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E the fraction of vertices that have degree greater

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Unformatted text preview: itation and Web e distribution of degrees, i.e., the fraction of vertices that have degree greater than or equal to k. The networks shown are: (a) the collaboration the cumulative of robability d(b) citations between 1981 Newman 03]. by the Institute for Scientific network pmathematicians [182]; istribution) [ and 1997 to all papers cataloged Information [351]; (c) a 300 million vertex subset of the World Wide Web, circa 1999 [74]; (d) the Internet at the level of Courtesy of Society for Industrial and Applied Mathematics. Used with permission. [416]; (f ) the interaction network of autonomous systems, April 1999 [86]; (e) the power grid of the western United States proteins in the metab "The the yeast S. Cerevisiae [212]. of Complex Networks." SIAM Review 45, no. 2 e Figure 6 in Mark E. J. Newman's. olism ofStructure and Function Of these networks, three of them, (c), (d) and (f ), appear to hav(2003): 167-256. power-law degree distributions, as indicated by their approximately straight-line forms on the doubly logarithmic scales, and one (b) has a power-law tail but deviates markedly from power-law behavior for small degree. Network (e) has an exponential 7 Networks: Lecture 6 History of Power Laws—1 Power laws had been observed in a variety of fields for some time. The earliest apparent reference is to the work by Pareto in 1897, who introduced the Pareto distribution to describe income distributions. When studying wealth distributions, Pareto observed power law features, where there were many more individuals who had large amounts of wealth than would appear in Gaussian or other distributions. Power laws also appeared in the work of Zipf in 1916, in describing word frequencies in documents and city sizes. The empirical principle, known as Zipf’s Law, states that the frequency of the j th most common word in English (or other common languages) is proportional to j −1 . These ideas were further developed in the work of Simon in 1955, who showed that power laws arise when “the rich get richer”, when the amount you get goes up with the amount you already have. 8 Networks: Lecture 6 History of Power Laws—2 Recall the examples: A city grows in proportion to its current size as a result of people having children. Gene copies arise in large part due mutational events in which a random segment of the DNA is accidentally duplicated (a gene which already has many copies more likely to be in a random stretch of DNA) All of these examples exhibit rich get richer effects. Rich get richer effects...
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This document was uploaded on 03/18/2014 for the course EECS 6.207J at MIT.

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