Emergence of Chaos and Complexity During System Growth

  • Andrzej Gecow
Part of the Understanding Complex Systems book series (UCS)


The main topic of this article is the emergence of chaos in networks describing adaptive systems.We investigate this process mainly during the system growth in dependency on network size when other parameters of the network do not change. However, we also compare the degree of chaos for different parameters and network types including random Erdős-Rényi and BA scale-free networks. We use Kauffman networks, and we follow Kauffmann using their parameters and the notion ‘chaos’ for them. However, we use more than two signal variants which we assume to be equally probable, therefore the Kauffman networks considered can become different from the Boolean networks. The terms ‘complex system’ and ‘complex network’ are commonly used but they have no common established definitions. We find that chaotic properties of networks well meet our intuition of complexity and that the appearance of chaotic features during system growth can be treated as complexity threshold. Crossing this threshold defines certain properties of system and mechanisms which create ‘structural tendencies’. These interesting phenomena, however, are described in another article in this book.


Kauffman network Boolean network damage spreading chaos adaptive system 


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  1. 1.
    Albert, R., Barabási, A.-L.: Dynamics of Complex Systems: Scaling Laws for the Period of Boolean Networks. Phys. Rev. Lett. 84(24), 5660–5663 (2000)CrossRefGoogle Scholar
  2. 2.
    Albert, R., Barabási, A.-L.: Statistical mechanics of complex networks. Rev. Mod. Phys. 74(1), 47–97 (2002)CrossRefGoogle Scholar
  3. 3.
    Aldana, M.: Dynamics of Boolean Networks with Scale Free Topology. Physica D 185, 45–66 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Ashby, W.R.: Design for a Brain, 2nd edn. Wiley, New York (1960)zbMATHGoogle Scholar
  5. 5.
    Ay, N., Olbrich, E., Bertschinger, N., Jost, J.: A unifying framework for complexity measures of finite systems. In: Proceedings of ECCS 2006 (2006),
  6. 6.
    Bak, P., Tang, C., Wiesenfeld, K.: Self-organized criticality. Phys. Rev. A 38, 364–374 (1988)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Bak, P.: How Nature Works. Springer, New York (1996)zbMATHGoogle Scholar
  8. 8.
    Ballesteros, F.J., Luque, B.: Phase transitions in random networks: Simple analytic determination of critical points. Phys. Rev. E 71, 031104 (2005)CrossRefGoogle Scholar
  9. 9.
    Barabási, A.-L., Albert, R., Jeong, H.: Mean-field theory for scale-free random networks. Physica A 272, 173–187 (1999)CrossRefGoogle Scholar
  10. 10.
    Barabási, A.-L., Bonabeau, E.: Scale-Free Networks. Scientific American, pp. 50–59 (2003),
  11. 11.
    Chatzimeletiou, K., Morrison, E.E., Prapas, N., Prapas, Y., Handyside, A.H.: Spindle abnormalities in normally developing and arrested human preimplantation embryos in vitro identified by confocal laser scanning microscopy. Hum. Reprod. 3, 672–682 (2005)CrossRefGoogle Scholar
  12. 12.
    Crucitti, P., Latora, V., Marchiori, M., Rapisarda, A.: Rapisarda: Error and attacktolerance of complex networks. Physica A 340, 388–394 (2004)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Derrida, B., Pomeau, Y.: Random Networks of Automata: A Simple Annealed Approximation. Europhys. Lett. 1(2), 45–49 (1986)CrossRefGoogle Scholar
  14. 14.
