Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-745bb68f8f-b6zl4 Total loading time: 0 Render date: 2025-02-06T16:55:05.425Z Has data issue: false hasContentIssue false

3 - Mostly linear dynamics

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
Get access

Summary

One strategy for predicting an improbable but potentially catastrophic future event is to construct a faithful model of the dynamics of the complex web of interest and study its extremal properties. But we must also keep in mind that the improbable is not the same as the impossible. The improbable is something that we know can happen, but our experience tells us that it probably will not happen, because it has not happened in the past. Most of what we consider to be common sense or probable is based on what has happened either to us in the past or to the people we know. In this and subsequent chapters we explore the probability of such events directly, but to set the stage for that discussion we examine some of the ways webs become dynamically complex, leading to an increase in the likelihood of the occurrence of improbable events. The extremes of a process determine the improbable and consequently it is at these extremes that failure occurs. Knowing a web's dynamical behavior can help us learn the possible ways in which it can fail and how long the recovery time from such failure may be. It will also help us answer such questions as the following. How much time does the web spend in regions where the likelihood of failure is high? Are the extreme values of dynamical variables really not important or do they actually dominate the asymptotic behavior of a complex web?

Type
Chapter
Information
Complex Webs
Anticipating the Improbable
 , pp. 105 - 165
Publisher: Cambridge University Press
Print publication year: 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] B.J., Adler and T. E., Wainwright, “Velocity autocorrelation function of hard spheres,” Phys. Rev. Lett. 18, 968 (1967); “Decay of the velocity autocorrelation function,” Phys. Rev. A1, 18 (1970).Google Scholar
[2] O. C., Akin, P., Paradisi and P., Grigolini, Physica A 371, 157–170 (2006).
[3] P., Allegrini, F., Barbi, P., Grigolini and P., Paradisi, “Renewal, modulation and superstatistics,” Phys. Rev. E 73, 046136 (2006).Google Scholar
[4] J.B., Bassingthwaighte, L.S., Liebovitch and B.J., West, Fractal Physiology, New York: Oxford University Press (1994).Google Scholar
[5] A. L., Barabási and R., Albert, “Emergence of scaling in random networks,” Science 286, 509 (1999).Google Scholar
[6] C., Beck, “Dynamical foundations of nonextensive statisical mechanics,” Phys.Rev. Lett. 87, 180601 (2001).Google Scholar
[7] R., Benzi, A., Sutera and A., Vulpiani, J. Phys. A 14, L453 (1981); R. Benzi, G. Parisi, A. Sutera and A. Vulpiani, Tellus34, 10 (1982).
[8] M., Bianucci, R., Mannella, B.J., West and P., Grigolini, “Chaos and linear response: analysis of the short-, intermediate-, and long-time regime,” Phys. Rev. E 50, 2630 (1994).Google Scholar
[9] B.S., Blanchard, Logistics Engineering and Management, 4th edn., Englewood Cliffs, NJ: Prentice Hall (1992), pp. 26-32.Google Scholar
[10] N., Bousquet, H., Bertholon and G., Cedeux, “An alternative competing risk model to the Weibull distribution for modeling aging in lifetime data analysis,” Lifetime Data Anal. 12, 481 (2006).Google Scholar
[11] R., Bogacz, E., Brown, J., Moehlis, P., Holmes and J.D., Cohen, “The physics of optimal decision making: a formal analysis of models of performance in two-alternative forced-choice tasks,” Psych. Rev. 113, 700 (2006).Google Scholar
[12] J. R., Busemeyer and J. T., Townsend, “Decision field theory: a dynamic-cognitive approach to decision making in an uncertain environment,” Psych. Rev. 100, 432 (1993).Google Scholar
[13] S. W., Chiu, J. A., Novotny and E., Jakobsson, “The nature of ion and water barrier crossings in a simulated ion channel,” Biophys. J. 64, 98 (1993).Google Scholar
[14] D.R., Cox, Renewal Theory, London: Metheun & Co. (1967); first printed in 1962.Google Scholar
[15] R., Dekeyser and H. H., Lee, “Nonequilibrium statistical mechanics of the spin-1/2 van der Walls model. II. Autocorrelation function of a single spin and long-time tails,” Phys. Rev. B 43, 8123-8147 (1991).Google Scholar
[16] W., Feller, An Introduction to Probability Theory and Its Applications, Vol. II, New York: John Wiley & Sons (1971).Google Scholar
[17] C.E., Finch, Longevity, Senescence and the Genome, Chicago, IL: University of Chicago Press (1990).Google Scholar
[18] M., Finkelstein, “Aging: damage accumulation versus increasing mortality rate,” Math. Biosci. 207, 104 (2007).Google Scholar
