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Power laws, Pareto distributions and Zipf's law
MEJ Newman a
a
Department of Physics and Center for the Study of Complex Systems, University of
Michigan, Ann Arbor, USA

Online Publication Date: 01 September 2005

To cite this Article: Newman, MEJ (2005) 'Power laws, Pareto distributions and Zipf's
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 Contemporary Physics, Vol. 46, No. 5, September–October 2005, 323 – 351
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Power laws, Pareto distributions and Zipf’s law

M.E.J. NEWMAN*

Department of Physics and Center for the Study of Complex Systems, University of Michigan, Ann Arbor,
MI 48109, USA

(Received 28 October 2004; in final form 23 November 2004)

When the probability of measuring a particular value of some quantity varies inversely as
a power of that value, the quantity is said to follow a power law, also known variously as
Zipf’s law or the Pareto distribution. Power laws appear widely in physics, biology, earth
and planetary sciences, economics and finance, computer science, demography and the
social sciences. For instance, the distributions of the sizes of cities, earthquakes, forest
fires, solar flares, moon craters and people’s personal fortunes all appear to follow power
laws. The origin of power-law behaviour has been a topic of debate in the scientific
community for more than a century. Here we review some of the empirical evidence for
the existence of power-law forms and the theories proposed to explain them.

But not all things we measure are peaked around a

1. Introduction
typical value. Some vary over an enormous dynamic range,
Many of the things that scientists measure have a typical sometimes many orders of magnitude. A classic example of
size or ‘scale’—a typical value around which individual this type of behaviour is the sizes of towns and cities. The
measurements are centred. A simple example would be the largest population of any city in the US is 8.00 million for
heights of human beings. Most adult human beings are New York City, as of the most recent (2000) census. The
about 180 cm tall. There is some variation around this town with the smallest population is harder to pin down,
figure, notably depending on sex, but we never see people since it depends on what you call a town. The author recalls
who are 10 cm tall, or 500 cm. To make this observation in 1993 passing through the town of Milliken, Oregon,
more quantitative, one can plot a histogram of people’s population 4, which consisted of one large house occupied
heights, as I have done in figure 1 (a). The figure shows the by the town’s entire human population, a wooden shack
heights in centimetres of adult men in the United States occupied by an extraordinary number of cats and a very
measured between 1959 and 1962, and indeed the distribu- impressive flea market. According to the Guinness Book,
tion is relatively narrow and peaked around 180 cm. however, America’s smallest town is Duffield, Virginia,
Another telling observation is the ratio of the heights of with a population of 52. Whichever way you look at it, the
the tallest and shortest people. The Guinness Book of ratio of largest to smallest population is at least 150000.
Records claims the world’s tallest and shortest adult men Clearly this is quite different from what we saw for heights
(both now dead) as having had heights 272 cm and 57 cm of people. And an even more startling pattern is revealed
respectively, making the ratio 4.8. This is a relatively low when we look at the histogram of the sizes of cities, which is
value; as we will see in a moment, some other quantities shown in figure 2.
have much higher ratios of largest to smallest. In the left panel of the figure, I show a simple histogram
Figure 1 (b) shows another example of a quantity with a of the distribution of US city sizes. The histogram is highly
typical scale: the speeds in miles per hour of cars on the right-skewed, meaning that while the bulk of the distribu-
motorway. Again the histogram of speeds is strongly tion occurs for fairly small sizes—most US cities have small
peaked, in this case around 75 mph. populations—there is a small number of cities with a

*Corresponding author. *E-mail: mejn@umich.edu

Contemporary Physics
ISSN 0010-7514 print/ISSN 1366-5812 online ª 2005 Taylor & Francis Group Ltd
http://www.tandf.co.uk/journals
DOI: 10.1080/00107510500052444
 324 M.E.J. Newman
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Figure 1. Left: histogram of heights in centimetres of American males. Data from the National Health Examination Survey,
1959 – 1962 (US Department of Health and Human Services). Right: histogram of speeds in miles per hour of cars on UK
motorways. Data from Transport Statistics 2003 (UK Department for Transport).

Figure 2. Left: histogram of the populations of all US cities with population of 10000 or more. Right: another histogram of
the same data, but plotted on logarithmic scales. The approximate straight-line form of the histogram in the right panel
implies that the distribution follows a power law. Data from the 2000 US Census.

population much higher than the typical value, producing when plotted in this fashion, follows quite closely a straight
the long tail to the right of the histogram. This right-skewed line. This observation seems first to have been made by
form is qualitatively quite different from the histograms of Auerbach [1], although it is often attributed to Zipf [2].
people’s heights, but is not itself very surprising. Given that What does it mean? Let p(x) dx be the fraction of cities with
we know there is a large dynamic range from the smallest to population between x and x + dx. If the histogram is a
the largest city sizes, we can immediately deduce that there straight line on log – log scales, then ln p(x) = – a ln x + c,
can only be a small number of very large cities. After all, in where a and c are constants. (The minus sign is optional,
a country such as America with a total population of 300 but convenient since the slope of the line in figure 2 is
million people, you could at most have about 40 cities the clearly negative.) Taking the exponential of both sides, this
size of New York. And the 2700 cities in the histogram of is equivalent to
figure 2 cannot have a mean population of more than
pðxÞ ¼ Cx
a ; ð1Þ
3 6 108/2700 = 110 000.
What is surprising on the other hand, is the right panel of
figure 2, which shows the histogram of city sizes again, but with C = exp(c).
this time replotted with logarithmic horizontal and vertical Distributions of the form (1) are said to follow a power
axes. Now a remarkable pattern emerges: the histogram, law. The constant a is called the exponent of the power law.
 Power laws, Pareto distributions and Zipfs law 325
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(The constant C is mostly uninteresting; once a is fixed, it is (a) of the figure shows a normal histogram of the numbers,
determined by the requirement that the distribution p(x) produced by binning them into bins of equal size 0.1. That
sum to 1; see section 3.1.) is, the first bin goes from 1 to 1.1, the second from 1.1 to
Power-law distributions occur in an extraordinarily 1.2, and so forth. On the linear scales used this produces a
diverse range of phenomena. In addition to city popula- nice smooth curve.
tions, the sizes of earthquakes [3], moon craters [4], solar To reveal the power-law form of the distribution it is
flares [5], computer files [6] and wars [7], the frequency of better, as we have seen, to plot the histogram on
use of words in any human language [2, 8], the frequency of logarithmic scales, and when we do this for the current
occurrence of personal names in most cultures [9], the data we see the characteristic straight-line form of the
numbers of papers scientists write [10], the number of power-law distribution, figure 3 (b). However, the plot is
citations received by papers [11], the number of hits on web in some respects not a very good one. In particular the
pages [12], the sales of books, music recordings and almost right-hand end of the distribution is noisy because of
every other branded commodity [13, 14], the numbers of sampling errors. The power-law distribution dwindles in
species in biological taxa [15], people’s annual incomes [16] this region, meaning that each bin only has a few samples
and a host of other variables all follow power-law in it, if any. So the fractional fluctuations in the bin
distributions*. counts are large and this appears as a noisy curve on the
Power-law distributions are the subject of this article. In plot. One way to deal with this would be simply to throw
the following sections, I discuss ways of detecting power- out the data in the tail of the curve. But there is often
law behaviour, give empirical evidence for power laws in a useful information in those data and furthermore, as we
variety of systems and describe some of the known will see in section 2.1, many distributions follow a power
mechanisms by which power-law behaviour can arise. law only in the tail, so we are in danger of throwing out
Readers interested in pursuing the subject further may the baby with the bathwater.
also wish to consult the recent reviews by Sornette [18] and An alternative solution is to vary the width of the bins
Mitzenmacher [19], as well as the bibliography compiled by in the histogram. If we are going to do this, we must also
Li{. normalize the sample counts by the width of the bins they
fall in. That is, the number of samples in a bin of width
Dx should be divided by Dx to get a count per unit interval
2. Measuring power laws
of x. Then the normalized sample count becomes
Identifying power-law behaviour in either natural or man- independent of bin width on average and we are free to
made systems can be tricky. The standard strategy makes vary the bin widths as we like. The most common choice
use of a result we have already seen: a histogram of a is to create bins such that each is a fixed multiple wider
quantity with a power-law distribution appears as a straight than the one before it. This is known as logarithmic
line when plotted on logarithmic scales. Just making a binning. For the present example, for instance, we might
simple histogram, however, and plotting it on log scales to choose a multiplier of 2 and create bins that span the
see if it looks straight is, in most cases, a poor way to intervals 1 to 1.1, 1.1 to 1.3, 1.3 to 1.7 and so forth (i.e.
proceed. the sizes of the bins are 0.1, 0.2, 0.4 and so forth). This
Consider figure 3. This example shows a fake data set: I means the bins in the tail of the distribution get more
have generated a million random real numbers drawn from samples than they would if bin sizes were fixed, and this
a power-law probability distribution p(x) = Cx – a with reduces the statistical errors in the tail. It also has the nice
exponent a = – 2.5, just for illustrative purposes{. Panel side-effect that the bins appear to be of constant width
when we plot the histogram on log scales.
*
Power laws also occur in many situations other than the statistical
I used logarithmic binning in the construction of figure 2
distributions of quantities. For instance, Newton’s famous 1/r2 law for (b), which is why the points representing the individual bins
gravity has a power-law form with exponent a = 2. While such laws are appear equally spaced. In figure 3 (c) I have done the same
certainly interesting in their own way, they are not the topic of this paper. for our computer-generated power-law data. As we can see,
Thus, for instance, there has in recent years been some discussion of the
the straight-line power-law form of the histogram is now
‘allometric’ scaling laws seen in the physiognomy and physiology of
biological organisms [17], but since these are not statistical distributions
much clearer and can be seen to extend for at least a decade
they will not be discussed here. further than was apparent in figure 3 (b).
{
http://linkage.rockefeller.edu/wli/zipf/. Even with logarithmic binning there is still some noise in
{
This can be done using the so-called transformation method. If we can the tail, although it is sharply decreased. Suppose the
generate a random real number r uniformly distributed in the range
bottom of the lowest bin is at xmin and the ratio of the
0 4 r 5 1, then x = xmin(1 – r)71/a71 is a random power-law-distributed
real number in the range xmin 4 x 5 ? with exponent a. Note that there
widths of successive bins is a. Then the kth bin extends
has to be a lower limit xmin on the range; the power-law distribution from xk – 1 = xminak – 1 to xk = xminak and the expected
diverges as x?0—see section 2.1. number of samples falling in this interval is
 326 M.E.J. Newman
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Figure 3. (a) Histogram of the set of 1 million random numbers described in the text, which have a power-law distribution
with exponent a = 2.5. (b) The same histogram on logarithmic scales. Notice how noisy the results get in the tail towards the
right-hand side of the panel. This happens because the number of samples in the bins becomes small and statistical
fluctuations are therefore large as a fraction of sample number. (c) A histogram constructed using ‘logarithmic binning’. (d) A
cumulative histogram or rank/frequency plot of the same data. The cumulative distribution also follows a power law, but
with an exponent of a – 1 = 1.5.

