Complexity - A Guided Tour Part 23

"the average path-length has shrunk to about 9": This value was estimated from the results given in Newman, M. E. J., Moore, C., and Watts, D. J., Mean-field solution of the small-world network model. Physical Review Letters, 84, 1999, pp. 32013204.

"only a few random links can generate a very large effect": Watts, D. J., Six Degrees: The Science of a Connected Age. New York: W. W. Norton, 2003, p. 89.

"small-world property": The formal definition of the small-world property is that, even though relatively few long-distance connections are present, the shortest path length (number of link hops) between two nodes scales logarithmically or slower with network size n for fixed average degree.

"Kevin Bacon game": See, e.g., [http://en.wikipedia.org/wiki/Six_Degrees_of_

Kevin_Bacon].

"neuroscientists had already mapped out every neuron and neural connection": For more information, see Achacoso, T. B. and Yamamoto, W. S., AY's Neuroanatomy of C. Elegans for Computation. Boca Raton, FL: CRC Press, 1991.

"do not actually have the kinds of degree distributions": The Watts-Strogatz model produces networks with exponential degree distributions, rather than the much more commonly observed power-law degree distributions in real-world networks. For details, see Albert, R. and Barabasi, A-L., (2002). Statistical mechanics of complex networks. Reviews of Modern Physics, 74, 2002, pp. 4897.

"report about Was.h.i.+ngton State apple prices": [http://www.americanfruitgrower.com/e_notes/page.php?page=news].

"information about the Great Huon Apple Race of Tasmania": [http://www.huonfranklincottage.com.au/events.htm].

"this rule actually fits the data": The in-link degree distribution of the Web is fit reasonably well with the power law k2.3 and cutoff kmin = 3684 (see Clauset, A., Shalizi, C. R., and Newman, M. E. J., Power-law distributions in empirical data. Preprint, 2007 [http://arxiv.org/abs/0706.1062].) The expression k2 given in this chapter is an approximation used for simplifying the discussion; plots of the k2.3 distribution look very similar to those given in the chapter.

Chapter 16.

"neuroscientists have mapped the connectivity structure": e.g., see Ba.s.sett, D. S. and Bullmore, D., Small-world brain networks. The Neuroscientist, 12, 2006, pp. 512523; and Stam, C. J. and Reijneveld, J. C., Graph theoretical a.n.a.lysis of complex networks in the brain. Nonlinear Biomedical Physics, 1(1), 2007, p. 3.

"Genetic Regulatory Networks": For more details on the application of network ideas to genetic regulation, see Barabasi, A.-L. and Oltvai, Z. N., Network biology: Understanding the cell's functional organization. Nature Reviews: Genetics, 5, 2004, pp. 101113.

"Albert-Laszlo Barabasi and colleagues looked in detail at the structure of metabolic networks": Jeong, H., Tombor, B., Albert, R., Oltvai, Z. N., and Barabasi, A.-L., The large-scale organization of metabolic networks. Nature, 407, 2000, pp. 651654. Examples of other work on the structure of metabolic networks are Fell, D. A. and Wagner, A., The small world of metabolism. Nature Biotechnology, 18, 2000, pp. 11211122; and Burgard, A. P., Nikolaev, E. V., Schilling, C. H., and Maranas, C. D., Flux coupling a.n.a.lysis of genome-scale metabolic network reconstructions. Genome Research, 14, 2004, pp. 301312.

"Although later studies debunked the theory": See, e.g., Robbins, K. E., Lemey, P., Pybus, O. G., Jaffe, H. W., Youngpairoj, A. S., Brown, T. M., Salemi, M., Vandamme, A. M., and Kalish, M. L., U.S. human immunodeficiency virus type 1 epidemic: Date of origin, population history, and characterization of early strains. Journal of Virology, 77 (11), 2003, pp. 63596366.

"Recently, a group consisting of": Liljeros, F., Edling, C. R., Nunes Amaral, L. A., Stanely, H. E., and Aberg, Y., The web of human s.e.xual contacts. Nature, 441, 2001, pp. 907908.

"similar results have been found in studies of other s.e.xual networks": e.g., Schneeberger, A., Mercer, C. H., Gregson, S. A., Ferguson, N. M., Nyamukapa, C. A., Anderson, R. M., Johnson, A. M., and Garnett, G. P., Scale-free networks and s.e.xually transmitted diseases: A description of observed patterns of s.e.xual contacts in Britain and Zimbabwe. s.e.xually Transmitted Diseases, 31 (6), 2004, pp. 380387.

