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 inlinks 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 dataset exhibits a
powerlaw distribution is to plot the (complementary) cumulative
distribution function or the density function on a loglog 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
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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
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Figure: Cumulative edegreenetwistributionsworks. thesvix daxis axis forcumulativetworks (degree k vs.
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 Fall '09
 Acemoglu
 Probability theory, Pareto distribution, power law, Pareto principle, Scalefree network, Complex Networks

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