Z xk Z xk The plot we get is no longer a simple representation of the

a
pðxÞ dx ¼ C x dx distribution of the data, but it is useful nonetheless. If the
xk
1 xk
1
ð2Þ distribution follows a power law p(x) = Cx – a, then

1a a
1
Z 1
¼C ðxmin ak Þ
aþ1 :
a C
ða
1Þ
a
1 PðxÞ ¼ C x0 dx0 ¼ x : ð4Þ
x a
1
Thus, so long as a 4 1, the number of samples per bin goes
down as k increases and the bins in the tail will have more Thus the cumulative distribution function P(x) also follows
statistical noise than those that precede them. As we will see a power law, but with a different exponent a – 1, which is 1
in the next section, most power-law distributions occurring less than the original exponent. Thus, if we plot P(x) on
in nature have 2 4 a 4 3, so noisy tails are the norm. logarithmic scales we should again get a straight line, but
Another, and in many ways a superior, method of with a shallower slope.
plotting the data is to calculate a cumulative distribution But notice that there is no need to bin the data at all to
function. Instead of plotting a simple histogram of the data, calculate P(x). By its definition, P(x) is well defined for
we make a plot of the probability P(x) that x has a value every value of x and so can be plotted as a perfectly normal
greater than or equal to x: function without binning. This avoids all questions about
Z 1 what sizes the bins should be. It also makes much better use
PðxÞ ¼ pðx0 Þ dx0 : ð3Þ of the data: binning of data lumps all samples within a
x given range together into the same bin and so throws out
 Power laws, Pareto distributions and Zipfs law 327
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any information that was contained in the individual values Applying equation (5) to our present data gives an
of the samples within that range. Cumulative distributions estimate of a = 2.500 + 0.002 for the exponent, which
do not throw away any information; it is all there in the agrees well with the known value of 2.5.
plot.
Figure 3 (d) shows our computer-generated power-law
2.1 Examples of power laws
data as a cumulative distribution, and indeed we again see
the tell-tale straight-line form of the power law, but with a In figure 4 we show cumulative distributions of twelve
shallower slope than before. Cumulative distributions like different quantities measured in physical, biological,
this are sometimes also called rank/frequency plots for technological and social systems of various kinds. All have
reasons explained in Appendix A. Cumulative distributions been proposed to follow power laws over some part of their
with a power-law form are sometimes said to follow Zipf’s range. The ubiquity of power-law behaviour in the natural
law or a Pareto distribution, after two early researchers who world has led many scientists to wonder whether there is a
championed their study. Since power-law cumulative single, simple, underlying mechanism linking all these
distributions imply a power-law form for p(x), ‘Zipf’s different systems together. Several candidates for such
law’ and ‘Pareto distribution’ are effectively synonymous mechanisms have been proposed, going by names like ‘self-
with ‘power-law distribution’. (Zipf’s law and the Pareto organized criticality’ and ‘highly optimized tolerance’.
distribution differ from one another in the way the However, the conventional wisdom is that there are
cumulative distribution is plotted—Zipf made his plots actually many different mechanisms for producing power
with x on the horizontal axis and P(x) on the vertical one; laws and that different ones are applicable to different
Pareto did it the other way around. This causes much cases. We discuss these points further in section 4.
confusion in the literature, but the data depicted in the The distributions shown in figure 4 are as follows.
plots are of course identical*.)
We know the value of the exponent a for our artificial (a) Word frequency: Estoup [8] observed that the
data set since it was generated deliberately to have a frequency with which words are used appears to
particular value, but in practical situations we would often follow a power law, and this observation was
like to estimate a from observed data. One way to do this famously examined in depth and confirmed by Zipf
would be to fit the slope of the line in plots like figures 3 (b), [2]. Panel (a) of figure 4 shows the cumulative
(c) or (d), and this is the most commonly used method. distribution of the number of times that words occur
Unfortunately, it is known to introduce systematic biases in a typical piece of English text, in this case the text
into the value of the exponent [20], so it should not be relied of the novel Moby Dick by Herman Melville{. Similar
upon. For example, a least-squares fit of a straight line to distributions are seen for words in other languages.
figure 3 (b) gives a = 2.26 + 0.02, which is clearly (b) Citations of scientific papers: As first observed by
incompatible with the known value of a = 2.5 from which Price [11], the numbers of citations received by
the data were generated. scientific papers appear to have a power-law distribu-
An alternative, simple and reliable method for extracting tion. The data in panel (b) are taken from the Science
the exponent is to employ the formula Citation Index, as collated by Redner [23], and are for
" #
1 papers published in 1981. The plot shows the
X
n
xi cumulative distribution of the number of citations
a¼1þn ln : ð5Þ
xmin received by a paper between publication and June
i¼1
1997.
(c) Web hits: The cumulative distribution of the number
Here the quantities xi, i = 1. . .n are the measured values of of ‘hits’ received by web sites (i.e. servers, not pages)
x and xmin is again the minimum value of x. (As discussed during a single day from a subset of the users of the
in the following section, in practical situations xmin usually AOL Internet service. The site with the most hits, by
corresponds not to the smallest value of x measured but to a long way, was yahoo.com. After Adamic and
the smallest for which the power-law behaviour holds.) The Huberman [12].
derivation of this formula is given in Appendix B. An error (d) Copies of books sold: The cumulative distribution of
estimate for a can be derived by a standard bootstrap or the total number of copies sold in America of the 633
jackknife resampling method [21]; for large data sets of the bestselling books that sold 2 million or more copies
type discussed in this paper, a bootstrap is normally the
more computationally economical of the two. {
The most common words in this case are, in order, ‘the’, ‘of’, ‘and’, ‘a’ and
‘to’, and the same is true for most written English texts. Interestingly,
*
See http://www.hpl.hp.com/research/idl/papers/ranking/ for a useful however, it is not true for spoken English. The most common words in
discussion of these and related points. spoken English are, in order, ‘I’, ‘and’, ‘the’, ‘to’ and ‘that’ [22].
 328 M.E.J. Newman
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Figure 4. Cumulative distributions or ‘rank/frequency plots’ of twelve quantities reputed to follow power laws. The
distributions were computed as described in Appendix A. Data in the shaded regions were excluded from the calculations of
the exponents in table 1. Source references for the data are given in the text. (a) Numbers of occurrences of words in the novel
Moby Dick by Hermann Melville. (b) Numbers of citations to scientific papers published in 1981, from time of publication
until June 1997. (c) Numbers of hits on web sites by 60000 users of the America Online Internet service for the day of 1
December 1997. (d) Numbers of copies of bestselling books sold in the US between 1895 and 1965. (e) Number of calls
received by AT&T telephone customers in the US for a single day. (f) Magnitude of earthquakes in California between
January 1910 and May 1992. Magnitude is proportional to the logarithm of the maximum amplitude of the earthquake, and
hence the distribution obeys a power law even though the horizontal axis is linear. (g) Diameter of craters on the moon.
Vertical axis is measured per square kilometre. (h) Peak gamma-ray intensity of solar flares in counts per second, measured
from Earth orbit between February 1980 and November 1989. (i) Intensity of wars from 1816 to 1980, measured as battle
deaths per 10000 of the population of the participating countries. (j) Aggregate net worth in dollars of the richest individuals
in the US in October 2003. (k) Frequency of occurrence of family names in the US in the year 1990. (l) Populations of US
cities in the year 2000.
 Power laws, Pareto distributions and Zipfs law 329
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between 1895 and 1965. The data were compiled CsI scintillation detector to measure gamma-rays
painstakingly over a period of several decades by from solar flares and the horizontal axis in the figure
Alice Hackett, an editor at Publisher’s Weekly [24]. is calibrated in terms of scintillation counts per
The best selling book during the period covered was second from this detector. The data are from the
Benjamin Spock’s The Common Sense Book of Baby NASA Goddard Space Flight Center, umbra.nas-
and Child Care. (The Bible, which certainly sold more com.nasa.gov/smm/hxrbs.html. See also Lu and
copies, is not really a single book, but exists in many Hamilton [5].
different translations, versions and publications, and (i) Intensity of wars: The cumulative distribution of the
was excluded by Hackett from her statistics.) intensity of 119 wars from 1816 to 1980. Intensity is
Substantially better data on book sales than Hack- defined by taking the number of battle deaths among
ett’s are now available from operations such as all participant countries in a war, dividing by the total
Nielsen BookScan, but unfortunately at a price this combined populations of the countries and multi-
author cannot afford. I should be very interested to plying by 10000. For instance, the intensities of the
see a plot of sales figures from such a modern source. First and Second World Wars were 141.5 and 106.3
(e) Telephone calls: The cumulative distribution of the battle deaths per 10000 respectively. The worst war of
number of calls received on a single day by 51 million the period covered was the small but horrifically
users of AT&T long distance telephone service in the destructive Paraguay-Bolivia war of 1932 – 1935 with
United States. After Aiello et al. [25]. The largest an intensity of 382.4. The data are from Small and
number of calls received by a customer on that day Singer [28]. See also Roberts and Turcotte [7].
was 375746, or about 260 calls a minute (obviously to (j) Wealth of richest Americans: The cumulative dis-
a telephone number that has many people manning tribution of the total wealth of the richest people in
the phones). Similar distributions are seen for the the United States. Wealth is defined as aggregate net
number of calls placed by users and also for the worth, i.e. total value in dollars at current market
numbers of e-mail messages that people send and prices of all an individual’s holdings, minus their
receive [26, 27]. debts. For instance, when the data were compiled in
(f) Magnitude of earthquakes: The cumulative distribu- 2003, America’s richest person, William H. Gates III,
tion of the Richter magnitude of earthquakes had an aggregate net worth of $46 billion, much of it
occurring in California between January 1910 and in the form of stocks of the company he founded,
May 1992, as recorded in the Berkeley Earthquake Microsoft Corporation. Note that net worth does not
Catalog. The Richter magnitude is defined as the actually correspond to the amount of money in-
logarithm, base 10, of the maximum amplitude of dividuals could spend if they wanted to: if Bill Gates
motion detected in the earthquake, and hence the were to sell all his Microsoft stock, for instance, or
horizontal scale in the plot, which is drawn as linear, otherwise divest himself of any significant portion of
is in effect a logarithmic scale of amplitude. The it, it would certainly depress the stock price. The data
power-law relationship in the earthquake distribution are from Forbes magazine, 6 October 2003.
is thus a relationship between amplitude and fre- (k) Frequencies of family names: Cumulative distribution
quency of occurrence. The data are from the National of the frequency of occurrence in the US of the 89000
Geophysical Data Center, www.ngdc.noaa.gov. most common family names, as recorded by the US
(g) Diameter of moon craters: The cumulative distribu- Census Bureau in 1990. Similar distributions are
tion of the diameter of moon craters. Rather than observed for names in some other cultures as well (for
measuring the (integer) number of craters of a given example in Japan [29]) but not in all cases. Korean
size on the whole surface of the moon, the vertical family names for instance appear to have an
axis is normalized to measure the number of craters exponential distribution [30].
per square kilometre, which is why the axis goes (l) Populations of cities: Cumulative distribution of the
below 1, unlike the rest of the plots, since it is entirely size of the human populations of US cities as
possible for there to be less than one crater of a given recorded by the US Census Bureau in 2000.
size per square kilometre. After Neukum and Ivanov
[4]. Few real-world distributions follow a power law over their
(h) Intensity of solar flares: The cumulative distribution entire range, and in particular not for smaller values of the
of the peak gamma-ray intensity of solar flares. The variable being measured. As pointed out in the previous
observations were made between 1980 and 1989 by section, for any positive value of the exponent a the function
the instrument known as the Hard X-Ray Burst p(x) = Cx – a diverges as x?0. In reality therefore, the
Spectrometer aboard the Solar Maximum Mission distribution must deviate from the power-law form below
satellite launched in 1980. The spectrometer used a some minimum value xmin. In our computer-generated
 330 M.E.J. Newman
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example of the last section we simply cut off the distribution Table 1. Parameters for the distributions shown in figure 4.
altogether below xmin so that p(x) = 0 in this region, but The labels on the left refer to the panels in the figure. Exponent
most real-world examples are not that abrupt. Figure 4 values were calculated using the maximum likelihood method
of equation (5) and Appendix B, except for the moon craters
shows distributions with a variety of behaviours for small (g), for which only cumulative data were available. For this
values of the variable measured; the straight-line power-law case the exponent quoted is from a simple least-squares fit and
form asserts itself only for the higher values. Thus one often should be treated with caution. Numbers in parentheses give
hears it said that the distribution of such-and-such a the standard error on the trailing figures.
quantity ‘has a power-law tail’.
Quantity Minimum Exponent
Extracting a value for the exponent a from distributions
like these can be a little tricky, since it requires us to make xmin a
a judgement, sometimes imprecise, about the value xmin (a) frequency of use of words 1 2.20(1)
(b) number of citations to papers 100 3.04(2)
above which the distribution follows the power law. Once (c) number of hits on web sites 1 2.40(1)
this judgement is made, however, a can be calculated (d) copies of books sold in the US 2000000 3.51(16)
simply from equation (5)*. (Care must be taken to use the (e) telephone calls received 10 2.22(1)
correct value of n in the formula; n is the number of (f) magnitude of earthquakes 3.8 3.04(4)
samples that actually go into the calculation, excluding (g) diameter of moon craters 0.01 3.14(5)
(h) intensity of solar flares 200 1.83(2)
those with values below xmin, not the overall total number (i) intensity of wars 3 1.80(9)
of samples.) (j) net worth of Americans $600m 2.09(4)
Table 1 lists the estimated exponents for each of the (k) frequency of family names 10000 1.94(1)
distributions of figure 4, along with standard errors (l) population of US cities 40000 2.30(5)
calculated by bootstrapping 100 times, and also the values
of xmin used in the calculations. Note that the quoted
errors correspond only to the statistical sampling error in
the estimation of a; I have included no estimate of any
2.2 Distributions that do not follow a power law
errors introduced by the fact that a single power-law
function may not be a good model for the data in some Power-law distributions are, as we have seen, impressively
cases or for variation of the estimates with the value ubiquitous, but they are not the only form of broad
chosen for xmin. distribution. Lest I give the impression that everything
In the author’s opinion, the identification of some of the interesting follows a power law—an opinion that has been
distributions in figure 4 as following power laws should be espoused elsewhere—let me emphasize that there are quite
considered unconfirmed. While the power law seems to be a number of quantities with highly right-skewed distribu-
an excellent model for most of the data sets depicted, a tions that nonetheless do not follow power laws. A few of
tenable case could be made that the distributions of web them, shown in figure 5, are the following.
hits and family names might have two different power-law
regimes with slightly different exponents. And the data for (a) The lengths of relationships between couples, which
the numbers of copies of books sold cover rather a small although they span more than four orders of
range—little more than one decade horizontally{. None- magnitude appear to be exponentially distributed.
theless, one can, without stretching the interpretation of the (b) The abundance of North American bird species,
data unreasonably, claim that power-law distributions have which spans over five orders of magnitude but is
been observed in language, demography, commerce, probably distributed according to a log-normal. A
information and computer sciences, geology, physics and log-normally distributed quantity is one whose
astronomy, and this on its own is an extraordinary logarithm is normally distributed; see section 4.7
statement. and [34] for further discussions.
(c) The number of entries in people’s email address
books, which spans about three orders of magnitude
but seems to follow a stretched exponential. A
*
Sometimes the tail is also cut off because there is, for one reason or
stretched exponential is a curve of the form exp( –
another, a limit on the largest value that may occur. An example is the axb) for some constants a, b.
finite-size effects found in critical phenomena—see section 4.5. In this case, (d) The distribution of the sizes of forest fires, which
equation (5) must be modified [20]. spans six orders of magnitude and could follow a
{
Significantly more tenuous claims to power-law behaviour for other
power law but with an exponential cut-off.
quantities have appeared elsewhere in the literature, for instance in the
discussion of the distribution of the sizes of electrical blackouts [31,32].
These however I consider insufficiently substantiated for inclusion in the This being an article about power laws, I will not discuss
present work. further the possible explanations for these distributions, but
 Power laws, Pareto distributions and Zipfs law 331
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Figure 5. Cumulative distributions of some quantities whose values span several orders of magnitude but that nonetheless do
not obey power laws. (a) The length in days of the most recent sexual relationship of 1013 men and women interviewed in the
study of Foxman et al. (unpublished). (b) The number of sightings of 591 species of birds in the North American Breeding
Bird Survey 2003. (c) The number of addresses in the e-mail address books of 16881 users of a large university computer
system [33]. (d) The size in acres of all wildfires occurring on US federal lands between 1986 and 1996 (National Fire
Occurrence Database, USDA Forest Service and Department of the Interior). Note that the horizontal axes in frames (a) and
(c) are linear but in (b) and (d) they are logarithmic.