"A very clever yet simple method was proposed": Cohen, R., ben-Avraham, D., and Havlin, S., Efficient immunization strategies for computer networks and populations. Physics Review Letters, 91 (24), 2003, p. 247901.

"one should target anti-virus methods": Newman, M. E. J., Forrest, S., and Balthrop, J., Email networks and the spread of computer viruses. Physical Review E, 66, 2002, p. 035101.

"the ecology research community has recently seen a lot of debate": See, e.g., Montoya, J. M. and Sole, R. V., Small world patterns in food webs. Journal of Theoretical Biology, 214 (3), 2002, pp. 405412; Dunne, J. A., Williams, R. J., and Martinez, N. D., Food-web structure and network theory: The role of connectance and size. Proceedings of the National Academy of Science, USA, 99 (20), 2002, pp. 1291712922; and Dunne, J. A., The network structure of food webs. In M. Pascual and J. A. Dunne (editors), Ecological Networks: Linking Structure to Dynamics in Food Webs. New York: Oxford University Press, 2006, pp. 2786.

"Where Do Scale-Free Networks Come From?" Parts of this section were adapted from Mitch.e.l.l, M., Complex systems: Network thinking. Artificial Intelligence, 170 (18), 2006, pp. 11941212.

"Albert-Laszlo Barabasi and Reka Albert proposed": Barabasi, A.-L. and Albert, R., Emergence of scaling in random networks, Science, 286, 1999, pp. 509512.

"this process and its power-law outcome had been discovered independently": Yule, G. U., A mathematical theory of evolution, based on the conclusions of Dr. J. C. Willis. Philosophical Transactions of the Royal Society of London, Ser. B 213, 1924, pp. 2187; Simon, H. A., On a cla.s.s of skew distribution functions." Biometrika 42 (3-4), 1955, p. 425; and Price, D. J., Networks of scientific papers. Science 149, 1965, pp. 510515.

"the growth of so-called scientific citation networks": e.g., see Redner, S., How popular is your paper? An empirical study of the citation distribution. European Physical Journal B, 4(2), 1998, pp. 131134.

"what the writer Malcolm Gladwell called tipping points": Gladwell, M., The Tipping Point: How Little Things Can Make a Big Difference. Boston: Little, Brown, 2000.

"A number of networks previously identified to be 'scale-free'": Clauset, A., Shalizi, C. R., and Newman, M. E. J., Power-law distributions in empirical data. Preprint, 2007, [http://arxiv.org/abs/0706.1062].

"Current a.s.sessments of the commonality of power-laws": Keller, E. F., Revisiting 'scale-free' networks. BioEssays, 27, 2005, pp. 10601068.

"Our tendency to hallucinate": Shalizi, C., Networks and Netwars, 2005. Essay at [http://www.cscs.umich.edu/~crshalizi/weblog/347.html].

"there turn out to be nine and sixty ways": Shalizi, C., Power Law Distributions, 1/f noise, Long-Memory Time Series, 2007. Essay at [http://cscs.umich.edu/~crshalizi/notebooks/power-laws.html].

"a new hypothesized mechanism that resulted in power law distributions": For surveys of some such mechanisms, see Mitzenmacher, M., A brief history of generative models for power law and lognormal distributions. Internet Mathematics, 1(2), 2003, pp. 226251; and Newman, M. E. J., Power laws, Pareto distributions and Zipf's law. Contemporary Physics, 46, 2005, pp. 323351.

"The reported cause of the shutdown": The cascading failure and its causes are described in detail in the U.S.-Canada Power System Outage Task Force's Final Report on the August 14, 2003 Blackout in the United States and Canada: Causes and Recommendations [https://reports.energy.gov/].

"The computer system of the US Customs and Border protection agency": see Schlossberg, D. "LAX Computer Crash Strands International Pa.s.sengers." ConsumerAffairs.com, August 13, 2007, [http://www.consumeraffairs.com/news04/

2007/08/lax_computers.html]; and Schwartz, J., "Who Needs Hackers?" New York Times, September 12, 2007.

"Long-Term Capital Management": see, e.g., Government Accounting Office, Long-Term Capital Management: Regulators Need to Focus Greater Attention on Systemic Risk. Report to Congressional Request, 1999, [http://www.gao.gov/cgi-bin/getrpt?GGD-00-3]; and Coy, P., Woolley, S., Spiro, L. N., and Glasgall, W., Failed wizards of Wall Street. Business Week, September 21, 1998.