    Dorogovtsev, S.N., Mendes, J.F.F.: Evolution of Networks: From Biological Nets to the Internet and WWW. Oxford University Press, New York (2003)zbMATHGoogle Scholar
  15. 15.
    Erdős, P., Rényi, A.: Random graphs. Publication of the Mathematical Institute of the Hungarian Academy of Science 5, 17–61 (1960)Google Scholar
  16. 16.
    Fortunato, S.: Damage spreading and opinion dynamics on scale-free networks. Physica A 348, 683–690 (2005)CrossRefGoogle Scholar
  17. 17.
    Gallos, L.K., Argyrakis, P., Bunde, A., Cohen, R., Havlin, S.: Tolerance of scale-free networks: from friendly to intentional attack strategies. Physica A 344, 504–509 (2004)CrossRefGoogle Scholar
  18. 18.
    Gecow, A.: A cybernetic model of improving and its application to the evolution and ontogenesis description. In: Proceedings of Fifth International Congress of Biomathematics, Paris (1975)Google Scholar
  19. 19.
    Gecow, A., Hoffman, A.: Self-improvement in a complex cybernetic system and its implication for biology. Acta Biotheoretica 32, 61–71 (1983)CrossRefGoogle Scholar
  20. 20.
    Gecow, A., Nowostawski, M., Purvis, M.: Structural tendencies in complex systems development and their implication for software systems. Journal of Universal Computer Science 11, 327–356 (2005), Google Scholar
  21. 21.
    Gecow, A.: From a “Fossil” Problem of Recapitulation Existence to Computer Simulation and Answer. Neural Network World 3, 189–201 (2005), Google Scholar
  22. 22.
    Gecow, A.: Structural Tendencies - effects of adaptive evolution of complex (chaotic) systems. Int. J. Mod. Phys. C 19(4), 647–664 (2008)zbMATHCrossRefGoogle Scholar
  23. 23.
    Gell-Mann, M.: What Is Complexity? John Wiley and Sons, Inc., Chichester (1995)Google Scholar
  24. 24.
    Grabowski, A., Kosiński, R.A.: Epidemic spreading in a hierarchical social network. Phys. Rev. E 70, 031908 (2004)CrossRefGoogle Scholar
  25. 25.
    Grabowski, A., Kosiński, R.A.: Ising-based model of opinion formation in a complex network of interpersonal interactions. Physica A 361, 651–664 (2006)CrossRefGoogle Scholar
  26. 26.
    Holyst, J.A., Fronczak, A., Fronczak, P.: Supremacy distribution in evolving networks. Phys. Rev. E 70, 046119 (2004)CrossRefGoogle Scholar
  27. 27.
    Iguchi, K., Kinoshita, S.I., Yamada, H.: Boolean dynamics of Kauffman models with a scale-free network. J. Theor. Biol. 247, 138–151 (2007)CrossRefMathSciNetGoogle Scholar
  28. 28.
    Jacobmeier, D.: Multidimensional Consensus Model on a Barabasi-Albert Network. Int. J. Mod. Phys. C 16(4), 633–646 (2005)CrossRefGoogle Scholar
  29. 29.
    Jan, N., de Arcangelis, L.: Computational Aspects of Damage Spreading. In: Stauffer, D. (ed.) Annual Reviews of Computational Physics I, pp. 1–16. World Scientific, Singapore (1994)Google Scholar
  30. 30.
    Jost, J.: External and internal complexity of complex adaptive systems. Theory Biosci. 123, 69–88 (2004)CrossRefGoogle Scholar
  31. 31.
    Kauffman, S.A.: Metabolic stability and epigenesis in randomly constructed genetic nets. J. Theor. Biol. 22, 437–467 (1969)CrossRefMathSciNetGoogle Scholar
  32. 32.
    Kauffman, S.A.: Gene regulation networks: a theory for their global structure and behaviour. Current topics in dev. biol. 6, 145 (1971)CrossRefGoogle Scholar
  33. 33.
    Kauffman, S.A.: The Origins of Order: Self-Organization and Selection in Evolution. Oxford University Press, New York (1993)Google Scholar
  34. 34.
    Kauffman, S.A., Peterson, C., Samuelsson, B., Troein, C.: Genetic networks with canalyzing Boolean rules are always stable. PNAS 101(49), 17102–17107 (2004)CrossRefGoogle Scholar
