 

  8. K.I. Goh, D.S. Lee, B. Kahng, D. Kim, Phys. Rev. Lett. 91, 148701 (2003)

    Article  ADS  Google Scholar 

  9. Y. Moreno, A. Vazquez, Europhys. Lett. 57, 765 (2002)

    Article  ADS  Google Scholar 

  10. A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, C. Zhou, Phys. Rep. 469, 93 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  11. C. Castellano, S. Fortunato, V. Loreto, Rev. Mod. Phys. 81, 591 (2008)

    Article  ADS  Google Scholar 

  12. M.J. Keeling, K.T.D. Eames, J. R. Soc. Interface 2, 295 (2005)

    Article  Google Scholar 

  13. F. Liljeros, C.R. Edling, L.A.N. Amaral, H.E. Stanley, Y. Aberg, Nature 411, 907 (2001)

    Article  ADS  Google Scholar 

  14. R. Pastor-Satorras, A. Vespignani, Phys. Rev. Lett. 86, 3200 (2001)

    Article  ADS  Google Scholar 

  15. A.L. Lloyd, R.M. May, Science 292, 1316 (2001)

    Article  Google Scholar 

  16. Y. Moreno, R. Pastor-Satorras, A. Vespignani, Eur. Phys. J. B 26, 521 (2002)

    ADS  Google Scholar 

  17. M.E.J. Newman, Phys. Rev. E 66, 016128 (2002)

    Article  MathSciNet  ADS  Google Scholar 

  18. R. Pastor-Satorras, A. Vespignani, Phys. Rev. E 65, 036104 (2001)

    Article  ADS  Google Scholar 

  19. R. Cohen, S. Havlin, D. ben-Avraham, Phys. Rev. Lett. 91, 247901 (2003)

    Article  ADS  Google Scholar 

  20. J.G. Gardenes, V. Latora, Y. Moreno, E. Profumo, Proc. Natl. Acad. Sci. USA 105, 1399 (2008)

    Article  ADS  Google Scholar 

  21. V. Colizza, A. Barrat, M. Barthelemy, A. Vespignani, Proc. Natl. Acad. Sci. USA 103, 2015 (2006)

    Article  ADS  Google Scholar 

  22. R.M. Anderson, R.M. May, Infectious diseases in humans (Oxford University Press, Oxford, 1992)

  23. D.J. Watts, S.H. Strogatz, Nature 393, 440 (1998)

    Article  ADS  Google Scholar 

  24. M. Boguñá, C. Castellano, R. Pastor-Satorras, Phys. Rev. E 79, 036110 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  25. P. Bak, K. Chen, C. Tang, Phys. Lett. A 147, 297 (1992)

    Article  ADS  Google Scholar 

  26. B. Drossel, F. Schwabl, Phys. Rev. Lett. 69, 1629 (1992)

    Article  ADS  Google Scholar 

  27. P. Grassberger, New J. Phys. 4, 17 (2002)

    Article  ADS  Google Scholar 

  28. J.A. Bonachela, M.A. Muñoz, JSTAT P09009 (2009)

  29. C. Rhodes, R. Anderson, Nature 381, 600 (1996)

    Article  ADS  Google Scholar 

  30. G. Abramson, M. Kuperman, Phys. Rev. Lett. 86, 2909 (2001)

    Article  ADS  Google Scholar 

  31. C.M.M. Müller-Linow, M.T. Hütt, Phys. Rev. E 74, 016112 (2006)

    Article  ADS  Google Scholar 

  32. J. Liu, Y. Tang, Z. Yang, J. Stat. Mech. P08008 (2004)

  33. S. Peng, Y. Li, B. Zheng, Steady states and critical behavior of epidemic spreading on complex networks, 7th World Congress on Intelligent Control and Automation (WCICA) (2008), pp. 3481–3486

  34. R. Pastor-Satorras, A. Vázquez, A. Vespignani, Phys. Rev. Lett. 87, 258701 (2001)

    Article  ADS  Google Scholar 

  35. H.J. Jensen, Self-Organized Criticality (Cambridge University Press, Cambridge, 1998)

  36. M. Boguñá, R. Pastor-Satorras, A. Vespignani, in Statistical Mechanics of Complex Networks, Lecture Notes in Physics, edited by R. Pastor-Satorras, J.M. Rubí, A. Díaz-Guilera (Springer Verlag, Berlin, 2003), Vol. 625

  37. R. Pastor-Satorras, A. Vespignani, Phys. Rev. E 63, 066117 (2001)

    Article  ADS  Google Scholar 

  38. M. Catanzaro, M. Boguñá, R. Pastor-Satorras, Phys. Rev. E 71, 056104 (2005)

    Article  ADS  Google Scholar 

  39. K. Christensen, H. Flyvbjerg, Z. Olami, Phys. Rev. Lett. 71, 2737 (1993)

    Article  ADS  Google Scholar 

  40. J. Marro, R. Dickman, Nonequilibrium phase transitions in lattice models (Cambridge University Press, Cambridge, 1999)

  41. M. Boguñá, R. Pastor-Satorras, Phys. Rev. E 66, 047104 (2002)

    Article  ADS  Google Scholar 

  42. M. Abramowitz, I.A. Stegun, Handbook of mathematical functions (Dover, New York, 1972)

  43. R.M. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey, D.E. Knuth, Adv. Comput. Math. 5, 329 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  44. R.M. Corless, D.J. Jeffrey, D.E. Knuth, A sequence of series for the Lambert W function, in ISSAC ’97: Proceedings of the 1997 international symposium on Symbolic and algebraic computation (ACM, New York, NY, USA, 1997), pp. 197–204

  45. A. Vázquez, R. Pastor-Satorras, A. Vespignani, Phys. Rev. E 65, 066130 (2002)

    Article  ADS  Google Scholar 

  46. A. Barrat, R. Pastor-Satorras, Phys. Rev. E 71, 036127 (2005)

    Article  ADS  Google Scholar 

  47. M. Catanzaro, M. Boguñá, R. Pastor-Satorras, Phys. Rev. E 71, 027103 (2005)

    Article  ADS  Google Scholar 

  48. M. Boguñá, R. Pastor-Satorras, A. Vespignani, Eur. Phys. J. B 38, 205 (2004)

    Article  ADS  Google Scholar 

  49. J.H. Lambert, Nouveaux mémoires de l’Académie royale des sciences et belles-lettres 1 (1772)

  50. E. Goursat, Course in Mathematical Analysis, Functions of a Complex Variable & Differential Equations (Dover, New York, 1959), Vol. 2

  51. J. Havil, Exploring Euler’s Constant (Princeton University Press, Princeton, NJ, 2003)

Download references

Author information

Authors and Affiliations

  1. Institut de Théorie des Phénomènes Physiques, École Polytechnique Fédérale de Lausanne, 1015, Lausanne, Switzerland

    J.-D. Bancal

  2. Department de Física Enginyeria Nuclear, Universitat Politécnica de Catalunya, Campus Nord, 08034, Barcelona, Spain

    R. Pastor-Satorras

Authors
  1. J.-D. Bancal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Pastor-Satorras
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. Pastor-Satorras.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bancal, JD., Pastor-Satorras, R. Steady-state dynamics of the forest fire model on complex networks. Eur. Phys. J. B 76, 109–121 (2010). https://doi.org/10.1140/epjb/e2010-00165-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1140/epjb/e2010-00165-7

Keywords

Search

Navigation