[19] M., Finkelstein, “Virtual age of non-reparable objects,” Reliability Eng. and System Safety 94, 666 (2009).Google Scholar
[20] D., Gross and C. M., Harris, Fundamentals of Queueing Theory, 3rd edn., New York: Wiley-Interscience (1998).Google Scholar
[21] H., Haken, Synergetics, an Introduction: Nonequilibrium Phase Transitions and Self-Organization in Physics, Chemistry and Biology, 3rd rev. edn., New York: Springer (1983).Google Scholar
[22] H., Haken, “Slaving principle revisited,” Physica D 97, 97 (1996).Google Scholar
[23] H., Haken, “Synergetics of brain function,” Int. J. Psychophys. 60, 110 (2006).Google Scholar
[24] H., Haken, Brain Dynamics, Synchronization and Activity Patterns in Pulse-Coupled Neural Nets with Delays and Noise, Berlin: Springer (2007).Google Scholar
[25] Hesoid, Works and Days, translated from the Greek, London (1728).
[26] M., Kijma and A., Sumita, “A useful generalization of renewal theory: counting processes governed by non-negative Markov increments,” J. Appl. Prob. 23, 71 (1986).Google Scholar
[27] A. N., Kolmogorov, “Local structure of turbulence in an incompressible liquid for very large Reynolds numbers,” Comptes Rendus (Dokl.) Acad. Sci. URSS (N.S.) 26, 115–118 (1941); “A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number,” J. FluidMech.13, 82–85 (1962).Google Scholar
[28] H. A., Kramers, “Brownian motion in a field of force and the diffusion model of chemical reactions,” Physica 7, 284 (1940).Google Scholar
[29] P., Langevin, Comptes Rendus Acad. Sci. Paris 146, 530 (1908).
[30] K., Lindenberg and B. J., West, The Nonequilibrium Statistical Mechanics of Open and Closed Systems, New York: VCH Publishers (1990).Google Scholar
[31] M., Longuet-Higgins, “The effect of non-linearities on statistical distributions in the theory of sea waves,” J. Fluid Mech. 17, 459 (1963); “Statistical properties of an isotropic random surface,” Phil. Trans. Roy. Soc. London A250, 158 (1957).Google Scholar
[32] R., Mannella, P., Grigolini and B. J., West, “A dynamical approach to fractional Brownian motion,” Fractals 2, 81–124 (1994).Google Scholar
[33] G., Margolin and E., Barkai, “Single-molecule chemical reactions: reexamination of the Kramers approach,” Phys. Rev. E 72, 025101 (2005).Google Scholar
[34] E. W., Montroll and B. J., West, “On an enriched collection of stochastic processes,” in Fluctuations Phenomena, 2nd edn., eds. E.W., Montroll and J.L., Lebowitz, Amsterdam: North-Holland (1987); first published in 1979.Google Scholar
[35] F., Moss and K., Wiesenfeld, “The benefits of background noise,” Sci. Am. 273, 66–69 (1995).Google Scholar
[36] F., Moss, L. M., Ward and W. G., Sannita, “Stochastic resonance and sensory information processing,” Clin. Neurophysiol. 115, 267–281 (2004).Google Scholar
[37] C., Nicolis and G., Nicolis, “Is there a climate attractor?,” Nature 311, 529–532 (1984).Google Scholar
[38] C. K., Peng, J., Mietus, J. M., Hausdorffet al., “Long-range autocorrelations and non- Gaussian behavior in the heartbeat,” Phys. Rev. Lett. 70, 1343 (1993).Google Scholar
[39] J., Perrin, Brownian Movement and Molecular Reality, London: Taylor and Francis (1910).Google Scholar
[40] J. W., Strutt (Lord Rayleigh), Phil. Mag. X, 73–78 (1880); in Scientific Papers, Vol.I, New York: Dover Publications (1880), pp. 491-496.
[41] L. F., Richardson, “Atmospheric diffusion shown on a distance-neighbor graph,” Proc. Roy. Soc. London Ser. A 110, 709–725 (1926).Google Scholar
[42] D., Ruelle, Chaotic Evolution and Strange Attractors, Cambridge: Cambridge University Press (1989).Google Scholar
[43] G. E., Uhlenbeck and L. S., Ornstein, “On the theory of Brownian motion,” Phys. Rev. 36, 823 (1930).Google Scholar
[44] B., Veber, M., Nagode and M., Fajdiga, “Generalized renewal process for repairable systems based on finite Weibull mixture,” Reliability Eng. and System Safety 93, 1461 (2008).Google Scholar
[45] J.N., Weiss, A., Garfinkel, M.L., Spano and W.L., Ditto, “Chaos control for cardiac arrhythmias,” in Proc. 2nd Experimental Chaos Conference, eds. W., Ditto, L., Pecora, M., Shlesinger, M., Spano and S., Vohra, Singapore: World Scientific (1995).Google Scholar
[46] B. J., West and A. L., Goldberger, “Physiology in fractal dimensions,” Am. Sci. 75, 354–364 (1987).Google Scholar
[47] B. J., West, “Physiology in fractal dimensions: error tolerance,” Ann. Biomed. Eng. 18, 135-149 (1990).Google Scholar
[48] B. J., West, Where Medicine Went Wrong, Singapore: World Scientific (2006).Google Scholar
[49] F. G., Zeng, Q., Fu and R., Morse, Brain Res. Inter. 869, 251 (2000).

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×