the scientist confronted with a new set of data having a

3.1 Normalization
broad dynamic range and a highly skewed distribution
should certainly bear in mind that a power-law model is The constant C in equation (6) is given by the normal-
only one of several possibilities for fitting it. ization requirement that
Z 1 Z 1
C

aþ1 1
1¼ pðxÞdx ¼ C x
a dx ¼ x : ð7Þ
xmin xmin 1
a xmin
3. The mathematics of power laws
A continuous real variable with a power-law distribution We see immediately that this makes sense only if a 4 1,
has a probability p(x) dx of taking a value in the interval since otherwise the right-hand side of the equation would
from x to x + dx, where diverge: power laws with exponents less than unity cannot
be normalized and do not normally occur in nature. If
pðxÞ ¼ Cx
a ; ð6Þ a 4 1 then equation (7) gives

with a 4 0. As we saw in section 2.1, there must be min ;

C ¼ ða
1Þxa
1 ð8Þ
some lowest value xmin at which the power law is
obeyed, and we consider only the statistics of x above and the correct normalized expression for the power law
this value. itself is
 332 M.E.J. Newman
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   2
a
1 x
a C

aþ3 1
pðxÞ ¼ : ð9Þ x ¼ x : ð12Þ
xmin xmin 3
a xmin

Some distributions follow a power law for part of their This diverges if a 4 3. Thus power-law distributions in this
range but are cut off at high values of x. That is, above range, which includes almost all of those in table 1, have no
some value they deviate from the power law and fall off finite mean square in the limit of a large data set, and thus
quickly towards zero. If this happens, then the distribution also no finite variance or standard deviation. We discuss
may be normalizable no matter what the value of the the meaning of this statement further below. If a 4 3, then
exponent a. Even so, exponents less than unity are rarely, if the second moment is finite and well defined, taking the
ever, seen. value
 2 a
1 2
x ¼ x : ð13Þ
a
3 min
3.2 Moments
The mean value of x in our power law is given by These results can easily be extended to show that in
Z 1 Z 1 general all moments hxmi exist for m 5 a 7 1 and all higher
hxi ¼ xpðxÞdx ¼ C x
aþ1 dx moments diverge. The ones that do exist are given by
xmin xmin
ð10Þ
C

aþ2 1 a
1
¼ x : hxm i ¼ xm : ð14Þ
2
a xmin a
1
m min

Note that this expression becomes infinite if a 4 2. Power

laws with such low values of a have no finite mean. The
3.3 Largest value
distributions of sizes of solar flares and wars in table 1 are
examples of such power laws. Suppose we draw n measurements from a power-law
What does it mean to say that a distribution has no distribution. What value is the largest of those measure-
finite mean? Surely we can take the data for real solar ments likely to take? Or, more precisely, what is the
flares and calculate their average? Indeed we can, but this probability p(x) dx that the largest value falls in the interval
is only because the data set is of finite size. Equation (10) between x and x + dx?
can be made to give a finite value of hxi if we cut the The definitive property of the largest value in a sample is
integral off at some upper limit, i.e. if there is a maximum that there are no others larger than it. The probability that
as well as a minimum value of x. In any real data set of a particular sample will be larger than x is given by the
finite size there is indeed such a maximum, which is just quantity P(x) defined in equation (3):
the largest value of x observed. But if we make more Z  
aþ1
1
measurements and generate a larger dataset, we have a C
aþ1 x
PðxÞ ¼ pðx0 Þdx0 ¼ x ¼ ; ð15Þ
non-negligible chance of getting a larger maximum value x a
1 xmin
of x, and this will make the value of hxi larger in turn.
The divergence of equation (10) is telling us that as we go so long as a 4 1. The probability that a sample is not
to larger and larger data sets, our estimate of the mean greater than x is 1 – P(x). Thus the probability that a
hxi will increase without bound. We discuss this more particular sample we draw, sample i, will lie between x and
below. x + dx and that all the others will be no greater than it is
For a 4 2 however, the mean does not diverge: the value p(x) dx 6 [1 – P(x)]n – 1. Then there are n ways to choose i,
of hxi will settle down to a normal finite value as the data giving a total probability
set becomes large, and that value is given by equation (10)
to be pðxÞ ¼ npðxÞ½1
PðxÞn
1 : ð16Þ

a
1
hxi ¼ xmin : ð11Þ Now we can calculate the mean value hxmaxi of the
a
2 largest sample thus:
Z 1 Z 1
We can also calculate higher moments of the distribution hxmax i ¼ xpðxÞdx ¼ n xpðxÞ½1
PðxÞn
1 dx:
p(x). For instance, the second moment, the mean square, is xmin xmin

given by ð17Þ
 Power laws, Pareto distributions and Zipfs law 333
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Using equations (9) and (15), this is As xmax becomes large this expression is dominated by the
upper limit, and using the result, equation (22), for xmax, we
hxmax i ¼ nða
1Þ get
Z 1  "   #n
1
x
aþ1 x
aþ1  2
 1
dx x  nð3
aÞ=ða
1Þ : ð24Þ
xmin xmin xmin
Z 1 ð18Þ
yn
1 So, for instance, if a ¼ 52, then the mean-square sample
¼ nxmin 1=ða
1Þ
dy
0 ð1
yÞ value, and hence also the sample variance, goes as n1/3 as
¼ nxmin Bðn; ða
2Þ=ða
1ÞÞ; the size of the data set gets larger.

3.4 Top-heavy distributions and the 80/20 rule

where I have made the substitution y = 1 – (x/xmin) – a+1
and B(a, b) is Legendre’s beta-function*, which is defined Another interesting question is where the majority of the
by distribution of x lies. For any power law with exponent
a 4 1, the median is well defined. That is, there is a point
ðaÞðbÞ
Bða; bÞ ¼ ; ð19Þ x1/2 that divides the distribution in half so that half the
ða þ bÞ measured values of x lie above x1/2 and half lie below. That
point is given by
with G(a) the standard G-function: Z 1 Z
1 1
Z 1 pðxÞdx ¼ pðxÞdx; ð25Þ
2 xmin
ðaÞ ¼ ta
1 expð
tÞdt: ð20Þ x1=2
0

or
The beta-function has the interesting property that for
large values of either of its arguments it itself follows a x1=2 ¼ 21=ða
1Þ xmin : ð26Þ
power law{. For instance, for large a and fixed b, B(a,
b)*a – b. In most cases of interest, the number n of samples So, for example, if we are considering the distribution of
from our power-law distribution will be large (meaning wealth, there will be some well-defined median wealth that
much greater than 1), so divides the richer half of the population from the poorer.
But we can also ask how much of the wealth itself lies in
Bðn; ða
2Þ=ða
1ÞÞ  n
ða
2Þ=ða
1Þ ð21Þ those two halves. Obviously more than half of the total
amount of money belongs to the richer half of the
and population. The fraction of the money in the richer half
is given by
hxmax i  n1=ða
1Þ : ð22Þ R1  
x xpðxÞdx x1=2
aþ2
R 11=2 ¼ ¼ 2
ða
2Þ=ða
1Þ ; ð27Þ
Thus hxmaxi always increases as n becomes larger so long as xmin xpðxÞdx
xmin
a 4 1.
This allows us to complete the calculation of the provided a 4 2 so that the integrals converge. Thus, for
moments in section 3.2. Consider for instance the second instance, if a = 2.1 for the wealth distribution, as indicated
moment, which is often of interest in power laws. For the in table 1, then a fraction 2 – 0.091^94% of the wealth is in
crucial case 2 5 a 4 3, which covers most of the power-law the hands of the richer 50% of the population, making the
distributions observed in real life, we saw in equation (12) distribution quite top-heavy.
that the second moment of the distribution diverges as the More generally, the fraction of the population whose
size of the data set becomes infinite. But in reality all data personal wealth exceeds x is given by the quantity P(x),
sets are finite and so have a finite maximum sample xmax. equation (15), and the fraction of the total wealth in the
This means that (12) becomes hands of those people is
 2 R1 0 0  
C

aþ3 xmax 0
x
aþ2
x ¼ : ð23Þ x x pðx Þdx
3
a
x xmin WðxÞ ¼ R 1 0 0 0
¼ ; ð28Þ
xmin x pðx Þdx
xmin

