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Adaptive Networks pp 73–106Cite as

Self-Organized Criticality and Adaptation in Discrete Dynamical Networks

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Part of the book series: Understanding Complex Systems ((UCS))

Abstract

It has been proposed that adaptation in complex systems is optimized at the critical boundary between ordered and disordered dynamical regimes. Here, we review models of evolving dynamical networks that lead to self-organization of network topology based on a local coupling between a dynamical order parameter and rewiring of network connectivity, with convergence towards criticality in the limit of large network size N. In particular, two adaptive schemes are discussed and compared in the context of Boolean Networks and Threshold Networks: (1) Active nodes loose links, frozen nodes aquire new links, (2) Nodes with correlated activity connect, de-correlated nodes disconnect. These simple local adaptive rules lead to co-evolution of network topology and -dynamics. Adaptive networks are strikingly different from random networks: They evolve inhomogeneous topologies and broad plateaus of homeostatic regulation, dynamical activity exhibits 1/f noise and attractor periods obey a scale-free distribution. The proposed co-evolutionary mechanism of topological self-organization is robust against noise and does not depend on the details of dynamical transition rules. Using finite-size scaling, it is shown that networks converge to a self-organized critical state in the thermodynamic limit. Finally, we discuss open questions and directions for future research, and outline possible applications of these models to adaptive systems in diverse areas.

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References

  1. Abraham, W.C., Mason-Parker, S.E., Bear, M.F., Webb, S., Tate, W.P.: Heterosynaptic metaplasticity in the hippocampus in vivo: A bcm-like modifiable threshold for ltp. Proc. Natl. Acad. Sci. USA 98, 10924–10929 (2001). DOI 10.1073/pnas.181342098

    Article  Google Scholar 

  2. Abraham, W.C., Tate, W.P.: Metaplasticity: A new vista across the field of synaptic plasticity. Prog. Neurobiol. 52, 303–323 (1997)

    Article  Google Scholar 

  3. Albert, R., Barabasi, A.L.: Dynamics of complex systems: Scaling laws for the period of boolean networks. Phys. Rev. Let. 84, 5660–5663 (2000)

    Article  Google Scholar 

  4. Andrecut, M.: Mean field dynamics of random boolean networks. J. Stat. Mech. (2005). DOI 10.1088/1742-5468/2005/02/P02003

    Google Scholar 

  5. Anirvan M. Sengupta, M.D., Shraiman, B.: Specificity and robustness in transcription control networks. Proc. Natl. Acad. Sci. 99, 2072–2077 (2002)

    Article  Google Scholar 

  6. Bak, P.: How Nature Works: The Science of Self-organized Criticality. Copernicus, New York (1996)

    MATH  Google Scholar 

  7. Bak, P., Sneppen, K.: Punctuated equilibrium and criticality in a simple model of evolution. Phys. Rev. Lett. 71, 4083 – 4086 (1993) DOI 10.1103/PhysRevLett.71.4083

    Article  Google Scholar 

  8. Bak, P., Tang, C., Wiesenfeld, K.: Self-organized criticality: An explanation of the 1/f noise. Phys. Rev. Lett. 59, 381–384 (1987)

    Article  MathSciNet  Google Scholar 

  9. Barkai, N., Leibler, S.: Robustness in simple biochemical networks. Nature 387, 913–916 (1997)

    Article  Google Scholar 

  10. Bastolla, U., Parisi, G.: Closing probabilities in the kauffman model: An annealed computation. Physica D 98, 1–25 (1996)

    Article  MATH  Google Scholar 

  11. Bastolla, U., Parisi, G.: Relevant elements, magnetization and dynamical properties in kauffman networks: A numerical study. Physica D 115, 203–218 (1998)

    Article  MATH  Google Scholar 

  12. Bastolla, U., Parisi, G.: The modular structure of kauffman networks. Physica D 115, 219–233 (1998a)

    Article  MATH  Google Scholar 

  13. Beggs, J.M., Plenz, D. Neuronal avalanches in neocortical circuits. J. Neurosci. 23 11167 (2003)

    Google Scholar 

  14. Beggs, J.M.: The criticality hypothesis: how local cortical networks might optimize information processing. Phil. Trans. Roy. Soc. A 366(1864), 329–343 (2008). DOI 10.1098/rsta.2007.2092

