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7 - Dynamics of chance

Published online by Cambridge University Press:  05 July 2014

Bruce J. West  and
Paolo Grigolini
Bruce J. West
Affiliation:
Army Research Office
Paolo Grigolini
Affiliation:
University of North Texas
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Summary

In this chapter we explore web dynamics using a master equation in which the rate of change of probability is determined by the probability flux into and out of the state of interest. The master equation captures the interactions among large numbers of elements each with the same internal dynamics; in this case the internal dynamics consist of the switching of a node between two states. A two-state node may be viewed as the possible choices of an individual, say whether or not that person will vote for a particular candidate in an election. This is one of the simplest dynamical webs which has been shown mathematically to result in synchronization under certain well-defined conditions. We focus on the intermittent fluctuations emerging from a phase-transition process that achieves synchronized behavior for the strength of the interaction exceeding a critical value. This model provides a first step towards proving that these intermittent fluctuations, rather than being a nuisance, are important channels of information transmission allowing communication within and between different complex webs. The crucial power-law index μ discussed earlier and there inserted for mathematical convenience is here determined by the web dynamics. This observation on the inverse power-law index leads us to define the network efficiency in a form that might not coincide with earlier definitions proposed through the observation of the network topology.

Both the discrete and the continuous master equation are discussed for modeling web dynamics. One of the most important aspects of the analysis concerns the perturbation of one complex network by another and the transfer of information between complex clusters. A cluster is a network with a uniformity of opinion due to a phase transition to a given state. This modeling strategy is not new, but in fact dates back to Yule [71], who used the master-equation approach, before the approach had been introduced, to obtain an inverse power-law distribution. We investigate whether the cluster’s opinion is robust or whether it can be easily changed by perturbing the way members of the cluster interact with one another.

Type
Chapter
Information
Complex Webs
Anticipating the Improbable
 , pp. 307 - 356
Publisher: Cambridge University Press
Print publication year: 2010

