2 - Webs, trees and branches
Published online by Cambridge University Press: 05 July 2014
Summary
In the previous chapter we examined examples of hyperbolic and inverse power-law distributions dating back to the beginning of the last century, addressing biodiversity, urban growth and scientific productivity. There was no discussion of physical structure in these examples because they concerned the counting of various quantities such as species, people, cities, words, and scientific publications. It is also interesting to examine the structure of complex physical phenomena, such as the familiar irregularity in the lightning flashes shown in Figure 1.13. The branches of the lightning bolt persist for fractions of a second and then blink out of existence. The impression left is verified in photographs, where the zigzag pattern of the electrical discharge is captured. The time scale for the formation of the individual zigs and zags is on the order of milliseconds and the space scales can be hundreds of meters. So let us turn our attention to webs having time scales of milliseconds, years or even centuries and spatial scales from millimeters to kilometers.
All things happen in space and time, and phenomena localized in space and time are called events. Publish a paper. Run a red light. It rains. The first two identify an occurrence at a specific point in time with an implicit location in space; the third implies a phenomenon extended in time over a confined location in space. But events are mental constructs that we use to delineate ongoing processes so that not everything happens at once. Publishing a paper is the end result of a fairly long process involving getting an idea about a possible research topic, doing the research, knowing when to gather results together into a paper, writing the paper, sending the manuscript to the appropriate journal, reading and responding to the referees’ criticism of the paper, and eventually having the paper accepted for publication.
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- Complex WebsAnticipating the Improbable, pp. 45 - 104Publisher: Cambridge University PressPrint publication year: 2010
References
[1] , “Zipf, power-laws, and Pareto - a ranking tutorial,” http://www.hpl.hp.com/research/idl/papers/ranking/ranking.html.
[2] , and , “Fractal nature of regional ventilation distribution,” J. Appl. Physiol. 88, 1551-1557 (2000).Google Scholar
[4] , “Das Gesetz der Bevolkerungskonzentration,” Petermanns Mitteilungen 59, 74 (1913).Google Scholar
[5] , Scaling Phenomena in Fluid Mechanics, Cambridge: Cambridge University Press (1994).Google Scholar
[6] and , “Lightning and the heart: fractal behavior in cardiac function,” Proc. IEEE 76, 693 (1988).Google Scholar
[10] and , “On the Weierstrass-Mandelbrot fractal function,” Proc. Roy. Soc. Lond. Ser. A 370, 459–184 (1980).Google Scholar
[11] and , “Postural stability and fractal dynamics,” Acta Neurobiol. Exp. 61, 105–112 (2001).Google Scholar
[13] , and , “A scale-free systems theory of motivation and addiction,” Neurosci. Biobehav. Rev. 31, 1017–1045 (2007).Google Scholar
[14] and , “Random walking during quiet standing,” Phys.Rev. Lett. 73, 764–767 (1994).Google Scholar
[15] , The Curves of Life, New York: Dover (1979); originally published by Constable and Co. (1914).Google Scholar
[17] and , “Unified view of scaling laws for river networks,” Phys. Rev. E 49, 4865 (1999).Google Scholar
[18] and , “Geometry of river networks II: Distribution of component size and number,” Phys. Rev. E 63, 016115 (2001).Google Scholar
[19] , and , “On the power-law relationships of the Internet dynamics,” Comput. Commun. Rev. 29, 251 (1999).Google Scholar
[21] and , “Stochastic models that separate fractal dimensions and the Hurst effect,” SIAM Rev. 46, 269–282 (2004).Google Scholar
[22] , , and , “Bronchial asymmetry and Fibonacci scaling,” Experientia 41, 1–5 (1985).Google Scholar
[23] , and , “Scale-invariant fluctuations of the dynamical synchronization in human brain electrical activity,” Phys. Rev. E 76, 011904 (2007).Google Scholar
[24] , , and , “Inverse cubic law for the distribution of stock price variations,” Eur. Phys. J. B 3, 139–140 (1998).Google Scholar
[25] , D. Leddon and N. Scafetta, “The diffusion entropy and waiting-time statistics and hard X-ray solar flares,” Phys. Rev. E 65, 046203 (2002).Google Scholar
[26] and , Seismicity of the Earth and Associated Phenomena, 2nd edn., Princeton, NJ: Princeton University Press (1954), pp. 17-19.Google Scholar