    Article  MATH  MathSciNet  Google Scholar 

  15. Bertschinger, N., Natschläger, T., Legenstein, R.A.: At the edge of chaos: Real-time computations and self-organized criticality in recurrent neural networks. In: L.K. Saul, Y. Weiss, L. Bottou (eds.) Advances in Neural Information Processing Systems 17, pp. 145–152. MIT Press, Cambridge, MA (2005)

    Google Scholar 

  16. Bhattacharjya, A., Liang, S.: Median attractor and transients in random boolean nets. Physica D 95, 29–34 (1996)

    Article  MATH  Google Scholar 

  17. Bhattacharjya, A., Liang, S.: Power-law distributions in some random boolean networks. Phys. Rev. Lett. 77, 1644–1647 (1996)

    Article  Google Scholar 

  18. Bornholdt, S., Röhl, T.: Self-organized critical neural networks. Phys. Rev. E 67, 066 118 (2003). DOI 10.1103/PhysRevE.67.066118

    Article  Google Scholar 

  19. Bornholdt, S., Rohlf, T.: Topological evolution of dynamical networks: Global criticality from local dynamics. Phys. Rev. Lett. 84, 6114–6117 (2000)

    Article  Google Scholar 

  20. Bornholdt, S., Sneppen, K.: Neutral mutations and punctuated equilibrium in evolving genetic networks. Phys. Rev. Lett. 81, 236–239 (1998)

    Article  Google Scholar 

  21. Bornholdt, S., Sneppen, K.: Robustness as an evolutionary principle. Proc. R. Soc. Lond. B 267, 2281–2286 (2000)

    Article  Google Scholar 

  22. Christensen, K., Donangelo, R., Koiller, B., Sneppen, K.: Evolution of random networks. Phys. Rev. Lett. 81, 2380 (1998)

    Article  Google Scholar 

  23. Correale, L., Leone, M., Pagnani, A., Weigt, M., Zecchina, R.: The computational core and fixed point organization in boolean networks. J. Stat. Mech. (2006). DOI 10.1088/1742-5468/2006/03/P03002

    Google Scholar 

  24. Correale, L., Leone, M., Pagnani, A., Weigt, M., Zecchina, R.: Core percolation and onset of complexity in boolean networks. Phys. Rev. Lett. 96, 018101 (2006). DOI 10.1103/PhysRevLett.96.018101

    Article  Google Scholar 

  25. Davidson, E.: Genomic Regulatory Systems. Development and Evolution. Academic Press, San Diego, CA (2001)

    Google Scholar 

  26. Derrida, B.: Dynamical phase transition in non-symmetric spin glasses. J. Phys. A 20, 721–725 (1987)

    Article  MathSciNet  Google Scholar 

  27. Derrida, B., Flyvbjerg, H.: Distribution of local magnetisations in random networks of automata. J. Phys. A 20, 1107–1112 (1987)

    Article  MathSciNet  Google Scholar 

  28. Derrida, B., Gardner, E., Zippelius, A.: An exactly solvable asymmetric neural network model. Europhys. Lett. 4, 167 (1987)

    Article  Google Scholar 

  29. Derrida, B., Pomeau, Y.: Random networks of automata: a simple annealed approximation. Europhys. Lett. 1, 45–49 (1986)

    Article  Google Scholar 

  30. Derrida, B., Stauffer, D.: Phase transitions in two-dimensional kauffman cellular automata. Europhys. Lett. 2, 739ff (1986)

    Article  Google Scholar 

  31. Drossel, B.: Extinction events and species lifetimes in a simple ecological model. Phys. Rev. Lett. 81, 5011 (1998)

    Article  Google Scholar 

  32. Drossel, B.: Number of attractors in random boolean networks. Phys. Rev. E 72(1, Part 2) (2005). DOI 10.1103/PhysRevE.72.016110

    Google Scholar 

  33. Engert, F., Bonhoeffer, T.: Dendritic spine changes associated with hippocampal long-term synaptic plasticity. Nature 399, 66–69 (1999)

    Article  Google Scholar 

  34. Erdős, P., Rényi, A.: On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci. 5, 17–61 (1960)

    Google Scholar 

  35. Flyvbjerg, H.: An order parameter for networks of automata. J. Phys. A 21, L955–L960 (1988)

    Article  MathSciNet  Google Scholar 

  36. Gireesh, E.D., Plenz, D.: Neuronal avalanches organize as nested theta- and beta/gamma-oscillations during development of cortical layer 2/3. Proc. Natl. Acad. Sci. USA 105, 7576-7581 (2008).

