Unformatted text preview: itation and Web
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 ; istribution) [ and 1997 to all papers cataloged
Information ; (c) a 300 million vertex subset of the World Wide Web, circa 1999 ; (d) the Internet at the level of Courtesy of Society for Industrial and Applied Mathematics. Used with permission. ; (f ) the interaction network of
autonomous systems, April 1999 ; (e) the power grid of the western United States
proteins in the metab "The the yeast S. Cerevisiae . 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
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
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...
View Full Document
This document was uploaded on 03/18/2014 for the course EECS 6.207J at MIT.
- Fall '09