*
Also called the Eulerian integral of the first kind.
assuming again that a 4 2. Eliminating x/xmin between (15)
{
This can be demonstrated by approximating the G-functions of equation and (28), we find that the fraction W of the wealth in the
(19) using Sterling’s formula. hands of the richest P of the population is
 334 M.E.J. Newman
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equation (27), tends to unity. In fact, the fraction of money

W ¼ Pða
2Þ=ða
1Þ ; ð29Þ in the top anything of the population, even the top 1%,
tends to unity, as equation (28) shows. In other words, for
of which equation (27) is a special case. This again has a distributions with a 5 2, essentially all of the wealth (or
power-law form, but with a positive exponent now. In other commodity) lies in the tail of the distribution. The
figure 6 I show the form of the curve of W against P for frequency of family names, which has an exponent a = 1.9,
various values of a. For all values of a the curve is concave is an example of this type of behaviour. For the data of
downwards, and for values only a little above 2 the curve figure 4 (k), about 75% of the population have names in the
has a very fast initial increase, meaning that a large fraction top 15000. Estimates of the total number of unique family
of the wealth is concentrated in the hands of a small names in the US put the figure at around 1.5 million. So in
fraction of the population. this case 75% of the population have names in the most
Using the exponents from table 1, we can for example common 1%—a very top-heavy distribution indeed. The
calculate that about 80% of the wealth should be in the line a = 2 thus separates the regime in which you will with
hands of the richest 20% of the population (the so-called some frequency meet people with uncommon names from
‘80/20 rule’, which is borne out by more detailed observa- the regime in which you will hardly ever meet such people.
tions of the wealth distribution), the top 20% of web sites
get about two-thirds of all web hits, and the largest 10% of
3.5 Scale-free distributions
US cities house about 60% of the country’s total
population. A power-law distribution is also sometimes called a scale-
If a 4 2 then the situation becomes even more extreme. free distribution. Why? Because a power law is the only
In that case, the integrals in equation (28) diverge at their distribution that is the same whatever scale we look at it on.
upper limits, meaning that in fact they depend on the value By this we mean the following.
xmax of the largest sample, as described in section 3.3. But Suppose we have some probability distribution p(x) for a
for a 4 1, equation (22) tells us that the expected value of quantity x, and suppose we discover or somehow deduce
xmax goes to ? as n becomes large, and in that limit the that it satisfies the property that
fraction of money in the top half of the population,
pðbxÞ ¼ gðbÞpðxÞ; ð30Þ

for any b. That is, if we increase the scale or units by which

we measure x by a factor of b, the shape of the distribution
p(x) is unchanged, except for an overall multiplicative
constant. Thus for instance, we might find that computer
files of size 2 kB are 14 as common as files of size 1 kB.
Switching to measuring size in megabytes we also find that
files of size 2 MB are 14 as common as files of size 1 MB.
Thus the shape of the file – size distribution curve (at least
for these particular values) does not depend on the scale on
which we measure file size.
This scale-free property is certainly not true of most
distributions. It is not true for instance of the exponential
distribution. In fact, as we now show, it is only true of one
type of distribution, the power law.
Starting from equation (30), let us first set x = 1, giving
p(b) = g(b)p(1). Thus g(b) = p(b)/p(1) and (30) can be
written as

pðbÞpðxÞ
pðbxÞ ¼ : ð31Þ
pð1Þ
Figure 6. The fraction W of the total wealth in a country
held by the fraction P of the richest people, if wealth is Since this equation is supposed to be true for any b, we can
distributed following a power law with exponent a. If differentiate both sides with respect to b to get
a = 2.1, for instance, as it appears to in the United States
p0 ðbÞpðxÞ
(table 1), then the richest 20% of the population hold about xp0 ðbxÞ ¼ ; ð32Þ
86% of the wealth (dashed lines). pð1Þ
 Power laws, Pareto distributions and Zipfs law 335
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where p’ indicates the derivative of p with respect to its for some constant exponent a. Clearly this distribution
argument. Now we set b = 1 and get cannot hold all the way down to k = 0, since it diverges
there, but it could in theory hold down to k = 1. If we
dp p0 ð1Þ
x ¼ pðxÞ: ð33Þ discard any data for k = 0, the constant C would then be
dx pð1Þ given by the normalization condition
X
1 X
1
This is a simple first-order differential equation which has 1¼ pk ¼ C k
a ¼ CzðaÞ; ð37Þ
the solution k¼1 k¼1

pð1Þ
ln pðxÞ ¼ ln x þ constant: ð34Þ where z(a) is the Riemann z-function. Rearranging,
p0 ð1Þ C=1/z(a) and
k
a
pk ¼ : ð38Þ
Setting x = 1 we find that the constant is simply ln p(1), and zðaÞ
then taking exponentials of both sides
If, as is usually the case, the power-law behaviour is seen
pðxÞ ¼ pð1Þx
a ; ð35Þ
only in the tail of the distribution, for values k 5 kmin, then
the equivalent expression is
where a = – p(1)/p’(1). Thus, as advertised, the power-law
k
a
distribution is the only function satisfying the scale-free pk ¼ ; ð39Þ
criterion (30). zða; kmin Þ
This fact is more than just a curiosity. As we will see
P
in section 4.5, there are some systems that become scale- where zða;kmin Þ ¼ 1 k¼kmin k

a
is the generalized or incom-
free for certain special values of their governing para- plete z-function.
meters. The point defined by such a special value is Most of the results of the previous sections can be
called a ‘continuous phase transition’ and the argument generalized to the case of discrete variables, although the
given above implies that at such a point the observable mathematics is usually harder and often involves special
quantities in the system should adopt a power-law functions in place of the more tractable integrals of the
distribution. This indeed is seen experimentally and the continuous case.
distributions so generated provided the original motiva- It has occasionally been proposed that equation (36) is
tion for the study of power laws in physics (although not the best generalization of the power law to the discrete
most experimentally observed power laws are probably case. An alternative and in many cases more convenient
not the result of phase transitions—a variety of other form is
mechanisms produce power-law behaviour as well, as we
ðkÞðaÞ
will shortly see). pk ¼ C ¼ CBðk; aÞ; ð40Þ
ðk þ aÞ

3.6 Power laws for discrete variables

where B(a, b) is, as before, the Legendre beta-function,
So far I have focused on power-law distributions for equation (19). As mentioned in section 3.3, the beta-
continuous real variables, but many of the quantities we function behaves as a power law pk*k – a for large k and so
deal with in practical situations are in fact discrete—usually the distribution has the desired asymptotic form. Simon
integers. For instance, populations of cities, numbers of [35] proposed that equation (40) be called the Yule
citations to papers or numbers of copies of books sold are distribution, after Udny Yule who derived it as the limiting
all integer quantities. In most cases, the distinction is not distribution in a certain stochastic process [36], and this
very important. The power law is obeyed only in the tail of name is often used today. Yule’s result is described in
the distribution where the values measured are so large section 4.4.
that, to all intents and purposes, they can be considered The Yule distribution is nice because sums involving it
continuous. Technically however, power-law distributions can frequently be performed in closed form, where sums
should be defined slightly differently for integer quantities. involving equation (36) can only be written in terms of
If k is an integer variable, then one way to proceed is to special functions. For instance, the normalizing constant C
declare that it follows a power law if the probability pk of for the Yule distribution is given by
measuring the value k obeys
X
1
1

a
1¼C Bðk; aÞ ¼ ; ð41Þ
pk ¼ Ck ; ð36Þ k¼1
a
1
 336 M.E.J. Newman
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and hence C = a – 1 and on a typewriter*, pressing the space bar with probability qs
per stroke and each letter with equal probability ql per
pk ¼ ða
1ÞBðk; aÞ: ð42Þ
stroke. If there are m letters in the alphabet then ql = (1 –
qs)/m. (In this simplest version of the argument we also type
The first and second moments (i.e. the mean and mean no punctuation, digits or other non-letter symbols.) Then
square of the distribution) are the frequency x with which a particular word with y letters
(followed by a space) occurs is
a
1  2 ða
1Þ2  
hki ¼ ; k ¼ ; ð43Þ 1
qs y
a
2 ða
2Þða
3Þ x¼ qs  exp ðbyÞ; ð47Þ
m
and there are similarly simple expressions corresponding to
many of our earlier results for the continuous case. where b = ln (1 – qs) – ln m. The number (or fraction) of
distinct possible words with length between y and y + dy
goes up exponentially as p(y)*my = exp (ay) with a = ln
4. Mechanisms for generating power-law distributions
m. Thus, following our argument above, the distribution of
In this section we look at possible candidate mechanisms by frequencies of words has the form p(x)*x – a with
which power-law distributions might arise in natural and
a 2 ln m
ln ð1
qs Þ
man-made systems. Some of the possibilities that have been a¼1
¼ : ð48Þ
suggested are quite complex—notably the physics of critical b ln m
ln ð1
qs Þ
phenomena and the tools of the renormalization group that
are used to analyse it. But let us start with some simple For the typical case where m is reasonably large and qs
algebraic methods of generating power-law functions and quite small this gives a^2 in approximate agreement with
progress to the more involved mechanisms later. table 1.
This is a reasonable theory as far as it goes, but real text
is not made up of random letters. Most combinations of
4.1 Combinations of exponentials
letters do not occur in natural languages; most are not even
A much more common distribution than the power law is pronounceable. We might imagine that some constant
the exponential, which arises in many circumstances, such fraction of possible letter sequences of a given length would
as survival times for decaying atomic nuclei or the correspond to real words and the argument above would
Boltzmann distribution of energies in statistical mechanics. then work just fine when applied to that fraction, but upon
Suppose some quantity y has an exponential distribution: reflection this suggestion is obviously bogus. It is clear for
instance that very long words simply do not exist in most
pðyÞ  exp ðayÞ: ð44Þ languages, although there are exponentially many possible
combinations of letters available to make them up. This
The constant a might be either negative or positive. If it is observation is backed up by empirical data. In figure 7 (a)
positive then there must also be a cut-off on the we show a histogram of the lengths of words occurring in
distribution—a limit on the maximum value of y—so that the text of Moby Dick, and one would need a particularly
the distribution is normalizable. vivid imagination to convince oneself that this histogram
Now suppose that the real quantity we are interested in is follows anything like the exponential assumed by Miller’s
not y but some other quantity x, which is exponentially argument. (In fact, the curve appears roughly to follow a
related to y thus: log-normal [34].)
There may still be some merit in Miller’s argument
x  exp ðbyÞ; ð45Þ
however. The problem may be that we are measuring word
‘length’ in the wrong units. Letters are not really the basic
with b another constant, also either positive or negative. units of language. Some basic units are letters, but some are
Then the probability distribution of x is groups of letters. The letters ‘th’ for example often occur
together in English and make a single sound, so perhaps
dy exp ðayÞ x
1þa=b they should be considered to be a separate symbol in their
pðxÞ ¼ pðyÞ  ¼ ; ð46Þ
dx b exp ðbyÞ b own right and contribute only one unit to the word length?
Following this idea to its logical conclusion we can
which is a power law with exponent a = 1 – a/b. imagine replacing each fundamental unit of the language—
A version of this mechanism was used by Miller [37] to
explain the power-law distribution of the frequencies of *
This argument is sometimes called the ‘monkeys with typewriters’
words as follows (see also [38]). Suppose we type randomly argument, the monkey being the traditional exemplar of a random typist.
 Power laws, Pareto distributions and Zipfs law 337
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whatever that is—by its own symbol and then measuring exponentially more distinct words in the language of high
lengths in terms of numbers of symbols. The pursuit of information content than of low. That this is the case is
ideas along these lines led Claude Shannon in the 1940s to experimentally verified by figure 7 (b), but the reason must
develop the field of information theory, which gives a be considered still a matter of debate. Some possibilities are
precise prescription for calculating the number of symbols discussed by, for instance, Mandelbrot [42] and more
necessary to transmit words or any other data [39, 40]. The recently by Mitzenmacher [19].
units of information are bits and the true ‘length’ of a word Another example of the ‘combination of exponentials’
can be considered to be the number of bits of information it mechanism has been discussed by Reed and Hughes [43].
carries. Shannon showed that if we regard words as the They consider a process in which a set of items, piles or
basic divisions of a message, the information y carried by groups each grows exponentially in time, having size
any particular word is x^(bt) with b 4 0. For instance, populations of organisms
reproducing freely without resource constraints grow
y ¼
k ln x; ð49Þ exponentially. Items also have some fixed probability of
dying per unit time (populations might have a stochasti-
where x is the frequency of the word as before and k is a cally constant probability of extinction), so that the times t
constant. (The reader interested in finding out more about at which they die are exponentially distributed p(t)^(at)
where this simple relation comes from is recommended to with a 5 0.
look at the excellent introduction to information theory by These functions again follow the form of equations (44)
Cover and Thomas [41].) and (45) and result in a power-law distribution of the sizes
But this has precisely the form that we want. Inverting it x of the items or groups at the time they die. Reed and
we have x = exp( – y/k) and if the probability distribution Hughes suggest that variations on this argument may
of the ‘lengths’ measured in terms of bits is also exponential explain the sizes of biological taxa, incomes and cities,
as in equation (44) we will get our power-law distribution. among other things.
Figure 7 (b) shows the latter distribution, and indeed it
follows a nice exponential—much better than figure 7 (a).
This is still not an entirely satisfactory explanation.
4.2 Inverses of quantities
Having made the shift from pure word length to informa-
tion content, our simple count of the number of words of Suppose some quantity y has a distribution p(y) that passes
length y—that it goes exponentially as my—is no longer through zero, so that y has both positive and negative
valid, and now we need some reason why there should be values. And suppose further that the quantity we are really
interested in is the reciprocal x = 1/y, which will have
distribution