[27] , “Studies of longitudinal stream profiles in Virginia and Maryland,” U.S. Geol. Surv. Prof. Paper 294-B, 45 (1957).Google Scholar
[28] , , et al., “Is walking a random walk? Evidence for long-range correlations in stride interval of human gait,” J. Appl. Physiol. 78 (1), 349-358 (1995).Google Scholar
[29] , “Relative and absolute strength of response as a function of frequency of reinforcement,” J. Exp. Anal. Behav. 29, 267–272 (1961).Google Scholar
[30] , “The fractal geometry of the convoluted brain,” J. Hirnforsch. 32, 103–111 (1991).Google Scholar
[31] , “Erosional development of streams and their drainage basins: hydrophysical approach to quantitative morphology,” Bull. Geol. Soc. Am. 56, 275–370 (1945).Google Scholar
[33] and , “Neurons and fractals: how reliable and useful are calculations of fractal dimensions?,” J. Neurosci. Methods 81, 9–18 (1998).Google Scholar
[34] , , , and , “The large-scale organization of metabolic networks,” Nature 407, 651 (2000).Google Scholar
[36] , “Small systems of neurons,” in Mind and Behavior, eds. and , San Francisco, CA: W. H. Freeman and Co. (1979).Google Scholar
[37] , , and , “Broadband criticality of human brain network synchronization,” PLoS Comp. Biol. 5, 1-13, www.ploscombiol.org (2009).Google Scholar
[38] , “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. Fluid Mech.13, 82–85 (1962).Google Scholar
[39] , “Random texts exhibit Zipf's-law-like word frequency distribution,” IEEE Trans. Information Theory 38 (6), 1842-1845 (1992).Google Scholar
[41] , “The frequency distribution of scientific productivity,” J. Wash. Acad. Sci. 16, 317 (1926).Google Scholar
[42] , A Wavelet Tour of Signal Processing, 2nd edn., San Diego, CA: Academic Press (1999).Google Scholar
[43] , “On the theory of word frequencies and on related Markovian models of discourse,” in Structures ofLanguage and Its Mathematical Aspects, ed. , New York: American Mathematical Society (1961).Google Scholar
[46] and , Foundations of Statistical Natural Language Processing, Cambridge, MA: MIT Press (1999), p. 24.Google Scholar
[47] and , “On the Hausdorff dimension of some graphs,” Trans. Am. Math. Soc. 298, 793 (1986).Google Scholar
[49] and , “Small world pattern in food webs,” arXiv: cond-mat/0011195.
[50] and , “On 1/f noise and other distributions with long tails,” PNAS 79, 337 (1982).Google Scholar
[51] , “A relationship between circumference and weight and its bearing on branching angles,” J. General Physiol. 10, 725–729 (1927).Google Scholar
[52] , , et al., “Biologically variable ventilation increases arterial oxygenation over that seen with positive end-expiratory pressure alone in a porcine model of acute respiratory distress syndrome,” Crit. Care Med. 28, 2457–2464 (2000).Google Scholar
[53] , “Power laws, Pareto distributions and Zipf's law,” Contemp. Phys. 46, 323-351 (2005).Google Scholar
[54] , “Mathematical model for the intermittence of turbulence flow,” Sov. Phys. Dokl. 11, 497–499 (1966).Google Scholar
[55] , “On the aftershocks of earthquakes,” J. College Sci., Imperial Univ. Tokyo 7, 111-200 (1894).Google Scholar
[57] , Manuale di Economia Politica, Milan (1906); French edn., revised, Manuel d'Economie Politique, Paris (1909); English translation, Manual of Political Economy (1971).Google Scholar
[58] , , et al., “Long-range anticorrelations and non-Gaussian behavior of the heartbeat,” Phys. Rev. Lett. 70, 1343–1346 (1993).Google Scholar
[59] , , et al. “Quantifying fractal dynamics of human respiration: age and gender effects,” Ann. Biomed. Eng. 30, 683–692 (2002).Google Scholar
[60] , The Rise of Statistical Thinking 1820-1900, Princeton, NJ: Princeton University Press (1986).Google Scholar
[61] , “Theory of physiological properties of dendrites,” Ann. New York Acad. Sci. 96, 1071 (1959).Google Scholar
[63] , “Atmospheric diffusion shown on a distance-neighbor graph,” Proc. Roy. Soc. Lond. Ser. A 110, 709–725 (1926).Google Scholar
[64] , The Notebooks of Leonardo da Vinci, Vol. 1, New York: Dover (1970); unabridged edition of the work first published in London in 1883.Google Scholar
[65] and , “Fractality and self-organized criticality of wars,” Fractals 6, 351–357 (1998).Google Scholar
[66] , Pflügers Arch. 162, 225-299; English translation “Flow resistance in human air passages and the effect of irregular branching of the bronchial system on the respiratory process in various regions of the lungs,” Translations in Respiratory Physiology, ed. J. B. West, Stroudsberg, PA: Dowden, Hutchinson and Ross Publishers (1975).