"The threat is complexity itself": Andreas Antonopoulos, quoted in Schwartz, J., "Who Needs Hackers?" New York Times, September 12, 2007.

"Self-Organized Criticality": for an introduction to SOC, see Bak, P., How Nature Works: The Science of Self-Organized Criticality. New York: Springer, 1996.

"Highly Optimized Tolerance": For an introduction to HOT, see Carlson, J. M. and Doyle, J., Complexity and robustness. Proceedings of the National Academy of Science, USA 99, 2002, pp. 25382545.

"Next to the mysteries of dynamics on a network": Watts, D. J., Six Degrees: The Science of a Connected Age. New York: W. W. Norton, 2003, p. 161.

Chapter 17.

"the surface area scales as the volume raised to the two-thirds power": Let V denote volume, S denote surface area, and r denote radius. V is proportional to r3, so cube-root (V) is proportional to the radius. Surface area is proportional to radius2, and thus to cube-root(V)2, which is .

"If you plot a power law on a double logarithmic plot, it will look like a straight line, and the slope of that line will be equal to the power law's exponent": In the example here, the power law is metabolic rate body ma.s.s3/4.

Taking the logarithm of both sides, we get log (metabolic rate) 3/4 log (body ma.s.s).

This is the equation of a straight line with slope 3/4, if we plot log (metabolic rate) against log (body ma.s.s), which is actually what is plotted in figure 16.2 "Enquist later described the group's math results as 'pyrotechnics'": Grant, B., The powers that be. The Scientist, 21 (3), 2007.

"You really have to think in terms of two separate scales": G. B. West, quoted in Mackenzie, D., Biophysics: New clues to why size equals destiny. Science, 284 (5420), 1999, pp. 16071609.

"The mathematical details of the model": A technical, but not too difficult to understand, description of the Metabolic Scaling model is given in West, G. B. and Brown, J. H., Life's universal scaling laws. Physics Today, 57 (9), 2004, pp. 3643.

"Although living things occupy a three-dimensional s.p.a.ce": West, G. B., Brown, J. H., and Enquist, B. J., The fourth dimension of life: Fractal geometry and allometric scaling of organisms. Science, 284, pp. 16771679.

"the potential to unify all of biology": Grant, B., The powers that be. The Scientist, 21 (3), 2007.

"as potentially important to biology as Newton's contributions are to physics": Niklas, K. J., Size matters! Trends in Ecology and Evolution 16 (8), 2001, p. 468.

"We see the prospects for the emergence of a general theory of metabolism": West, G. B. and Brown, J. H., The origin of allometric scaling laws in biology from genomes to ecosystems: Towards a quant.i.tative unifying theory of biological structure and organization. Journal of Experimental Biology 208, 2005, pp. 15751592.

"Still others argue that Kleiber was wrong all along": A review of various critiques of metabolic scaling theory is given in Agutter P. S and Wheatley, D. N., Metabolic scaling: Consensus or Controversy? Theoretical Biology and Medical Modeling, 18, 2004, pp. 283289.

"The more detail that one knows about the particular physiology involved": H. Horn, quoted in Whitfield, J., All creatures great and small. Nature, 413, 2001, pp. 342344.

"It's nice when things are simple": H. Muller-Landau, quoted in Grant, B., The powers that be. The Scientist, 21 (3), 2007.

"There have been arguments that the mathematics in metabolic scaling theory is incorrect": e.g., Kozlowski, J. and Konarzweski, M., Is West, Brown and Enquist's model of allometric scaling mathematically correct and biologically relevant? Functional Ecology, 18, 2004, pp. 283289.

"The authors of metabolic scaling theory have strongly stood by their work": E.g., see West, G. B., Brown, J. H., and Enquist, B. J., Yes, West, Brown and Enquist's model of allometric scaling is both mathematically correct and biologically relevant. (Reply to Kozlowski and Konarzweski, 2004.) Functional Ecology, 19, 2005, pp. 735738; and Borrell, B., Metabolic theory spat heats up. The Scientist (News), November 8, 2007. [http://www.the-scientist.com/news/display/53846/].

"Part of me doesn't want to be cowered": G. West, quoted in Grant, B., The powers that be. The Scientist, 21 (3), 2007.

"I suspect that West, Enquist et al. will continue repeating their central arguments": H. Muller-Landau, quoted in Borrell, B., Metabolic theory spat heats up. The Scientist (News), November 8, 2007. [http://www.the-scientist.com/news/display/53846/].