dy pðyÞ
pðxÞ ¼ pðyÞ ¼
2 : ð50Þ
dx x

The large values of x, those in the tail of the distribution,

correspond to the small values of y close to zero and thus
the large-x tail is given by

pðxÞ  x
2 ; ð51Þ

where the constant of proportionality is p(y = 0).

More generally, any quantity x = y – g for some g will
have a power-law tail to its distribution p(x)*x – a, with
a = 1 + 1/g. The first clear description of this mechanism of
which I am aware is that of Jan et al. [44]; a good discussion
Figure 7. (a) Histogram of the lengths in letters of all has also been given by Sornette [45].
distinct words in the text of the novel Moby Dick. (b) One might argue that this mechanism merely generates a
Histogram of the information content a la Shannon of power law by assuming another one: the power-law
words in Moby Dick. The former does not, by any stretch relationship between x and y generates a power-law
of the imagination, follow an exponential, but the latter distribution for x. This is true, but the point is that the
could easily be said to do so. (Note that the vertical axes are mechanism takes some physical power-law relationship
logarithmic.) between x and y—not a stochastic probability distribu-
 338 M.E.J. Newman
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tion—and from that generates a power-law probability

distribution. This is a non-trivial result.
One circumstance in which this mechanism arises is in
measurements of the fractional change in a quantity. For
instance, Jan et al. [44] consider one of the most famous
systems in theoretical physics, the Ising model of a magnet.
In its paramagnetic phase, the Ising model has a
magnetization that fluctuates around zero. Suppose we
measure the magnetization m at uniform intervals and
calculate the fractional change d = (Dm)/m between each
successive pair of measurements. The change Dm is roughly
normally distributed and has a typical size set by the width
of that normal distribution. The 1/m on the other hand
produces a power-law tail when small values of m coincide
with large values of Dm, so that the tail of the distribution
of d follows p(d)*d – 2 as above.
In figure 8 I show a cumulative histogram of measure-
ments of d for simulations of the Ising model on a square
lattice and the power-law distribution is clearly visible.
Using equation (5), the value of the exponent is
a = 1.98 + 0.04, in good agreement with the expected Figure 8. Cumulative histogram of the magnetization
value of 2. fluctuations of a 128 6 128 nearest-neighbour Ising model
on a square lattice. The model was simulated at a
temperature of 2.5 times the spin – spin coupling for
4.3 Random walks 100000 time steps using the cluster algorithm of Swendsen
Many properties of random walks are distributed accord- and Wang [46] and the magnetization per spin measured at
ing to power laws, and this could explain some power-law intervals of ten steps. The fluctuations were calculated as
distributions observed in nature. In particular, a randomly the ratio di = 2(mi + 1 – mi)/(mi + 1 + mi).
fluctuating process that undergoes ‘gambler’s ruin’*, i.e.
that ends when it hits zero, has a power-law distribution of
1;
P if n ¼ 0;
possible lifetimes. u2n ¼ n ð52Þ
Consider a random walk in one dimension, in which a m¼1 f2m u2n
2m ; if n  1;
walker takes a single step randomly one way or the other
along a line in each unit of time. Suppose the walker starts where m is also an integer and we define f0 = 0. This
at position 0 on the line and let us ask what the probability equation can conveniently be solved for f2n using a
is that the walker returns to position 0 for the first time at generating function approach. We define
time t (i.e. after exactly t steps). This is the so-called first
X
1 X
1
return time of the walk and represents the lifetime of a UðzÞ ¼ u2n zn ; FðzÞ ¼ f2n zn : ð53Þ
gambler’s ruin process. A trick for answering this question n¼0 n¼1
is depicted in figure 9. We consider first the unconstrained
problem in which the walk is allowed to return to zero as Then, multiplying equation (52) throughout by zn and
many times as it likes, before returning there again at time summing, we find
t. Let us denote the probability of this event as ut. Let us 1 X
X n
also denote by ft the probability that the first return time is UðzÞ ¼ 1 þ f2m u2n
2m zn
t. We note that both of these probabilities are non-zero n¼1 m¼1
only for even values of their arguments since there is no X1 X
1
ð54Þ
way to get back to zero in any odd number of steps. ¼1þ f2m zm u2n
2m zn
m
m¼1 n¼m
As figure 9 illustrates, the probability ut = u2n, with n
integer, can be written ¼ 1 þ FðzÞUðzÞ:

So
*
Gambler’s ruin is so called because a gambler’s night of betting ends when 1
his or her supply of money hits zero (assuming the gambling establishment FðzÞ ¼ 1
: ð55Þ
declines to offer him or her a line of credit).
UðzÞ
 Power laws, Pareto distributions and Zipfs law 339
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Now consider the form of f2n for large n. Writing out the
2
n ¼ ð2nÞ!=ðn!Þ , we take logs thus:
binomial coefficient as 2n

ln f2n ¼ ln ð2nÞ!
2 ln n!
2n ln 2
ln ð2n
1Þ; ð61Þ

and use Sterling’s formula ln n! ’ n ln n
n þ 12 ln n to

get ln f2n ’ 12 ln 2
12 ln n
ln ð2n
1Þ, or

Figure 9. The position of a one-dimensional random walker !1=2

(vertical axis) as a function of time (horizontal axis). The 2
f2n ’ : ð62Þ
probability u2n that the walk returns to zero at time t = 2n nð2n
1Þ2
is equal to the probability f2m that it returns to zero for the
first time at some earlier time t = 2m, multiplied by the In the limit n??, this implies that f2n*n – 3/2, or
probability u2n – 2m that it returns again a time 2n – 2m later, equivalently
summed over all possible values of m. We can use this
observation to write a consistency relation, equation (52), ft  t
3=2 : ð63Þ
which can be solved for ft, equation (60).
So the distribution of return times follows a power law with
exponent a ¼
32. Note that the distribution has a divergent
The function U(z) however is quite easy to calculate. The mean (because a 4 2). As discussed in section 3.3, in
probability u2n that we are at position zero after 2n steps is practice this implies that the mean is determined by the size
  of the sample. If we measure the first return times of a large
2n
u2n ¼ 2
2n ; ð56Þ number of random walks, the mean will of course be finite.
n But the more walks we measure, the larger that mean will
become, without bound.
so{ As an example application, the random walk can be
1   considered a simple model for the lifetime of biological
X 2n zn 1
UðzÞ ¼ ¼ pffiffiffiffiffiffiffiffiffiffiffi ; ð57Þ taxa. A taxon is a branch of the evolutionary tree, a group
n¼0
n 4n 1
z of species all descended by repeated speciation from a
common ancestor. The ranks of the Linnean hierarchy—
and hence genera, families, orders and so forth—are examples of
pffiffiffiffiffiffiffiffiffiffiffi taxa*. If a taxon gains and loses species at random over
F ð zÞ ¼ 1
1
z: ð58Þ time, then the number of species performs a random walk,
the taxon becoming extinct when the number of species
Expanding this function using the binomial theorem thus: reaches zero for the first (and only) time. (This is one
example of ‘gambler’s ruin’.) Thus the time for which taxa
1 1
1 1
13 live should have the same distribution as the first return
FðzÞ ¼ z þ 2 2 z2 þ 2 2 2 z3 þ   
2 2! 3! times of random walks.
 
2n ð59Þ In fact, it has been argued that the distribution of the
X1
n lifetimes of genera in the fossil record does indeed follow a
¼ zn
n¼1
ð 2n
1 Þ2 2n
power law [47]. The best fits to the available fossil data put
the value of the exponent at a = 1.7 + 0.3, which is in
and comparing this expression with equation (53), we agreement with the simple random walk model [48]{.
immediately see that
2n *
Modern phylogenetic analysis, the quantitative comparison of species’
f2n ¼ n
; ð60Þ genetic material, can provide a picture of the evolutionary tree and hence
ð2n
1Þ22n
allow the accurate ‘cladistic’ assignment of species to taxa. For prehistoric
species, however, whose genetic material is not usually available,
and we have our solution for the distribution of first return determination of evolutionary ancestry is difficult, so classification into
times. taxa is based instead on morphology, i.e. on the shapes of organisms. It is
widely accepted that such classifications are subjective and that the
taxonomic assignments of fossil species are probably riddled with errors.
{
To be fair, I consider the power law for the distribution of genus lifetimes
{
The enthusiastic
pffiffiffiffiffiffiffiffiffiffiffireader can easily derive this result for him or herself by to fall in the category of ‘tenuous’ identifications to which I alluded in the
expanding 1
z using the binomial theorem. second footnote on p. 9. This theory should be taken with a pinch of salt.
 340 M.E.J. Newman
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of mechanisms, including competition for resources, spatial

