Power-Law Size Distributions - University of Vermont€¦ · Your turn—ideal: 0 20 40 60 80 100...
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Power-Law Size Distributions Our Intuition Definition Examples Wild vs. Mild CCDFs Zipf’s law Zipf ⇔ CCDF References 1 of 43 Power-Law Size Distributions Principles of Complex Systems CSYS/MATH 300, Spring, 2013 Prof. Peter Dodds @peterdodds Department of Mathematics & Statistics | Center for Complex Systems | Vermont Advanced Computing Center | University of Vermont Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. Power-Law Size Distributions Our Intuition Definition Examples Wild vs. Mild CCDFs Zipf’s law Zipf ⇔ CCDF References 2 of 43 Outline Our Intuition Definition Examples Wild vs. Mild CCDFs Zipf’s law Zipf ⇔ CCDF References Power-Law Size Distributions Our Intuition Definition Examples Wild vs. Mild CCDFs Zipf’s law Zipf ⇔ CCDF References 3 of 43 Let’s test our collective intuition: Money ≡ Belief Two questions about wealth distribution in the United States: 1. Please estimate the percentage of all wealth owned by individuals when grouped into quintiles. 2. Please estimate what you believe each quintile should own, ideally. 3. Extremes: 100, 0, 0, 0, 0 and 20, 20, 20, 20, 20 Power-Law Size Distributions Our Intuition Definition Examples Wild vs. Mild CCDFs Zipf’s law Zipf ⇔ CCDF References 4 of 43 Wealth distribution in the United States: [8] “Building a better America—One wealth quintile at a time” Norton and Ariely, 2011. [8] Power-Law Size Distributions Our Intuition Definition Examples Wild vs. Mild CCDFs Zipf’s law Zipf ⇔ CCDF References 5 of 43 Wealth distribution in the United States: [8] Power-Law Size Distributions Our Intuition Definition Examples Wild vs. Mild CCDFs Zipf’s law Zipf ⇔ CCDF References 6 of 43 Your turn—estimates: 0 20 40 60 80 100
Transcript of Power-Law Size Distributions - University of Vermont€¦ · Your turn—ideal: 0 20 40 60 80 100...
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
1 of 43
Power-Law Size DistributionsPrinciples of Complex SystemsCSYS/MATH 300, Spring, 2013
Prof. Peter Dodds@peterdodds
Department of Mathematics & Statistics | Center for Complex Systems |Vermont Advanced Computing Center | University of Vermont
Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
2 of 43
Outline
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf⇔ CCDF
References
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
3 of 43
Let’s test our collective intuition:
Money≡
Belief
Two questions about wealth distribution in the UnitedStates:
1. Please estimate the percentage of all wealth ownedby individuals when grouped into quintiles.
2. Please estimate what you believe each quintileshould own, ideally.
3. Extremes: 100, 0, 0, 0, 0 and 20, 20, 20, 20, 20
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
4 of 43
Wealth distribution in the United States: [8]
“Building a better America—One wealth quintile at a time”Norton and Ariely, 2011. [8]
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
5 of 43
Wealth distribution in the United States: [8]
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
6 of 43
Your turn—estimates:
0 20 40 60 80 100
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
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Your turn—ideal:
0 20 40 60 80 100
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
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The sizes of many systems’ elements appear to obey aninverse power-law size distribution:
P(size = x) ∼ c x−γ
where 0 < xmin < x < xmax and γ > 1.
I Exciting class exercise: sketch this function.
I xmin = lower cutoff, xmax = upper cutoffI Negative linear relationship in log-log space:
log10 P(x) = log10 c − γ log10 x
I We use base 10 because we are good people.
I power-law decays in probability:The Statistics of Surprise.
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
9 of 43
Size distributions:
Usually, only the tail of the distribution obeys apower law:
P(x) ∼ c x−γ for x large.
I Still use term ‘power-law size distribution.’I Other terms:
I Fat-tailed distributions.I Heavy-tailed distributions.
Beware:I Inverse power laws aren’t the only ones:
lognormals (), Weibull distributions (), . . .
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
10 of 43
Size distributions:
Many systems have discrete sizes k :I Word frequencyI Node degree in networks: # friends, # hyperlinks, etc.I # citations for articles, court decisions, etc.
P(k) ∼ c k−γ
where kmin ≤ k ≤ kmax
I Obvious fail for k = 0.I Again, typically a description of distribution’s tail.
