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# Some physicists think these correspond to some

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Unformatted text preview: wer law distribution. Some physicists think these correspond to some “universal laws”, as illustrated by the following quote from Barabasi that appeared in the April 2002 issue of the Scientist: “What do proteins in our bodies, the Internet, a cool collection of atoms, and sexual networks have in common? One man thinks he has the answer and it is going to transform the way we view the world.” A nonnegative random variable X is said to have a power law distribution if P(X ≥ x ) ∼ cx −α , for constants c > 0 and α > 0. (Here f (x ) ∼ g (x ) represents that the limit of the ratios goes to 1 as x grows large.) Roughly speaking, in a power law distribution, asymptotically, the tails fall of polynomially with power α. 5 Networks: Lecture 6 Power Law Distribution—2 Such a distribution leads to much heavier tails than other common models, such as Gaussian and exponential distributions. In the context of the WWW, this implies that pages with large numbers of in-links are much more common than we’d expect in a Gaussian distribution. This accords well with our intuitive notion of popularity exhibiting extreme imbalances. One speciﬁc commonly used power law distribution is the Pareto distribution, which satisﬁes � x �−α P(X ≥ x ) = , t for some α > 0 and t > 0. The Pareto distribution requires X ≥ t . The density function for the Pareto distribution is f (x ) = αt α x −α−1 . For a power law distribution, usually α falls in the range 0 < α ≤ 2, in which case X has inﬁnite variance. If α ≤ 1, then X also has inﬁnite mean. 6 Networks: Lecture 6 Examples A simple method for providing a quick test for whether a data-set exhibits a power-law distribution is to plot the (complementary) cumulative distribution function or the density function on a log-log scale. 14 The structure and function of complex networks 0 0 0 10 10 10 -2 10 -2 -2 10 10 -4 10 -4 -4 10 10 (a) collaborations in mathematics -6 10 (b) citations (c) World Wide Web -8 1 10 100 0 1 10 100 10 1000 10 10 -2 10 -1 10 -2 10 4 10 6 10 -1 10 -3 10 (d) Internet 2 -2 -3 10 10 10 10 -1 0 0 0 10 10 (e) power grid (f) protein interactions -3 10 -4 10 1 10 100 1000 0 10 20 1 10 FIG. 6 Cumulativ degree distributions for six diﬀerent net forThe horizoniﬀerenth n is vertex Figure: Cumulative edegreenetwistributionsworks. thesvix daxis axis forcumulativetworks (degree k vs. d orks, which are directed) and ertical tal is the eac panelprobabilitydegree k (or indegree for the c...
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