    Article  Google Scholar 

  37. Gross, T., Blasius, B.: Adaptive coevolutionary networks: a review. J. Roy. Soc. Interface 5, 259–271 (2008). DOI 10.1098/rsif.2007.1229

    Article  Google Scholar 

  38. Hebb, D.O.: The Organization of Behavior. Wiley, New York (1949)

    Google Scholar 

  39. Kauffman, S.: Metabolic stability and epigenesis in randomly constructed genetic nets. J. Theor. Biol. 22, 437–467 (1969)

    Article  MathSciNet  Google Scholar 

  40. Kauffman, S.: The Origins of Order: Self-Organization and Selection in Evolution. Oxford University Press, Oxford (1993)

    Google Scholar 

  41. Kaufman, V., Drossel, B.: Relevant components in critical random boolean networks. New J. Phys. 8 (2006). DOI 10.1088/1367-2630/8/10/228

    Google Scholar 

  42. Kaufman, V., Mihaljev, T., Drossel, B.: Scaling in critical random boolean networks. Phys. Rev. E 72, 046124 (2005). DOI 10.1103/PhysRevE.72.046124

    Article  Google Scholar 

  43. Kelso, J.A.S., Bressler, S.L., Buchanan, S., DeGuzman, G.C., Ding, M., et al.: A phase transition in human brain and behavior. Phys. Lett. A 169, 134–144 (1992)

    Article  Google Scholar 

  44. Kesseli, J., Ramo, P., Yli-Harja, O.: Iterated maps for annealed boolean networks. Phys. Rev. E 74(4, Part 2), 046 104 (2006). DOI 10.1103/PhysRevE.74.046104

    Article  MathSciNet  Google Scholar 

  45. Klemm, K., Bornholdt, S.: Stable and unstable attractors in boolean networks. Phys. Rev. E 72(5, Part 2) (2005). DOI 10.1103/PhysRevE.72.055101

    Google Scholar 

  46. Krawitz, P., Shmulevich, I.: Basin entropy in boolean network ensembles. Phys. Rev. Lett. 98(15) (2007). DOI 10.1103/PhysRevLett.98.158701

    Google Scholar 

  47. Kree, R., Zippelius, A.: Continuous-time dynamics of asymmetrically diluted neural networks. Phys. Rev. A 36(9), 4421–4427 (1987). DOI 10.1103/PhysRevA.36.4421

    Article  MathSciNet  Google Scholar 

  48. Kürten, K.: Critical phenomena in model neural networks. Phys. Lett. A 129, 156–160 (1988)

    Article  Google Scholar 

  49. Kürten, K.: Correspondence between neural threshold networks and kauffman boolean cellular automata. J. Phys. A 21, L615–L619 (1988b)

    Article  Google Scholar 

  50. Langton, C.: Life at the edge of chaos. In: Artificial Life, vol. II, pp. 255–276. Addison-Wesley, Boston, MA (1991)

    Google Scholar 

  51. Levina, A., Herrmann, J.M., Geisel, T. Dynamical synapses causing self-organized criticality in neural networks. Nat. Phys. 3 857–860 (2007)

    Article  Google Scholar 

  52. Linkenkaer-Hansen, K., Nikouline, V.V., Palva, J.M., Ilmoniemi, R.J.: Long-range temporal correlations and scaling behavior in human brain oscillations. J. Neurosci. 21, 1370–1377 (2001)

    Google Scholar 

  53. Liu, M., Bassler, K.E.: Emergent criticality from coevolution in random boolean networks. Phys. Rev. E 74, 041910 (2006). DOI 10.1103/PhysRevE.74.041910

    Article  Google Scholar 

  54. Luque, B., Ballesteros, F.J., Muro, E.M.: Self-organized critical random boolean networks. Phys. Rev. E 63, 051913 (2001). DOI 10.1103/PhysRevE.63.051913

    Article  Google Scholar 

  55. Luque, B., Ferrera, A.: Measuring mutual information in random boolean networks. Complex Syst. 12, 241–252 (2000)

    Google Scholar 

  56. Luque, B., Sole, R.: Lyapunov exponents in random boolean networks. Physica A 284(1–4), 33–45 (2000)

    Article  Google Scholar 

  57. Luque, B., Sole, R.V.: Phase transitions in random networks: simple analytic determination of critical points. Phys. Rev. E 55, 257–260 (1996)