[67] and , “Laws governing the fineness of powdered coal,” J. Inst. Fuel 7, 29–36 (1933).Google Scholar
[69] and , “Multiscaling comparative analysis of time series and a discussion on ‘earthquake conversations’ in California,” Phys. Rev. Lett. 92, 138501 (2004).Google Scholar
[70] , “Über spontane Stromschwankungen in verschiedenen Elektrizitätsleitern,” Ann. Phys. 362, 541–567 (1918).Google Scholar
[71] , “Discrimination reaction time for a 1023-alternative task,” J. Exp. Psych. 66, 215-226 (1963).Google Scholar
[72] and , “Complex fractal dimension of the bronchial tree,” Phys. Rev. Lett. 67, 2106 (1991).Google Scholar
[73] Southern California Earthquake Center, http://www.sceddc.scec.org/ftp/SCNS/.
[74] , Introduction to Phase Transitions and Critical Phenomena, Oxford: Oxford University Press (1979).Google Scholar
[76] , , and , “Estimation of intravascular blood pressure by mathematical analysis of arterial casts,” Tokoku J. Exp. Med. 79, 168–198 (1963).Google Scholar
[77] , , et al., “Fractal properties of fetal breathing dynamics,” Am.J.Physiol. 262 (Regulatory Integrative Comp. Physiol.32) R141-R147 (1992).
[79] and , “Aggregation, migration and population mechanics,” Nature 265, 415–121 (1977).Google Scholar
[80] and , “Temporal stability as a density-dependent species characteristic,” J. Animal Ecol. 49, 209–224 (1980).Google Scholar
[81] , “Fractal character of auditory neural spike trains,” IEEE Trans. Biomed. Eng. 36, 150–160 (1989).Google Scholar
[82] , and , “The fractal doubly stochastic Poisson point process as a model for the cochlear neural spike train,” in The Mechanics and Biophysics of Hearing, eds. , , , and , New York: Springer (1990), pp. 345-361.Google Scholar
[83] , On Growth and Form, Cambridge: Cambridge University Press (1917); 2nd edn. (1963).Google Scholar
[84] and , “Landslides, forest ires, and earthquakes: examples of self-organized critical behavior,” PhysicaA 340, 580–589 (2004).Google Scholar
[85] , Über das Gehen des Menschen in gesunden und kranken Zuständen nach selbstregistrierenden Methoden, Tübingen (1881).Google Scholar
[86] and , “1/f Noise in music: music from 1/f noise,” J. Acoust. Soc. Am. 63, 258–263 (1978).Google Scholar
[87] , “Evolution of long-range fractal correlations and 1/f noise in DNA base sequences,” Phys.Rev. Lett. 68, 3805 (1992).Google Scholar
[92] , and , “Beyond the principle of similitude: renormalization in the bronchial tree,” J. Appl. Physiol. 60, 1089–1098 (1986).Google Scholar
[95] and , “Fractal physiology for physicists: Lévy statistics,” Phys. Rep. 246, 1–100 (1994).Google Scholar
[97] , “Physiology in fractal dimensions: error tolerance,” Ann. Biomed. Eng. 18, 135-149 (1990).Google Scholar
[98] and , “Allometric control of human gait,” Fractals 6, 101–108 (1998); “Allometric control, inverse power laws and human gait,” Chaos, Solitons & Fractals10, 1519-1527 (1999).
[99] , , and , “The independently fractal nature of respiration and heart rate during exercise under normobaric and hyperbaric conditions,” Respiratory Physiol. & Neurobiol. 145, 219–233 (2005).Google Scholar
[101] , Age and Area: A Study in Geographical Distribution and Origin of Species, Cambridge: Cambridge University Press (1922).Google Scholar
[103] , “Skew distributions in biomedicine including some with negative power of time,” in Statistical Distributions in Scientific Work, Vol. 2, eds. , , and , Dordrecht: D. Reidel (1975), pp. 241-262.Google Scholar
[104] , Human Behavior and the Principle of Least Effort: An Introduction to Human Ecology, Cambridge, MA: Addison-Wesley (1949).Google Scholar