Editing Power law (section)

==== Log-log plots ====
[[File:Log-log plot example.svg|thumb|A straight line on a log–log plot is necessary but insufficient evidence for power-laws, the slope of the straight line corresponds to the power law exponent.]]

[[Log–log plot]]s are an alternative way of graphically examining the tail of a distribution using a random sample. Taking the logarithm of a power law of the form <math>f(x) = ax^{k}</math> results in:<ref>http://www.physics.pomona.edu/sixideas/old/labs/LRM/LR05.pdf</ref>

:<math>\begin{align}
 \log(f(x)) &= \log(ax^{k}) \\
 &= \log(a) + \log(x^k) \\
 &= \log(a) + k \cdot \log(x),
\end{align}</math>

which forms a straight line with slope <math>k</math> on a log-log scale. Caution has to be exercised however as a log–log plot is necessary but insufficient evidence for a power law relationship, as many non power-law distributions will appear as straight lines on a log–log plot.{{sfn|Clauset|Shalizi|Newman|2009}}<ref>{{cite web|url=http://bactra.org/weblog/491.html|title=So You Think You Have a Power Law — Well Isn't That Special?|website=bactra.org|access-date=27 March 2018}}</ref> This method consists of plotting the logarithm of an estimator of the probability that a particular number of the distribution occurs versus the logarithm of that particular number. Usually, this estimator is the proportion of times that the number occurs in the data set. If the points in the plot tend to converge to a straight line for large numbers in the x axis, then the researcher concludes that the distribution has a power-law tail. Examples of the application of these types of plot have been published.<ref>{{cite journal |last1=Jeong |first1=H. |last2=Tombor |first2= B. Albert |last3=Oltvai |first3=Z.N. |last4=Barabasi |first4= A.-L. |year=2000 |title=The large-scale organization of metabolic networks |journal=Nature |volume=407 |issue=6804| pages=651–654 |doi=10.1038/35036627 |pmid=11034217 |arxiv=cond-mat/0010278 |bibcode=2000Natur.407..651J |s2cid=4426931}}</ref> A disadvantage of these plots is that, in order for them to provide reliable results, they require huge amounts of data. In addition, they are appropriate only for discrete (or grouped) data.