4.4 The Yule process
separation of breeding populations and genetic drift. If we
One of the most convincing and widely applicable assume that this happens at some stochastically constant
mechanisms for generating power laws is the Yule process, rate, then it follows that a genus with k species in it will gain
which was invented in the 1920s by G. Udny Yule and, new species at a rate proportional to k, since each of the k
coincidentally, also inspired by observations of the statistics species has the same chance per unit time of dividing in
of biological taxa as discussed in the previous section. In two. Let us further suppose that occasionally, say once
addition to having a (possibly) power-law distribution of every m speciation events, the new species produced is, by
lifetimes, biological taxa also have a very convincing chance, sufficiently different from the others in its genus as
power-law distribution of sizes. That is, the distribution to be considered the founder member of an entire new
of the number of species in a genus, family or other genus. (To be clear, we define m such that m species are
taxonomic group appears to follow a power law quite added to pre-existing genera and then one species forms a
closely. This phenomenon was first reported by J.C. Willis new genus. So m + 1 new species appear for each new
in 1922. His impressive plot of the distribution of the genus and there are m + 1 species per genus on average.)
numbers of species in genera of flowering plants is Thus the number of genera goes up steadily in this model,
reproduced in its original form in figure 10. (To the as does the number of species within each genus.
author’s knowledge, this is the first published graph We can analyse this Yule process mathematically as
showing a power-law statistical distribution using the follows*. Let us measure the passage of time in the model
modern logarithmic scales, preceding even Alfred Lotka’s by the number of genera n. At each time step one new
remarkable 1926 discovery of the so-called ‘law of scientific species founds a new genus, thereby increasing n by 1, and
productivity’, i.e. the apparent power-law distribution of m other species are added to various pre-existing genera
the numbers of papers that scientists write [10].) which are selected in proportion to the number of species
Yule offered an explanation for the observations of they already have. We denote by pk,n the fraction of genera
Willis using a simple—almost trivial—model that has since that have k species when the total number of genera is n.
found wide application in other areas. He argued as Thus the number of such genera is npk,n. We now ask what
follows. Suppose first that new species appear but they the probability is that the next species added to the system
never die; species are only ever added to genera and never happens to be added to a particular genus i having ki
removed. This differs from the random walk model of the species in it already. This probability is proportional to ki,
last section, and certainly from reality as well. It is believed and so when properly normalized is just ki/Siki. But Siki is
that in practice all species and all genera become extinct in simply the total number of species, which is n(m + 1).
the end. But let us persevere; there is nonetheless much of Furthermore, between the appearance of the nth and the
worth in Yule’s simple model. (n + 1)th genera, m other new species are added, so the
Species are added to genera by speciation, the splitting of probability that genus i gains a new species during this
one species into two, which is known to happen by a variety interval is mki/(n(m + 1)). And the total expected number
of genera of size k that gain a new species in the same
interval is
mk m
 npk;n ¼ kpk;n : ð64Þ
nðm þ 1Þ mþ1

Now we observe that the number of genera with k species

will decrease on each time step by exactly this number, since
by gaining a new species they become genera with k + 1
instead. At the same time the number increases because of
species that previously had k – 1 species and now have an
extra one. Thus we can write a master equation for the new
number (n + 1)pk,n + 1 of genera with k species thus:

*
Figure 10. J.C. Willis’s 1922 plot of the cumulative Yule’s analysis of the process was considerably more involved than the
one presented here, essentially because the theory of stochastic processes as
distribution of the number of species in genera of flowering we now know it did not yet exist in his time. The master equation method
plants [49,15]. (Reproduced with permission from Nature, we employ is a relatively modern innovation, introduced in this context by
vol. 109, pp. 177 – 179 http://www.nature.com/). Simon [35].
 Power laws, Pareto distributions and Zipfs law 341
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m
 1
ðn þ 1Þpk;nþ1 ¼ npk;n þ ðk
1Þpk
1;n
kpk;n : ð65Þ a¼2þ : ð72Þ
mþ1 m

The only exception to this equation is for genera of size The mean number m + 1 of species per genus for the
1, which instead obey the equation example of flowering plants is about 3, making m^2 and
m a^2.5. The actual exponent for the distribution in figure 10
ðn þ 1Þp1;nþ1 ¼ np1;n þ 1
p1;n ; ð66Þ is a = 2.5 + 0.1, which is in excellent agreement with the
mþ1
theory.
Most likely this agreement is fortuitous, however. The
since by definition exactly one new such genus appears on Yule process is probably not a terribly realistic explanation
each time step. for the distribution of the sizes of genera, principally
Now we ask what form the distribution of the sizes of because it ignores the fact that species (and genera) become
genera takes in the limit of long times. To do this we extinct. However, it has been adapted and generalized by
allow n?? and assume that the distribution tends to others to explain power laws in many other systems, most
some fixed value pk = limn??pn,k independent of n. Then famously city sizes [35], paper citations [50, 51], and links to
equation (66) becomes p1 = 1 – mp1/(m + 1), which has the pages on the world wide web [52, 53]. The most general
solution form of the Yule process is as follows.
Suppose we have a system composed of a collection of
mþ1
p1 ¼ : ð67Þ objects, such as genera, cities, papers, web pages and so
2m þ 1 forth. New objects appear every once in a while as cities
grow up or people publish new papers. Each object also has
And equation (65) becomes some property k associated with it, such as number of
m species in a genus, people in a city or citations to a paper,
pk ¼ ½ðk
1Þpk
1
kpk ; ð68Þ which is reputed to obey a power law, and it is this power
mþ1
law that we wish to explain. Newly appearing objects have
some initial value of k which we will denote k0. New genera
which can be rearranged to read initially have only a single species k0 = 1, but new towns or
cities might have quite a large initial population—a single
k
1
pk ¼ pk
1 ; ð69Þ person living in a house somewhere is unlikely to constitute
k þ 1 þ 1=m a town in their own right but k0 = 100 people might do so.
The value of k0 can also be zero in some cases: newly
and then iterated to get published papers usually have zero citations for instance.
In between the appearance of one object and the next, m
ðk
1Þðk
2Þ . . . 1
pk ¼ p1 new species/people/citations etc. are added to the entire
ðk þ 1 þ 1=mÞðk þ 1=mÞ . . . ð3 þ 1=mÞ system. That is some cities or papers will get new people or
ð70Þ
ðk
1Þ . . . 1 citations, but not necessarily all will. And in the simplest
¼ ð1 þ 1=mÞ ;
ðk þ 1 þ 1=mÞ . . . ð2 þ 1=mÞ case these are added to objects in proportion to the number
that the object already has. Thus the probability of a city
where I have made use of equation (67). This can be gaining a new member is proportional to the number
simplified further by making use of a handy property of the already there; the probability of a paper getting a new
G-function, equation (20), that G(a) = (a – 1)G(a – 1). Using citation is proportional to the number it already has. In
this, and noting that G(1) = 1, we get many cases this seems like a natural process. For example,
a paper that already has many citations is more likely to be
ðkÞð2 þ 1=mÞ
pk ¼ ð1 þ 1=mÞ discovered during a literature search and hence more likely
ðk þ 2 þ 1=mÞ ð71Þ to be cited again. Simon [35] dubbed this type of ‘rich-get-
¼ ð1 þ 1=mÞBðk; 2 þ 1=mÞ; richer’ process the Gibrat principle. Elsewhere it also goes
by the names of the Matthew effect [54], cumulative
where B(a, b) is again the beta-function, equation (19). advantage [50], or preferential attachment [52].
This, we note, is precisely the distribution defined in There is a problem however when k0 = 0. For example, if
equation (40), which Simon called the Yule distribution. new papers appear with no citations and garner citations in
Since the beta-function has a power-law tail B(a, b)*a – b, proportion to the number they currently have, which is
we can immediately see that pk also has a power-law tail zero, then no paper will ever get any citations! To overcome
with an exponent this problem one typically assigns new citations not in
 342 M.E.J. Newman
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proportion simply to k, but to k + c, where c is some citations Price [50] assumed that c = 1, so that paper
constant. Thus there are three parameters k0, c and m that citations have the same exponent a = 2 + 1/m as the
control the behaviour of the model. standard Yule process, although there does not seem to
By an argument exactly analogous to the one given be any very good reason for making this assumption. As we
above, one can then derive the master equation saw in table 1 (and as Price himself also reported), real
citations seem to have an exponent a^3, so we should
k
1þc
ðn þ 1Þpk;nþ1 ¼ npk;n þ m pk
1;n expect c^. For the data from the Science Citation Index
k0 þ c þ m
ð73Þ examined in section 2.1, the mean number m of citations
kþc

m pk;n ; for k4k0 ; per paper is 8.6. So we should put c^8.6 too if we want the
k0 þ c þ m Yule process to match the observed exponent.
The most widely studied model of links on the web, that
and of Barabási and Albert [52], assumes c = m so that a = 3,
but again there does not seem to be a good reason for this
k0 þ c
ðn þ 1Þpk0 ;nþ1 ¼ npk0 ;n þ 1
m pk ;n ; for k ¼ k0 : assumption. The measured exponent for numbers of links
k0 þ c þ m 0 to web sites is about a = 2.2, so if the Yule process is to
ð74Þ match the data in this case, we should put c^0.2m.
However, the important point is that the Yule process is
(Note that k is never less than k0, since each object appears a plausible and general mechanism that can explain a
with k = k0 initially.) number of the power-law distributions observed in nature
Looking for stationary solutions of these equations as and can produce a wide range of exponents to match the
before, we define pk = limn??pn,k and find that observations by suitable adjustments of the parameters.
For several of the distributions shown in figure 4, especially
k0 þ c þ m
pk0 ¼ ; ð75Þ citations, city populations and personal income, it is now
ðm þ 1Þðk0 þ cÞ þ m the most widely accepted theory.

and

ðk
1 þ cÞðk
2 þ cÞ . . . ðk0 þ cÞ 4.5 Phase transitions and critical phenomena

pk ¼ pk
ðk
1 þ c þ aÞðk
2 þ c þ aÞ . . . ðk0 þ c þ aÞ 0 A completely different mechanism for generating power
ðk þ cÞðk0 þ c þ aÞ laws, one that has received a huge amount of attention over
¼ pk ; ð76Þ
ðk0 þ cÞðk þ c þ aÞ 0 the past few decades from the physics community, is that of
critical phenomena.
where I have made use of the G-function notation Some systems have only a single macroscopic length-
introduced for equation (71) and, for reasons that will scale, size-scale or time-scale governing them. A classic
become clear in just a moment, I have defined example is a magnet, which has a correlation length that
a = 2 + (k0 + c)/m. As before, this expression can also be measures the typical size of magnetic domains. Under
written in terms of the beta-function, equation (19): certain circumstances this length-scale can diverge, leaving
the system with no scale at all. As we will now see, such a
Bðk þ c; aÞ
pk ¼ pk : ð77Þ system is ‘scale-free’ in the sense of section 3.5 and hence
Bðk0 þ c; aÞ 0 the distributions of macroscopic physical quantities have to
follow power laws. Usually the circumstances under which
Since the beta-function follows a power law in its tail, B(a, the divergence takes place are very specific ones. The
b)*a – b, the general Yule process generates a power-law parameters of the system have to be tuned very precisely to
distribution pk*k – a with the exponent related to the three produce the power-law behaviour. This is something of a
parameters of the process according to disadvantage; it makes the divergence of length-scales an
unlikely explanation for generic power-law distributions of
k0 þ c
a¼2þ : ð78Þ the type highlighted in this paper. As we will shortly see,
m however, there are some elegant and interesting ways
around this problem.
For example, the original Yule process for number of The precise point at which the length-scale in a system
species per genus has c = 0 and k0 = 1, which reproduces diverges is called a critical point or a phase transition. More
the result of equation (72). For citations of papers or links specifically it is a continuous phase transition. (There are
to web pages we have k0 = 0 and we must have c 4 0 to get other kinds of phase transitions too.) Things that happen in
any citations or links at all. So a = 2 + c/m. In his work on the vicinity of continuous phase transitions are known as
 Power laws, Pareto distributions and Zipfs law 343
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critical phenomena, of which power-law distributions are And what happens in between these two extremes? As we
one example. increase p from small values, the value of hsi also increases.
To better understand the physics of critical phenomena, But at some point we reach the start of the regime in which
let us explore one simple but instructive example, that of hsi goes up with system size instead of staying constant. We
the ‘percolation transition’. Consider a square lattice like now know that this point is at p = 0.5927462. . ., which is
the one depicted in figure 11 in which some of the squares called the critical value of p and is denoted pc. If the size of
have been coloured in. Suppose we colour each square with the lattice is large, then hsi also becomes large at this point,
independent probability p, so that on average a fraction p and in the limit where the lattice size goes to infinity hsi
of them are coloured in. Now we look at the clusters of actually diverges. To illustrate this phenomenon, I show in
coloured squares that form, i.e. the contiguous regions of figure 13 a plot of hsi from simulations of the percolation
adjacent coloured squares. We can ask, for instance, what model and the divergence is clear.
the mean area hsi is of the cluster to which a randomly Now consider not just the mean cluster size but the entire
chosen square belongs. If that square is not coloured in distribution of cluster sizes. Let p(s) be the probability that
then the area is zero. If it is coloured in but none of the a randomly chosen square belongs to a cluster of area s. In
adjacent ones is coloured in then the area is one, and so general, what forms can p(s) take as a function of s? The
forth. important point to notice is that p(s), being a probability
When p is small, only a few squares are coloured in and distribution, is a dimensionless quantity—just a number—
most coloured squares will be alone on the lattice, or maybe but s is an area. We could measure s in terms of square
grouped in twos or threes. So hsi will be small. This metres, or whatever units the lattice is calibrated in. The
situation is depicted in figure 12 for p = 0.3. Conversely, if average hsi is also an area and then there is the area of a
p is large—almost 1, which is the largest value it can have— unit square itself, which we will denote a. Other than these
then most squares will be coloured in and they will almost three quantities, however, there are no other independent
all be connected together in one large cluster, the so-called parameters with dimensions in this problem. (There is the
spanning cluster. In this situation we say that the system area of the whole lattice, but we are considering the limit
percolates. Now the mean size of the cluster to which a where that becomes infinite, so it is out of the picture.)
vertex belongs is limited only by the size of the lattice itself If we want to make a dimensionless function p(s) out of
and as we let the lattice size become large hsi also becomes these three dimensionful parameters, there are three
large. So we have two distinctly different behaviours, one dimensionless ratios we can form: s/a, a/hsi and s/hsi (or
for small p in which hsi is small and does not depend on the their reciprocals, if we prefer). Only two of these are
size of the system, and one for large p in which hsi is much independent however, since the last is the product of the
larger and increases with the size of the system. other two. Thus in general we can write
 