"more normal than 'normal'": "The presence of [power-law] distributions": Willinger, W., Alderson, D., Doyle, J. C., and Li, L., More 'normal' than normal: Scaling distributions and complex systems. In R. G. Ingalls et al., Proceedings of the 2004 Winter Simulation Conference, pp. 130141. Piscataway, NJ: IEEE Press, 2004.

"This relation is now called Zipf's law": Zipf's original publication on this work is a book: Zipf, G. K., Selected Studies of the Principle of Relative Frequency in Language. Cambridge, MA: Harvard University Press, 1932.

"Benoit Mandelbrot ... had a somewhat different explanation": Mandelbrot. B., An informational theory of the statistical structure of languages. In W. Jackson (editor), Communicaiton Theory, Woburn, MA: b.u.t.terworth, 1953, pp. 486502.

"Herbert Simon proposed yet another explanation": Simon, H. A., On a cla.s.s of skew distribution functions." Biometrika 42 (34), 1955, p. 425.

"Evidently Mandelbrot and Simon had a rather heated argument": Mitzenmacher, M., A brief history of generative models for power law and lognormal distributions. Internet Mathematics, 1 (2), 2003, pp. 226251.

"the psychologist George Miller showed": Miller, G. A., Some effects of intermittent silence. The American Journal of Psychology, 70, 1957, pp. 311314.

Chapter 18.

" 'mobile genetic elements'": A technical article on the proposed role of mobile genetic elements on brain diversity is Muotri, A. R., Chu, V. T., Marchetto, M. C. N., Deng, W., Moran, J. V. and Gage, F. H., Somatic mosaicism in neuronal precursor cells mediated by L1 retrotransposition. Nature, 435, 2005, pp. 903910.

"Recently a group of science philosophers and biologists performed a survey": Reported in Pearson, H., What is a gene? Nature, 441, 2006, pp. 399401.

"The more expert scientists become in molecular genetics": Ibid.

"the absence of necessary methylation": See, e.g., Dean, W., Santos, F., Stojkovic, M., Zakhartchenko, V., Walter, J., Wolf, E., and Reik, W., Conservation of methylation reprogramming in mammalian development: Aberrant reprogramming in cloned embryos. Proceedings of the National Academy of Science, USA, 98 (24), 2001, pp. 1373413738.

"a large proportion of the DNA that is transcribed into RNA is not subsequently translated into proteins": See, e.g., Mattick, J. S., RNA regulation: A new genetics? Nature Reviews: Genetics, 5, 2004, pp. 316323; Grosshans, H. and Filipowicz, W., The expanding world of small RNAs. Nature, 451, 2008, pp. 414416.

"The significance of non-coding RNA": For a discussion of some of the current research and controversies in this area, see Huttenhofer, A., Scattner, P., and Polacek, N., Non-coding RNAs: Hope or Hype? Trends in Genetics, 21 (5), 2005, pp. 289297.

"The presumption that genes operate independently": Caruso, D., A challenge to gene theory, a tougher look at biotech. New York Times, July 1, 2007.

"encode a specific functional product": U.S. patent law, quoted in ibid.

"Evidence of a networked genome shatters the scientific basis": Ibid.

"more than 90% of our DNA is shared with mice": Published estimates of DNA overlap between humans and different species differ. However, it seems that "over 90% shared" is a fairly safe statement.

"a 'black box' that somehow transformed genetic information": Carroll, S. B., Endless Forms Most Beautiful: The New Science of Evo Devo and the Making of the Animal Kingdom. New York: W. W. Norton, 2005, p. 7.

"The irony ... is that what was dismissed as junk": In the video Gene Regulation; Science, 319, no 5871, 2008.

"To verify that the BMP4 gene itself could indeed trigger the growth": From Yoon, C. K., From a few genes, life's myriad shapes. New York Times, June 26, 2007.

"In a strange but revealing experiment": This work is described in Travis, J., Eye-opening gene. Science News Online, May 10, 1997.

"This conclusion is still quite controversial among evolutionary biologists": For discussion of this controversy, see, e.g., Erwin, D. H., The developmental origins of animal bodyplans. In S. Xiao and A. J. Kaufman (editors), Neoproterozoic Geobiology and Paleobiology, New York: Springer, 2006, pp. 159197.

"a world-cla.s.s intellectual riffer": Horgan, J., From complexity to perplexity. Scientific American, 272, June 1995, pp. 7479.

"the genomic networks that control development": Kauffman, S. A., At Home in the Universe. New York: Oxford University Press, 1995, p. 26.