Abstract
In this paper we show that a well-known model of genetic regulatory networks, namely that of Random Boolean Networks (RBNs), allows one to study in depth the relationship between two important properties of complex systems, i.e. dynamical criticality and power-law distributions. The study is based upon an analysis of the response of a RBN to permanent perturbations, that may lead to avalanches of changes in activation levels, whose statistical properties are determined by the same parameter that characterizes the dynamical state of the network (ordered, critical or disordered). Under suitable approximations, in the case of large sparse random networks an analytical expression for the probability density of avalanches of different sizes is proposed, and it is shown that for not-too-small avalanches of critical systems it may be approximated by a power law. In the case of small networks the above-mentioned formula does not maintain its validity, because of the phenomenon of self-interference of avalanches, which is also explored by numerical simulations.
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References
Kauffman, S.A.: The Origins of Order. Oxford University Press, New York (1993)
Kauffman, S.A.: At Home in the Universe. Oxford University Press, New York (1995)
Packard, N.H.: Adaptation toward the edge of chaos. In: Kelso, J.A.S., Mandell, A.J., Shlesinger, M.F. (eds.) Dynamic Patterns in Complex Systems, pp. 293–301. World Scientific, Singapore (1988)
Langton, C.G.: Computation at the edge of chaos. Physica D 42, 12–37 (1990)
Langton, C.G.: Life at the edge of chaos. In: Langton, C.G., Taylor, C., Farmer, J.D., Rasmussen, S. (eds.) Artificial Life II, pp. 41–91. Addison-Wesley, Reading MA (1992)
Benedettini, S., Villani, M., Roli, A., Serra, R., Manfroni, M., Gagliardi, A., Pinciroli, C., Birattari, M.: Dynamical regimes and learning properties of evolved Boolean networks. Neurocomputing 99, 111–123 (2013)
Bak, P., Tang, C., Wiesenfeld, K.: Self-organized criticality: an explanation of 1/f noise. Phys. Rev. Lett. 59, 381–384 (1987)
Kauffman, S.A.: Metabolic stability and epigenesis in randomly constructed nets. J. Theor. Biol. 22, 437–467 (1969)
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)
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. 249, 449–460 (2007)
Ramo, P., Kesseli, J., Yli-Harja, O.: Perturbation avalanches and criticality in gene regulatory networks. J. Theor. Biol. 242, 164–170 (2006)
Derrida, B., Pomeau, Y.: Random networks of automata: a simple annealed approximation. Europhys. Lett. 1, 45–49 (1986)
Aldana, M., Coppersmith, S., Kadanoff, L.P.: Boolean dynamics with random couplings. In: Kaplan, E., Marsden, J.E., Sreenivasan, K.R. (eds.) Perspectives and Problems in Nonlinear Science, pp. 23–89. Springer, Heidelberg (2003)
Harris, S.E., Sawhill, B.K., Wuensche, A., Kauffman, S.A.: A model of transcriptional regulatory networks based on biases in the observed regulation rules. Complexity 7(4), 23–40 (2001)
Di Stefano, M.L.: Perturbazioni in reti booleane casuale. Master thesis, Department of Physics, Informatics and Mathematics, University of Modena and Reggio Emilia (2015)
Miller, R.G.: The jackknife—a review. Biometrika 61, 1–15 (1974)
Efron, B.: The Jackknife, the Bootstrap, and Other Resampling Plans. SIAM, Philadelphia (1982)
Kass, R.E., Raftery, A.E.: Bayes factors. J. Am. Statist. Assoc. 90, 773–795 (1995)
Robert, C.P.: The Bayesian Choice. Springer, New York (2007)
Villani, M., Serra, R., Ingrami, P., Kauffman, S.A.: Coupled random boolean network forming an artificial tissue. In: El Yacoubi, S., Chopard, B., Bandini, S. (eds.) ACRI 2006. LNCS, vol. 4173, pp. 548–556. Springer, Heidelberg (2006)
Serra, R., Villani, M., Damiani, C., Graudenzi, A., Colacci, A.: The diffusion of perturbations in a model of coupled random boolean networks. In: Umeo, H., Morishita, S., Nishinari, K., Komatsuzaki, T., Bandini, S. (eds.) ACRI 2008. LNCS, vol. 5191, pp. 315–322. Springer, Heidelberg (2008)
Damiani, C., Kauffman, S.A., Serra, R., Villani, M., Colacci, A.: Information transfer among coupled random boolean networks. In: Bandini, S., Manzoni, S., Umeo, H., Vizzari, G. (eds.) ACRI 2010. LNCS, vol. 6350, pp. 1–11. Springer, Heidelberg (2010)
Damiani, C., Serra, R., Villani, M., Kauffman, S.A., Colacci, A.: Cell-cell interaction and diversity of emergent behaviours. IET Syst. Biol. 5(2), 137–144 (2011)
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Useful discussions with Alex Graudenzi, Chiara Damiani and Alessandro Filisetti are gratefully acknowledged.
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Di Stefano, M.L., Villani, M., La Rocca, L., Kauffman, S.A., Serra, R. (2016). Dynamically Critical Systems and Power-Law Distributions: Avalanches Revisited. In: Rossi, F., Mavelli, F., Stano, P., Caivano, D. (eds) Advances in Artificial Life, Evolutionary Computation and Systems Chemistry. WIVACE 2015. Communications in Computer and Information Science, vol 587. Springer, Cham. https://doi.org/10.1007/978-3-319-32695-5_3
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