s a
pðsÞ ¼ Cf ; ; ð79Þ
a hsi

where f is a dimensionless mathematical function of its

dimensionless arguments and C is a normalizing constant
chosen so that Ssp(s) = 1.
But now here’s the trick. We can coarse-grain or rescale
our lattice so that the fundamental unit of the lattice
changes. For instance, we could double the size of our unit
square a. The kind of picture I am thinking of is shown in
figure 14. The basic percolation clusters stay roughly the
same size and shape, although I have had to fudge things
around the edges a bit to make it work. For this reason this
argument will only be strictly correct for large clusters s
whose area is not changed appreciably by the fudging. (And
the argument thus only tells us that the tail of the
distribution is a power law, and not the whole distribution.)
The probability p(s) of getting a cluster of area s is
unchanged by the coarse-graining since the areas them-
Figure 11. The percolation model on a square lattice: selves are, to a good approximation, unchanged, and the
squares on the lattice are coloured in independently at mean cluster size is thus also unchanged. All that has
random with some probability p. In this example p = 12. changed, mathematically speaking, is that the unit area a
 344 M.E.J. Newman
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Figure 12. Three examples of percolation systems on 100 6 100 square lattices with p = 0.3, p = pc = 0.5927. . . and p = 0.9.
The first and last are well below and above the critical point respectively, while the middle example is precisely at it.

still sums to unity, and that this change will depend on the
value we choose for the rescaling factor b.
But now we notice that there is one special point at which
this rescaling by definition does not result in a change in hsi
or a corresponding change in the site occupation prob-
ability, and that is the critical point. When we are precisely
at the point at which hsi??, then bhsi = hsi by definition.
Putting hsi?? in equations (79) and (80), we then get
p(s) = C’f(bs/a, 0) = (C’/C)p(bs). Or equivalently

pðbsÞ ¼ gðbÞpðsÞ; ð81Þ

where g(b) = C/C’. Comparing with equation (30) we see

that this has precisely the form of the equation that defines
a scale-free distribution. The rest of the derivation below
Figure 13. The mean area of the cluster to which a equation (30) follows immediately, and so we know that
randomly chosen square belongs for the percolation model p(s) must follow a power law.
described in the text, calculated from an average over 1000 This in fact is the origin of the name ‘scale-free’ for a
simulations on a 1000 6 1000 square lattice. The dotted distribution of the form (30). At the point at which hsi
line marks the known position of the phase transition. diverges, the system is left with no defining size-scale, other
than the unit of area a itself. It is ‘scale-free’, and by the
argument above it follows that the distribution of s must
has been rescaled a?a/b for some constant rescaling factor obey a power law.
b. The equivalent of equation (79) in our coarse-grained In figure 15 I show an example of a cumulative
system is distribution of cluster sizes for a percolation system right
    at the critical point and, as the figure shows, the
0 s a=b 0 bs a
pðsÞ ¼ C f ; ¼Cf ; : ð80Þ distribution does indeed follow a power law. Technically
a=b hsi a bhsi the distribution cannot follow a power law to arbitrarily
large cluster sizes since the area of a cluster can be no bigger
Comparing with equation (79), we can see that this is equal, than the area of the whole lattice, so the power-law
to within a multiplicative constant, to the probability p(bs) distribution will be cut off in the tail. This is an example of
of getting a cluster of size bs, but in a system with a a finite-size effect. This point does not seem to be visible in
different mean cluster size of bhsi. Thus we have related the figure 15 however.
probabilities of two different sizes of clusters to one The kinds of arguments given in this section can be
another, but on systems with different average cluster size made more precise using the machinery of the renorma-
and hence presumably also different site occupation lization group. The real-space renormalization group
probability. Note that the normalization constant must in makes use precisely of transformations such as that
general be changed in equation (80) to make sure that p(s) shown in figure 14 to derive power-law forms and their
 Power laws, Pareto distributions and Zipfs law 345
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Figure 14. A site percolation system is coarse-grained, so that the area of the fundamental square is (in this case) quadrupled.
The occupation of the squares in the coarse-grained lattice (right) is chosen to mirror as nearly as possible that of the squares
on the original lattice (left), so that the sizes and shapes of the large clusters remain roughly the same. The small clusters are
mostly lost in the coarse-graining, so that the arguments given in the text are valid only for the large-s tail of the cluster size
distribution.

4.6 Self-organized criticality

As discussed in the preceding section, certain systems
develop power-law distributions at special ‘critical’ points
in their parameter space because of the divergence of some
characteristic scale, such as the mean cluster size in the
percolation model. This does not, however, provide a
plausible explanation for the origin of power laws in most
real systems. Even if we could come up with some model of
earthquakes or solar flares or web hits that had such a
divergence, it seems unlikely that the parameters of the real
world would, just coincidentally, fall precisely at the point
where the divergence occurred.
As first proposed by Bak et al. [57], however, it is possible
that some dynamical systems actually arrange themselves
so that they always sit at the critical point, no matter what
state we start off in. One says that such systems self-
organize to the critical point, or that they display self-
organized criticality. A now-classic example of such a
system is the forest fire model of Drossel and Schwabl [58],
Figure 15. Cumulative distribution of the sizes of clusters which is based on the percolation model we have already
for (site) percolation on a square lattice of 40000 6 40000 seen.
sites at the critical site occupation probability Consider the percolation model as a primitive model of a
pc = 0.592746. . .. forest. The lattice represents the landscape and a single tree
can grow in each square. Occupied squares represent trees
and empty squares represent empty plots of land with no
exponents for distributions at the critical point. An trees. Trees appear instantaneously at random at some
example application to the percolation problem is given constant rate and hence the squares of the lattice fill up at
by Reynolds et al. [55]. A more technically sophisticated random. Every once in a while a wildfire starts at a random
technique is the k-space renormalization group, which square on the lattice, set off by a lightning strike perhaps,
makes use of transformations in Fourier space to and burns the tree in that square, if there is one, along with
accomplish similar aims in a particularly elegant formal every other tree in the cluster connected to it. The process is
environment [56]. illustrated in figure 16. One can think of the fire as leaping
 346 M.E.J. Newman
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from tree to adjacent tree until the whole cluster is burned, estimates give a value of a = 1.19 + 0.01 [59], meaning that
but the fire cannot cross the firebreak formed by an empty the distribution has an infinite mean in the limit of large
square. If there is no tree in the square struck by the system size. For all real systems however the mean is finite:
lightning, then nothing happens. After a fire, trees can grow the distribution is cut off in the large-size tail because fires
up again in the squares vacated by burnt trees, so the cannot have a size any greater than that of the lattice as a
process keeps going indefinitely. whole and this makes the mean well behaved. This cut-off is
If we start with an empty lattice, trees will start to appear clearly visible in figure 17 as the drop in the curve towards
but will initially be sparse and lightning strikes will either the right of the plot. What is more the distribution of the
hit empty squares or if they do chance upon a tree they will sizes of fires in real forests, figure 5 (d), shows a similar cut-
burn it and its cluster, but that cluster will be small and off and is in many ways qualitatively similar to the
localized because we are well below the percolation distribution predicted by the model. (Real forests are
threshold. Thus fires will have essentially no effect on the obviously vastly more complex than the forest fire model,
forest. As time goes by however, more and more trees will and no one is seriously suggesting that the model is an
grow up until at some point there are enough that we have accurate representation of the real world. Rather it is a
percolation. At that point, as we have seen, a spanning guide to the general type of processes that might be going
cluster forms whose size is limited only by the size of the on in forests.)
lattice, and when any tree in that cluster gets hit by the There has been much excitement about self-organized
lightning the entire cluster will burn away. This gets rid of criticality as a possible generic mechanism for explaining
the spanning cluster so that the system does not percolate where power-law distributions come from. Per Bak, one of
any more, but over time as more trees appear it will the originators of the idea, wrote an entire book about it
presumably reach percolation again, and so the scenario [60]. Self-organized critical models have been put forward
will play out repeatedly. The end result is that the system not only for forest fires, but for earthquakes [61, 62], solar
oscillates right around the critical point, first going just flares [5], biological evolution [63], avalanches [57] and
above the percolation threshold as trees appear and then many other phenomena. Although it is probably not the
being beaten back below it by fire. In the limit of large universal law that some have claimed it to be, it is certainly
system size these fluctuations become small compared to a powerful and intriguing concept that potentially has
the size of the system as a whole and to an excellent applications to a variety of natural and man-made systems.
approximation the system just sits at the threshold
indefinitely. Thus, if we wait long enough, we expect the
4.7 Other mechanisms for generating power laws
forest fire model to self-organize to a state in which it has a
power-law distribution of the sizes of clusters, or of the In the preceding sections I have described the best known
sizes of fires. and most widely applied mechanisms that generate power-
In figure 17 I show the cumulative distribution of the law distributions. However, there are a number of others
sizes of fires in the forest fire model and, as we can see, it that deserve a mention. One that has been receiving some
follows a power law closely. The exponent of the attention recently is the highly optimized tolerance
distribution is quite small in this case. The best current mechanism of Carlson and Doyle [64, 65]. The classic