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References

[1] P., Allegrini, G., Aquino, P., Grigoliniet al., “Correlation function and generalized master equation of any age,” Phys. Rev. E 71, 066109 (2005).Google Scholar
[2] P., Allegrini, F., Barbi, P., Grigolini and P., Paradisi, “Renewal, modulation and superstatistics,” Phys. Rev. E 73, 046136 (2006).Google Scholar
[3] P., Allegrini, V., Benci, P., Grigoliniet al., “Compression and diffusion: a joint approach to detect complexity,” Chaos, Solitons & Fractals, 15, 517–535 (2003).Google Scholar
[4] P., Allegrini, M., Bologna, P., Grigolini and B.J., West, “Fluctuation-dissipation theorem for event-dominated processes,” Phys. Rev. Lett. 99, 010603 (2007).Google Scholar
[5] P., Allegrini, M., Bologna, L., Fronzoni, P., Grigolini and L., Silvestri, “Experimental quenching of harmonic stimuli: universality of linear response theory,” Phys. Rev. Lett. 103, 030602 (2009).Google Scholar
[6] G., Aquino, M., Bologna, P., Grigolini and B. J., West, “Beyond the death of linear response theory: 1//-optimal information transport,” Phys. Rev. Lett. 105, 040601 (2010).Google Scholar
[7] G., Aquino, P., Grigolini and B. J., West, “Linear response and fluctuation-dissipation theorem for non-Poissonian renewal processes,” Europhys. Lett. 80, 10002 (1-6) (2007).Google Scholar
[8] F., Barbi, M., Bologna and P., Grigolini, “Linear response to perturbation of nonexponential renewal processes,” Phys.Rev.Lett. 95, 22061 (2005).Google Scholar
[9] J.B., Bassingthwaighte, L. S., Liebovitch and B.J., West, Fractal Physiology, New York: Oxford University Press (1994).Google Scholar
[10] C., Beck, “Dynamical foundations of nonextensive statistical mechanics,” Phys. Rev. Lett. 87, 180601 (1-4) (2001).Google Scholar
[11] J.M., Beggs and D., Plenz, “Neuronal avalanches in neocortical circuits,” J. Neurosci. 23, 11167 (2003).Google Scholar
[12] R., Benzi, A., Sutera and A. Vulpiani, “The mechanism of stochastic resonance,” J. Phys. A 14, L453 (1981).Google Scholar
[13] R., Benzi, G., Parisi, A., Sutera and A., Vulpiani, “Stochastic resonance in climate change,” Tellus 34, 10 (1982).Google Scholar
[14] R., Benzi, “Stochastic resonance: from climate to biology,” arXiv:nlin/0702008.
[15] S., Bianco, E., Geneston, P., Grigolini and M., Ignaccolo, “Renewal aging as emerging property of phase sychronization,” Physica A 387, 1387 (2008).Google Scholar
[16] M., Bologna and P., Grigolini, “Levy density: the density versus trajectory approach,” J. Statist. Mech.P03005 (2009).Google Scholar
[17] M. D., Breed, www.animalbehavioronline.com/habituation.html.
[18] H.-P., Breuer and B., Vacchini, “Structure of completely positive quantum master equation with memory kernel,” Phys. Rev. E 79, 041147 (2009).Google Scholar
[19] X., Brokmann, J.-P., Hermier, G., Messinet al., “Statistical aging and nonergodicity in the fluorescence of single nanocrystals,” Phys. Rev. Lett. 90, 120601 (2003).Google Scholar
[20] T. E., Cohen, S.W., Kaplan, E.R., Kandel and R. D., Hawkins, “A simplified preparation for relating cellular events to behavior: mechanisms contributing to habituation, dishabituation, and sensitization of the Aplysia gill-withdrawal reflex,” J. Neurosci. 17, 2886–2899 (1997).Google Scholar
[21] F., Corberi, E., Lipiello and M., Zannetti, “Fluctuation dissipation relations far from equilibrium,” J. Statist. Mech.P07002 (2007).Google Scholar
[22] J., Correll, “1/f -Noise and effort on implicit measures of bias,” J. Personal. and Social Psych. 94, 48 (2008).Google Scholar
[23] D. R., Cox, Renewal Theory, London: Methuen (1962).Google Scholar
[24] O., Deniz, J., Lorenzo and M., Hernández, “A simple habituation mechanism for perceptual user interfaces,” Inteligencia Artificial, Revista Iberoamericana de Inteligencia Artificial 23, 87 (2004).Google Scholar
[25] S.S., Deshmukh and U.S., Bhalla, “Representation of odor habituation and timing in the hippocampus,” J. Neurosci. 23, 1903–1915 (2003).Google Scholar
