Complexity_ A Guided Tour - Melanie Mitchell [159]
“No one descends with such fury”: Watts, D. J., Six Degrees: The Science of a Connected Age. New York: W. W. Norton & Co, 2003, p. 32.
“network thinking is poised”: Barabási, A.-L. Linked: The New Science of Networks. Cambridge, MA: Perseus, 2002, p. 222.
“123 incoming links”: All the in-link counts in this chapter were obtained from [http://www.microsoft-watch.org/cgi-bin/ranking.htm]. The count includes only in-links from outside the Web page’s domain.
“mathematically define the concept of ‘small-world network’”: See Watts, D. J. and Strogatz, S. H., Collective dynamics of ‘small world’ networks. Nature, 393, 1998, pp. 440–442.
“The average path length of the regular network of figure 15.4 turns out to be 15”: This value was calculated using the formula l = N/2k. Here l is the average path length, N is the number of nodes, and k is the degree of each node (here 2). See Albert, R. and Barabási, A-L., Statistical mechanics of complex networks. Reviews of Modern Physics, 74, 2002, pp. 48–97.
“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. 3201–3204.
“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 Barabási, A-L., (2002). Statistical mechanics of complex networks. Reviews of Modern Physics, 74, 2002, pp. 48–97.
“report about Washington 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 k−2.3 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 k−2 given in this chapter is an approximation used for simplifying the discussion; plots of the k−2.3 distribution look very similar to those given in the chapter.
Chapter 16
“neuroscientists have mapped the connectivity structure”: e.g., see Bassett, D. S. and Bullmore, D., Small-world brain networks. The Neuroscientist, 12, 2006, pp. 512–523; and Stam, C. J. and Reijneveld, J. C., Graph theoretical analysis 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 Barabási, A.-L. and Oltvai, Z. N., Network biology: Understanding the cell’s functional organization. Nature Reviews: Genetics, 5, 2004, pp. 101–113.
“Albert-László Barabási and colleagues looked in detail at the structure of metabolic networks”: Jeong, H., Tombor, B., Albert, R., Oltvai, Z. N., and Barabási, A.-L., The large-scale organization of metabolic networks. Nature, 407, 2000, pp. 651–654. Examples of other work on the structure