Figure 16. Lightning strikes at random positions in the forest fire model, starting fires that wipe out the entire cluster to which
a struck tree belongs.
 Power laws, Pareto distributions and Zipfs law 347
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Another mechanism, which is mathematically similar to

that of Carlson and Doyle but quite different in motivation,
is the coherent noise mechanism proposed by Sneppen and
Newman [66] as a model of biological extinction. In this
mechanism a number of agents or species are subjected to
stresses of various sizes, and each agent has a threshold for
stress above which an applied stress will wipe that agent
out—the species becomes extinct. Extinct species are
replaced by new ones with randomly chosen thresholds.
The net result is that the system self-organizes to a state
where most of the surviving species have high thresholds,
but the exact distribution depends on the distribution of
stresses in a way very similar to the relation between block
sizes and fire frequency in highly optimized tolerance. No
conscious optimization is needed in this case, but the end
result is similar: the overall distribution of the numbers of
species becoming extinct as a result of any particular stress
approximately follows a power law. The power-law form is
not exact, but it is as good as that seen in real extinction
data. Sneppen and Newman have also suggested that their
Figure 17. Cumulative distribution of the sizes of ‘fires’ in a
mechanism could be used to model avalanches and
simulation of the forest fire model of Drossel and Schwabl
earthquakes.
[58] for a square lattice of size 5000 6 5000.
One of the broad distributions mentioned in section 2.2
as an alternative to the power law was the log-normal. A
log-normally distributed quantity is one whose logarithm is
example of this mechanism is again a model of forest fires normally distributed. That is
and is based on the percolation process. Suppose again !
that fires start at random in a grid-like forest, just as we ðln x
mÞ2
pðln xÞ  exp
; ð82Þ
considered in section 4.6, but suppose now that instead of 2s2
appearing at random, trees are deliberately planted by a
knowledgeable forester. One can ask what the best for some choice of the mean m and standard deviation s of
distribution of trees is to optimize the amount of lumber the distribution. Distributions like this typically arise when
the forest produces, subject to random fires that could we are multiplying together random numbers. The log of
start at any place. The answer turns out to be that one the product of a large number of random numbers is the
should plant trees in blocks, with narrow firebreaks sum of the logarithms of those same random numbers, and
between them to prevent fires from spreading. Moreover, by the central limit theorem such sums have a normal
one should make the blocks smaller in regions where fires distribution essentially regardless of the distribution of the
start more often and larger where fires are rare. The individual numbers.
reason for this is that we waste some valuable space by But equation (82) implies that the distribution of x itself
making firebreaks, space in which we could have planted is
more trees. If fires are rare, then on average it pays to put !
the breaks further apart—more trees will burn if there is a d ln x 1 ðln x
mÞ2
pðxÞ ¼ pðln xÞ ¼ exp
: ð83Þ
fire, but we also get more lumber if there is not. dx x 2s2
Carlson and Doyle show both by analytic arguments and
by numerical simulation that for quite general distributions To see how this looks if we were to plot it on log scales, we
of starting points for fires this process leads to a take logarithms of both sides, giving
distribution of fire sizes that approximately follows a
power law. The distribution is not a perfect power law in ðln x
mÞ2
ln pðxÞ ¼
ln x

this case, but on the other hand neither are many of those 2s2 ð84Þ
ðln xÞ 2 h m i m2
seen in the data of figure 4, so this is not necessarily a
¼
2
þ 2
1 ln x
2 ;
disadvantage. Carlson and Doyle have proposed that 2s s 2s
highly optimized tolerance could be a model not only for
forest fires but also for the sizes of files on the world wide which is quadratic in ln x. However, any quadratic curve
web, which appear to follow a power law [6]. looks straight if we view a sufficiently small portion of it, so
 348 M.E.J. Newman
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p(x) will look like a power-law distribution when we look at could produce a power-law distribution of the sizes of
a small portion on log scales. The effective exponent a of meteor craters similar to the one in figure 4 (g).
the distribution is in this case not fixed by the theory—it In fact, as discussed by a number of authors [67 – 69],
could be anything, depending on which part of the random multiplication processes can also generate perfect
quadratic our data fall on. power-law distributions with only a slight modification: if
On larger scales the distribution will have some down- there is a lower bound on the value that the product of a set
ward curvature, but so do many of the distributions of numbers is allowed to take (for example if there is a
claimed to follow power laws, so it is possible that these ‘reflecting boundary’ on the lower end of the range, or an
distributions are really log-normal. In fact, in many cases additive noise term as well as a multiplicative one) then the
we do not even have to restrict ourselves to a particularly behaviour of the process is modified to generate not a log-
small portion of the curve. If s is large then the quadratic normal, but a true power law.
term in equation (84) will vary slowly and the curvature of Finally, some processes show power-law distributions of
the line will be slight, so the distribution will appear to times between events. The distribution of times between
follow a power law over relatively large portions of its earthquakes and their aftershocks is one example. Such
range. This situation arises commonly when we are power-law distributions of times are observed in critical
considering products of random numbers. models and in the coherent noise mechanism mentioned
Suppose for example that we are multiplying together above, but another possible explanation for their occur-
100 numbers, each of which is drawn from some distribu- rence is a random extremal process or record dynamics. In
tion such that the standard deviation of the logs is around this mechanism we consider how often a randomly
1—i.e. the numbers themselves vary up or down by about a fluctuating quantity will break its own record for the
factor of e. Then, by the central limit theorem, the standard highest value recorded. For a quantity with, say, a
deviation for ln x will be s^10 and ln x will have to vary Gaussian distribution, it is always in theory possible for
by about + 10 for changes in (ln x)2/s2 to be apparent. But the record to be broken, no matter what its current value,
such a variation in the logarithm corresponds to a variation but the more often the record is broken the higher the
in x of more than four orders of magnitude. If our data record will get and the longer we will have to wait until it is
span a domain smaller than this, as many of the plots in broken again. As shown by Sibani and Littlewood [70], this
figure 4 do, then we will see a measured distribution that non-stationary process gives a distribution of waiting times
looks close to a power law. And the range will get quickly between the establishment of new records that follows a
larger as the number of numbers we are multiplying grows. power law with exponent a = – 1. Interestingly, this is
One example of a random multiplicative process might precisely the exponent observed for the distribution of
be wealth generation by investment. If a person invests waiting times for aftershocks of earthquakes. The record
money, for instance in the stock market, they will get a dynamics has also been proposed as a model for the
percentage return on their investment that varies over time. lifetimes of biological taxa [71].
In other words, in each period of time their investment is
multiplied by some factor which fluctuates from one period
5. Conclusions
to the next. If the fluctuations are random and uncorre-
lated, then after many such periods the value of the In this review I have discussed the power-law statistical
investment is the initial value multiplied by the product of a distributions seen in a wide variety of natural and man-
large number of random numbers, and therefore should be made phenomena, from earthquakes and solar flares to
distributed according to a log-normal. This could explain populations of cities and sales of books. We have seen
why the tail of the wealth distribution, figure 4 (j), appears many examples of power-law distributions in real data and
to follow a power law. seen how to analyse those data to understand the behaviour
Another example is fragmentation. Suppose we break a and parameters of the distributions. I have also described a
stick of unit length into two parts at a position which is a number of physical mechanisms that have been proposed to
random fraction z of the way along the stick’s length. Then explain the occurrence of power laws. Perhaps the two most
we break the resulting pieces at random again and so on. important of these are the following.
After many breaks, the length of one of the remaining
pieces will be Pizi, where zi is the position of the ith break. (a) The Yule process, a rich-get-richer mechanism in
This is a product of random numbers and thus the resulting which the most populous cities or best-selling books
distribution of lengths should follow a power law over a get more inhabitants or sales in proportion to the
portion of its range. A mechanism like this could, for number they already have. Yule and later Simon
instance, produce a power-law distribution of meteors or showed mathematically that this mechanism pro-
other interplanetary rock fragments, which tend to break duces what is now called the Yule distribution, which
up when they collide with one another, and this in turn follows a power law in its tail.
 Power laws, Pareto distributions and Zipfs law 349
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(b) Critical phenomena and the associated concept of [14] R. Kohli and R. Sah, Market shares: some power law results and
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self-organized criticality, in which a scale-factor of a
University of Chicago, 2003).
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to that point by some dynamical process. The [18] D. Sornette, Critical Phenomena in Natural Sciences (Springer, Berlin,
2000), chapter 14.
divergence can leave the system with no appropriate
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and as we have seen the quantity must then follow a (2004).
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The study of power-law distributions is an area in which
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experimentally and theoretically before we can say we ACM Symposium on Theory of Computing (Association of Comput-
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really understand the physical processes driving these
[26] H. Ebel, L.-I. Mielsch and S. Bornholdt, Phys. Rev. E 66 035103
systems. Without doubt there are many exciting discoveries (2002).
still waiting to be made. [27] B.A. Huberman and L.A. Adamic, in Complex Networks, No. 650,
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Z. Toroczkai (Springer, Berlin, 2004), pp. 371 – 398.
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and Erik van Nimwegen for useful conversations, and Lada Conference on System Sciences (IEEE Computer Society, New York,
2001).
Adamic for the Web site hit data. This work was funded in
[32] B.A. Carreras, D.E. Newman, I. Dobson, et al., in 34th Hawaii
part by the National Science Foundation under grant International Conference on System Sciences (IEEE Computer
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Appendix B: Maximum likelihood estimate of exponents

Consider the power-law distribution
 
a
1 x
a
Appendix A: Rank/frequency plots pðxÞ ¼ Cx
a ¼ ; ðB1Þ
xmin xmin
Suppose we wish to make a plot of the cumulative
distribution function P(x) of a quantity such as, for where we have made use of the value of the normalization
example, the frequency with which words appear in a body constant C calculated in equation (8).
of text (figure 4 (a)). We start by making a list of all the Given a set of n values xi, the probability that those
words along with their frequency of occurrence. Now the values where generated from this distribution is propor-
cumulative distribution of the frequency is defined such tional to
that P(x) is the fraction of words with frequency greater
Y Y  
than or equal to x. Or alternatively one could simply plot
n n
a
1 xi
a
PðxjaÞ ¼ pðxi Þ ¼ : ðB2Þ
the number of words with frequency greater than or equal i¼1
x
i¼1 min
xmin
to x, which differs from the fraction only in its normal-
ization. This quantity is called the likelihood of the data set. What
Now consider the most frequent word, which is ‘the’ in we really want to know however is the probability of a
most written English texts. If x is the frequency with particular value of a given the observed {xi}, which is given
which this word occurs, then clearly there is exactly one by Bayes’ law thus:
word with frequency greater than or equal to x, since no
PðaÞ
other word is more frequent. Similarly, for the frequency PðajxÞ ¼ PðxjaÞ : ðB3Þ
of the second most common word—usually ‘of’—there PðxÞ
are two words with that frequency or greater, namely ‘of’
and ‘the’. And so forth. In other words, if we rank the The prior probability of the data P(x) is fixed—it is 1 for
words in order, then by definition there are n words with the set of observations we made and zero for every other—
frequency greater than or equal to that of the nth most and it is usually assumed, in the absence of any information
common word. Thus the cumulative distribution P(x) is to the contrary, that the prior probability of the exponent
simply proportional to the rank n of a word. This means P(a) is uniform, i.e. a constant. Thus P(ajx)!P(xja). For
that to make a plot of P(x) all we need do is sort the convenience we typically work with the logarithm of P(ajx),
words in decreasing order of frequency, number them which, to within an additive constant, is equal to the log L
starting from 1, and then plot their ranks as a function of the likelihood, given by
 Power laws, Pareto distributions and Zipfs law 351
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n 
X  X
n
xi n xi
L ¼ ln PðxjaÞ ¼ ln ða
1Þ
ln xmin
a ln
ln ¼ 0; ðB5Þ
i¼1
xmin a
1 i¼1 xmin
X
n
xi
¼ n ln ða
1Þ
n ln xmin
a ln : ðB4Þ or
i¼1
xmin
" #
1
X xi
a¼1þn ln : ðB6Þ
Now we calculate the most likely value of a by maximizing i
xmin
the likelihood with respect to a, which is the same as
maximizing the log likelihood, since the logarithm is a
monotonic increasing function. Setting dL/da = 0, we find