[26] V., Dragoi, “A feedforward model of suppressive and facilitatory habituation effects,” Biol. Cybern. 86, 419 (2002).Google Scholar
[27] P. J., Drew and L. F., Abbott, “Models and properties of power-law adaptation in neural systems,” J. Neurophysiol. 96, 826 (2006).Google Scholar
[28] W., Feller, An Introduction to Probability Theory and Its Applications, Vol. II, New York: John Wiley & Sons (1971).Google Scholar
[29] J., Feng, “Is the integrate-and-fire model good enough?,” Neural Networks 14, 955 (2001).Google Scholar
[30] L., Gammaitoni, P., Hanggi, P., Jung and F., Marchesoni, “Stochastic resonance,” Rev. Mod. Phys. 70, 223–287 (1998).Google Scholar
[31] D. L., Gilden, T., Thornton and M. W., Mallon, “1/ f-Noise in human cognition,” Science 267, 1837 (1995).Google Scholar
[32] P., Gong, A. R., Nikolaev and C., van Leeuwen, “Intermittent dynamics underlying the intrinsic fluctuations of the collective synchronization patterns in electrocortical activity,” Phys. Rev. 76, 011904 (2007).Google Scholar
[33] P., Grigolini, G., Aquino, M., Bologna, M., Lukovic and B.J., West, “A theory of 1/f -noise in human cognition,” PhysicaA 388, 4129–1204 (2009).Google Scholar
[34] C., Haldeman and J. M., Beggs, “Critical branching captures activity in living neural networks and maximizes the number of metastable states,” Phys. Rev. Lett. 94, 058101 (2005).Google Scholar
[35] E., Heinsalu, M., Patriarca, I., Goychuk and P., Hanggi, “Use and abuse of a fractional Fokker-Planck dynamics for time-dependent driving,” Phys. Rev. Lett. 99, 120602 (2007).Google Scholar
[36] C. T., Kello, B. C., Beltz, J. G., Holden and G. C., Van Orden, “The emergent coordination of cognitive function,” J. Expl. Psych.: Gen. 136, 551–568 (2007).Google Scholar
[37] H. A., Kramers, “Brownian motion in a field of force and the diffusion model of chemical reactions,” PhysicaA 7, 284–340 (1940).Google Scholar
[38] R., Kubo, “A general expression for the conductivity tensor,” Canadian J. Phys. 34, 12741277 (1956).Google Scholar
[39] Q. S., Li and Y., Liu, “Enhancement and sustainment of internal stochastic resonance in unidirectional coupled neural systems,” Phys. Rev. E 73, 016218 (2006).Google Scholar
[40] M. H., Lee, “Can the velocity autocorrelation function decay exponentially?,” Phys. Rev. Lett. 51, 1277 (1983).Google Scholar
[41] M., Lukovic and P., Grigolini, “Power spectra for both interrupted and perennial aging processes,” J. Chem. Phys. 129, 184102 (2008).Google Scholar
[42] M., Lukovic, L., Fronzoni, M., Ignaccolo and P., Grigolini, “The rate matching effect: a hidden property of aperiodic stochastic resonance,” Phys. Lett. A 372, 2608–2613 (2008).Google Scholar
[43] M., Magdziarz, A., Weron and K., Weron, “Fractional Fokker-Planck dynamical stochastic representation and computer simulation,” Phys. Rev. E 75, 016708 (2007).Google Scholar
[44] M., Magdziarz, A., Weron and J., Klafter, “Equivalence of the fractional Fokker-Planck equation and the subordinated Langevin equation: the case of a time-dependent force,” Phys. Rev. Lett. 101, 210601 (2008).Google Scholar
[45] G., Margolin and E., Barkai, “Nonergodicity of a time series obeying Levy statistics,” J. Statist. Phys. 122, 137–167 (2006); G. Margolin and E. Barkai, “Nonergodicity of blinking nanocrystals and other Levy-walk processes,” Phys. Rev. Lett.94, 080601 (2005).
[46] B., McNamara and K., Wiesenfeld, “Theory of stochastic resonance,” Phys. Rev. A 39, 4854 (1989).Google Scholar
[47] C., Nicolis and G., Nicolis, “Is there a climate attractor?,” Nature 34, 529–532 (1984).Google Scholar
[48] A., Neiman, L., Schimansky-Reier, A., Cornall-Bell and F., Moss, “Noise-enhanced phase synchronization in excitable media,” Phys. Rev. Lett. 83, 4896–1899 (1999).Google Scholar
[49] M.E.J., Newman, “Power laws, Pareto distributions and Zipf's law,” Contemp. Phys. 46, 323–351 (2005).Google Scholar