    Article  Google Scholar 

  58. Luscombe, N.M., et al.: M.M.B.: Genomic analysis of regulatory network dynamics reveals large topological changes. Nature 431, 308–312 (2004)

    Article  Google Scholar 

  59. Lux, T., Marchesi, M.: Scaling and criticality in a stochastic multi-agent model of a financial market. Nature 397, 498–500 (1999). DOI 10.1038/17290

    Article  Google Scholar 

  60. Malamud, B.D., Morein, G., Turcotte, D.L.: Forest fires: An example of self-organized critical behavior. Science 281, 1840–1842 (1998)

    Article  Google Scholar 

  61. McGuire, P.C., Bohr, H., Clark, J.W., Haschke, R., Pershing, C.L., Rafelski, J.: Threshold disorder as a source of diverse and complex behavior in random nets. Neural Networks 15, 1243–1258 (2002)

    Article  Google Scholar 

  62. li Ming, G., Wong, S.T., Henley, J., bing Yuan, X., jun Song, H., Spitzer, N.C., Poo, M.m.: Adaptation in the chemotactic guidance of nerve growth cones. Nature 417, 411–418 (2002)

    Article  Google Scholar 

  63. Molgedey, L., Schuchard, J., Schuster, H.G.: Suppressing chaos in neural networks by noise. Phys. Rev. Lett. 69, 3717 (1992)

    Article  Google Scholar 

  64. Murray, J.: Mathematical Biology. Springer, New York (2002)

    MATH  Google Scholar 

  65. Nakamura, I.: Dynamics of threshold network on non-trivial distribution degree. Eur. Phys. J. B 40, 217–221 (2004). DOI 10.1140/epjb/e2004-00260-4

    Article  Google Scholar 

  66. Nicolis, G., Prigogine, I.: Self-Organization in Nonequilibrium Systems: From Dissipative Structures to Order Through Fluctuations. Wiley, New York (1977)

    MATH  Google Scholar 

  67. Nykter, M., Price, N.D., Aldana, M., Ramsey, S.A., Kauffman, S.A., Hood, L.E., Yli-Harja, O., Shmulevich, I.: Gene expression dynamics in the macrophage exhibit criticality. Proc. Natl. Acad. Sci. USA 105(6), 1897–1900 (2008). DOI 10.1073/pnas.0711525105

    Article  Google Scholar 

  68. Nykter, M., Price, N.D., Larjo, A., Aho, T., Kauffman, S.A., Yli-Harja, O., Shmulevich, I.: Critical networks exhibit maximal information diversity in structure-dynamics relationships. Phys. Rev. Lett. 1(5) (2008). DOI 10.1103/PhysRevLett.100.058702

    Google Scholar 

  69. van Ooyen, A.: Competition in the development of nerve connections: a review of models. Network: Computation in Neural Systems 12, R1–R47 (2001)

    Article  MATH  Google Scholar 

  70. Paczuski, M., Bassler, K.E., Corral, A.: Self-organized networks of competing boolean agents. Phys. Rev. Lett. 84, 3185–3188 (2000). DOI 10.1103/PhysRevLett.84.3185

    Article  Google Scholar 

  71. Paczuski, M., Maslov, S., Bak, P.: Avalance dynamics in evolution, growth and depinning models. Phys. Rev. E 53, 414–443 (1996)

    Article  Google Scholar 

  72. Rämö, P., Kesseli, J., Yli-Harja, O.: Perturbation avalanches and criticality in gene regulatory networks. J. Theor. Biol. 242, 164–170 (2006). DOI 10.1016/j.jtbi.2006.02.011

    Article  Google Scholar 

  73. Ribeiro, A.S., Kauffman, S.A., Lloyd-Price, J., Samuelsson, B., Socolar, J.E.S.: Mutual information in random boolean models of regulatory networks. Phys. Rev. E 77(1, Part 1) (2008). DOI 10.1103/PhysRevE.77.011901

    Google Scholar 

  74. Rohlf, T.: Networks and Self-Organized Criticality. Master’s thesis, Christian-Albrechts-Universität Kiel (Germany) (2000)

    Google Scholar 

  75. Rohlf, T.: Critical line in random threshold networks with inhomogeneous thresholds.Phys. Rev. E 78, 066118 (2008)

    Google Scholar 

  76. Rohlf, T.: Self-organization of heterogeneous topology and symmetry breaking in networks with adaptive thresholds and rewiring. Europhys. Lett. 84, 10004 (2008)