  35. 35.
    Luque, B., Solé, R.V.: Phase transitions in random networks: Simple analytic determination of critical points. Phys. Rev. E 55(1), 257–260 (1997)CrossRefGoogle Scholar
  36. 36.
    Luque, B., Ballesteros, F.J.: Random walk networks. Physica A 342, 207–213 (2004)CrossRefGoogle Scholar
  37. 37.
    Luttge, U., Beck, F.: Endogenous rhythms and chaos in crassulcean acid metabolism. Planta 188, 28–38 (1992)CrossRefGoogle Scholar
  38. 38.
    Li, M., Vitányi, P.: An Introduction to Kolmogorov Complexity and Its Applications. Springer, Heidelberg (1997)zbMATHGoogle Scholar
  39. 39.
    Nowostawski, M., Purvis, M.: Evolution and Hypercomputing in Global Distributed Evolvable Virtual Machines Environment. In: Brueckner, S.A., Hassas, S., Jelasity, M., Yamins, D. (eds.) Engineering Self-Organising Systems, pp. 176–191. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  40. 40.
    Olbrich, E., Brertschinger, N., Jost, J.: How should complexity scale with system size? In: Jost, J., Helbing, D. (eds.) Proceedings of ECCS 2007: European Conference on Complex Systems, paper #276 (2007) (CD-Rom)Google Scholar
  41. 41.
    Peliti, L., Vulpiani, A.: Measures of Complexity. Lect. Notes in Phys. 314 (1988)Google Scholar
  42. 42.
    Schuster, H.: Deterministic Chaos: An Introduction. Physik-Verlag (1984)Google Scholar
  43. 43.
    Serra, R., Villani, M., Semeria, A.: Genetic network models and statistical properties of gene expression data in knock-out experiments. J. Theor. Biol. 227, 149–157 (2004)CrossRefMathSciNetGoogle Scholar
  44. 44.
    Serra, R., Villani, M., Agostini, L.: On the dynamics of random Boolean networks with scale-free outgoing connections. Physica A 339, 665–673 (2004)CrossRefMathSciNetGoogle Scholar
  45. 45.
    Serra, R., Villani, M., Graudenzi, A., Kauffman, S.A.: Why a simple model of genetic regulatory networks describes the distribution of avalanches in gene expression data. J. Theor. Biol. 246, 449–460 (2007), CrossRefMathSciNetGoogle Scholar
  46. 46.
    Serra, R., Villani, M., Damiani, C., Graudenzi, A., Colacci, A., Kauffamn, S.A.: Interacting random boolean networks. In: Jost, J., Helbing, D. (eds.) Proceedings of ECCS 2007: European Conference on Complex Systems, paper #165 (2007) (CD-Rom)Google Scholar
  47. 47.
    Sole, R.V., Luque, B., Kauffman, S.: Phase transitions in random networks with multiple states. Technical Report 00-02-011, Santa Fe Institute (2000)Google Scholar
  48. 48.
    Sousa, A.O.: Consensus formation on a triad scale-free network. Physica A 348, 701–710 (2005)CrossRefGoogle Scholar
  49. 49.
    Stauffer, D., Sousa, A., Schulze, C.: Discretized Opinion Dynamics of The Deffuant Model on Scale-Free Networks. Journal of Artificial Societies and Social Simulation 7,
  50. 50.
    Stauffer, D., Moss de Oliveira, S., de Oliveira, P.M.C., Sa Martins, J.S.: Biology, Sociology, Geology by Computational Physicists, 276 + IX pages. Elsevier, Amsterdam (2006)Google Scholar
  51. 51.
    Wagner, A.: Estimating coarse gene network structure from large-scale gene perturbation data. Santa Fe Institute Working Paper, 01-09-051 (2001)Google Scholar
  52. 52.
    Yamano, T.: A statistical measure of complexity with nonextensive entropy. Physica A 340, 131–137 (2004)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Andrzej Gecow
    • 1
  1. 1.Institute of PaleobiologyPolish Academy of ScienceWarsawPoland

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