[50] A., Nordsieck, W. W., Lamb and G. E., Uhlenbeck, “On the theory of cosmic-ray showers: I. The Furry model and the fluctuation problem,” Physica 7, 344 (1940).Google Scholar
[51] I., Oppenheim, K. E., Shuler and G. H., Weiss, Stochastic Processes in Chemical Physics: The Master Equation, Cambridge, MA: MIT Press (1977).Google Scholar
[52] M., Perc, “Stochastic resonance on weakly paced scale-free networks,” Phys. Rev. E 78, 036105 (2008).Google Scholar
[53] D., Plenz and T. C., Thiagarajan, “The organizing principle of neuronal avalanche activity: cell assemblies in the cortex?,” Trends Neurosci. 30, 101 (2007).Google Scholar
[54] A., Rebenshtok and E., Barkai, “Weakly nonergodic statistical physics,” J. Statist. Phys. 133, 565 (2008).Google Scholar
[55] L. E., Reichl, A Modern Course in Statistical Physics, Austin, TX: University of Texas Press (1980).Google Scholar
[56] B. R., Sheth, J., Sharma, S. C., Rao and M., Sur, “Orientation maps of subjective contours in visual cortex,” Science 274, 2110 (1996).Google Scholar
[57] L., Silvestri, L., Fronzoni, P., Grigolini and P., Allegrini, “Event-driven power-law relaxation in weak turbulence,” Phys. Rev. Lett. 102, 014502 (1-4) (2009).Google Scholar
[58] H. A., Simon, “On a class of skew distribution functions,” Biometrika 42, 425–440 (1935).Google Scholar
[59] I. M., Sokolov, “Linear response to perturbation of nonexponential renewal process: a generalized master equation approach,” Phys. Rev. E. 73, 067102 (2006).Google Scholar
[60] I. M., Sokolov and J., Klafter, “Field-induced dispersion and subdiffusion,” Phys. Rev. Lett. 97, 140602 (2006).Google Scholar
[61] I. M., Sokolov and J., Klafter, “Continuous time random walks in an oscillating field: field-induced dispersion and the death of linear response,” Chaos, Solitons and Fractals 34, 81–86 (2007).Google Scholar
[62] L., Silvestri, L., Fronzoni, P., Grigolini and P., Allegrini, “Event driven power-law relaxation in weak turbulence,” Phys. Rev. Lett. 102, 014502 (2009).Google Scholar
[63] R. G., Turcott, P. D., Barker and M. C., Teich, “Long-duration correlation in the sequence of action potentials in an insect visual interneuron,” J. Statist. Comput. & Simul. 52, 253–271 (1995).Google Scholar
[64] M., Turalska, M., Lukovic, B.J., West and P., Grigolini, “Complexity and synchronization,” Phys. Rev.E 80, 021110–1 (2009).Google Scholar
[65] D., Vitali and P., Grigolini, “Subdynamics, Fokker-Planck equation, and exponential decay of relaxation processes,” Phys. Rev. A 39, 1486–1499 (1989).Google Scholar
[66] L. M., Ward, “Physics of neural synchronization mediated by stochastic resonance,” Con- temp. Phys. 50, 563–574 (2009).Google Scholar
[67] A., Weron, M., Magdziarz and K., Weron, “Modeling of subdiffusion in space-time-dependent force fields beyond the fractional Fokker-Planck equation,” Phys. Rev. E 77, 036704 (2008).Google Scholar
[68] B. J., West, E. L., Geneston and P. Grigolini, “Maximizing information exchange between complex networks,” Phys. Rep. 468, 1–99 (2008).Google Scholar
[69] B.J., West and P., Grigolini, “Statistical habituation model and 1/f noise,” Physica A doi: 10.1016/j.physa.2010.08.033 (2010).Google Scholar
[70] J. C., Willis and G. U., Yule, “Some statistics of evolution and geographical distribution in plants and animals, and their signature,” Nature 109, 177–179 (1922).Google Scholar
[71] G. U., Yule, “A mathematical theory of evolution based on the conclusions of Dr. J. C. Willis,” Philos. Trans. R. Soc. London B 213, 21–87 (1925).Google Scholar
[72] E., Zohary, M. N., Shadlen and W. T., Newsome, “Correlated neuronal discharge rate and its implications for psychophysical performance,” Nature 370, 140 (1994).Google Scholar
[73] R., Zwanzig, Nonequilibrium Statistical Mechanics, New York: Oxford University Press (2001).Google Scholar

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