    Article  MathSciNet  Google Scholar 

  77. Rohlf, T., Bornholdt, S.: Criticality in random threshold networks: Annealed approximation and beyond. Physica A 310, 245–259 (2002)

    Article  MATH  Google Scholar 

  78. Rohlf, T., Bornholdt, S.: Gene regulatory networks: A discrete model of dynamics and topological evolution. In: A. Deutsch, J. Howard, M. Falcke, W. Zimmermann (eds.) Function and Regulation of Cellular Systems: Experiments and Models. Birkhäuser Basel (2004)

    Google Scholar 

  79. Rohlf, T., Gulbahce, N., Teuscher, C.: Damage spreading and criticality in finite dynamical networks. Phys. Rev. Lett. 99, 248701 (2007). DOI 10.1103/PhysRevLett.99.248701

    Article  Google Scholar 

  80. Schmoltzi, K., Schuster, H.G.: Introducing a real time scale into the Bak-Sneppen model. Phys. Rev. E 52, 5273–5280 (1995).

    Article  Google Scholar 

  81. Schroeder, B.C., Kubisch, C., Stein, V., Jentsch, T.: Moderate loss of function of cyclic-amp-modulated kcnq2/kcnq3 k+ channels causes epilepsy. Nature 396, 687–690 (1998). DOI 10.1038/25367

    Article  Google Scholar 

  82. Shmulevich, I., Kauffman, S.A., Aldana, M.: Eukaryotic cells are dynamically ordered or critical but not chaotic. Proc. Natl. Acad. Sci. USA 102, 13439–13444 (2005)

    Article  Google Scholar 

  83. Sole, R., Luque, B.: Phase transitions and antichaos in generalized kauffman networks. Phys. Lett. A 196, 331–334 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  84. Sole, R.V., Manrubia, S.C.: Criticality and unpredictability in macroevolution. Phys. Rev. E 55, 4500–4507 (1997)

    Article  Google Scholar 

  85. Sornette, D.: Critical phase transitions made self-organized: a dynamical system feedback mechanism for self-organized criticality. J. Phys. I France 2, 2065–2073 (1992)

    Article  Google Scholar 

  86. Stewart, C.V., Plenz, D. Homeostasis of neuronal avalanches during postnatal cortex development in vitro. J. Neurosci. 169, 405–416 (2008)

    Google Scholar 

  87. Teuscher, C., Sanchez, E.: Self-organizing topology evolution of turing neural networks. Artificial Neural Networks - ICANN 2001, Proceedings 2130, 820–826 (2001)

    Google Scholar 

  88. Thomas, R.: Boolean formalization of genetic control circuits. J. Theor. Biol. 42, 563–585 (1973)

    Article  Google Scholar 

  89. Trachtenberg, J.T., Chen, B.E., Knott, G.W., Feng, G., Sanes, J.R., Walker, E., Svoboda, K.: Long-term in vivo imaging of experience-dependent synaptic plasticity in adult cortex. Nature 420, 788–794 (2002)

    Article  Google Scholar 

  90. Wagner, A.: Robustness against mutations in genetic networks of yeast. Nat. Genet. 24, 355–361 (2000)

    Article  Google Scholar 

  91. Wooters, W.K., Langton, C.G.: Is there a sharp phase transition for deterministic cellular automata? Physica D 45, 95–104 (1990)

    Article  Google Scholar 

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Authors and Affiliations

  1. Max-Planck-Institute for Mathematics in the Sciences, Inselstrasse 22, 04103, Leipzig, Germany

    Thimo Rohlf

  2. Institute for Theoretical Physics, University Bremen, Otto-Hahn-Allee, 28359, Bremen, Germany

    Stefan Bornholdt

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  1. Thimo Rohlf
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  2. Stefan Bornholdt
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  1. MPI für Physik Komplexer Systeme, Nöthnitzer Str. 38, Dresden, 01187, Germany

    Thilo Gross

  2. Binghamton University, State University of New York, Binghamton, 13902-6000, U.S.A.

    Hiroki Sayama

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Rohlf, T., Bornholdt, S. (2009). Self-Organized Criticality and Adaptation in Discrete Dynamical Networks. In: Gross, T., Sayama, H. (eds) Adaptive Networks. Understanding Complex Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01284-6_5

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