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
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The statistics of surprise—words:Brown Corpus () (∼ 106 words):
rank word % q1. the 6.88722. of 3.58393. and 2.84014. to 2.57445. a 2.29966. in 2.10107. that 1.04288. is 0.99439. was 0.9661
10. he 0.939211. for 0.934012. it 0.862313. with 0.717614. as 0.713715. his 0.6886
rank word % q1945. apply 0.00551946. vital 0.00551947. September 0.00551948. review 0.00551949. wage 0.00551950. motor 0.00551951. fifteen 0.00551952. regarded 0.00551953. draw 0.00551954. wheel 0.00551955. organized 0.00551956. vision 0.00551957. wild 0.00551958. Palmer 0.00551959. intensity 0.0055
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
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Jonathan Harris’s Wordcount: ()
A word frequency distribution explorer:
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
13 of 43
The statistics of surprise—words:
First—a Gaussian example:
P(x)dx =1√2πσ
e−(x−µ)2/2σdx
linear:
0 5 10 15 200
0.1
0.2
0.3
0.4
x
P(x
)
log-log
−3 −2 −1 0 1 2−25
−20
−15
−10
−5
0
log10 x
log 1
0P
(x)
mean µ = 10, variance σ2 = 1.
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
14 of 43
The statistics of surprise—words:
Raw ‘probability’ (binned) for Brown Corpus:
linear:
0 2 4 6 80
200
400
600
800
1000
1200
q
Nq
log-log
−3 −2 −1 0 10
1
2
3
4
log10
q
log 10
Nq
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
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The statistics of surprise—words:
‘Exceedance probability’ N>q:
linear:
0 2 4 6 80
500
1000
1500
2000
2500
q
N>
q
log-log
−3 −2 −1 0 10
1
2
3
4
log10
q
log 10
N>
q
Power-Law SizeDistributions
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Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
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My, what big words you have...
I Test capitalizes on word frequency following aheavily skewed frequency distribution with adecaying power-law tail.
I Let’s do it collectively... ()
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
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The statistics of surprise:
Gutenberg-Richter law ()
I Log-log plotI Base 10I Slope = -1
N(M > m) ∝ m−1
I From both the very awkwardly similar Christensen etal. and Bak et al.:“Unified scaling law for earthquakes” [4, 2]
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
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The statistics of surprise:
From: “Quake Moves Japan Closer to U.S. andAlters Earth’s Spin” () by Kenneth Chang, March13, 2011, NYT:‘What is perhaps most surprising about the Japanearthquake is how misleading history can be. In the past300 years, no earthquake nearly that large—nothinglarger than magnitude eight—had struck in the Japansubduction zone. That, in turn, led to assumptions abouthow large a tsunami might strike the coast.’
“‘It did them a giant disservice,” said Dr. Stein of thegeological survey. That is not the first time that theearthquake potential of a fault has been underestimated.Most geophysicists did not think the Sumatra fault couldgenerate a magnitude 9.1 earthquake, . . . ’
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
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Well, that’s just great:
Two things we have poor cognitive understanding of:1. Probability
I Ex. The Monty Hall Problem ()I Ex. Daughter/Son born on Tuesday () (see asides;
Wikipedia entry Boy or Girl Paradox ()here).
2. Logarithmic scales.
On counting and logarithms:
I Listen to Radiolab’s“Numbers.” ().
I Later: Benford’s Law ().
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
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6
100 102 104
word frequency
100
102
104
100 102 104
citations
100
102
104
106
100 102 104
web hits
100
102
104
106 107
books sold
1
10
100
100 102 104 106
telephone calls received
100
103
106
2 3 4 5 6 7earthquake magnitude
102
103
104
0.01 0.1 1crater diameter in km
10-4
10-2
100
102
102 103 104 105
peak intensity
101
102
103
104
1 10 100intensity
1
10
100
109 1010
net worth in US dollars
1
10
100
104 105 106
name frequency
100
102
104
103 105 107
population of city
100
102
104
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
(j) (k) (l)
FIG. 4 Cumulative distributions or “rank/frequency plots” of twelve quantities reputed to follow power laws. The distributionswere computed as described in Appendix A. Data in the shaded regions were excluded from the calculations of the exponentsin Table I. Source references for the data are given in the text. (a) Numbers of occurrences of words in the novel Moby Dickby Hermann Melville. (b) Numbers of citations to scientific papers published in 1981, from time of publication until June1997. (c) Numbers of hits on web sites by 60 000 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&Ttelephone 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 apower law even though the horizontal axis is linear. (g) Diameter of craters on the moon. Vertical axis is measured per squarekilometre. (h) Peak gamma-ray intensity of solar flares in counts per second, measured from Earth orbit between February1980 and November 1989. (i) Intensity of wars from 1816 to 1980, measured as battle deaths per 10 000 of the population of theparticipating countries. (j) Aggregate net worth in dollars of the richest individuals in the US in October 2003. (k) Frequencyof occurrence of family names in the US in the year 1990. (l) Populations of US cities in the year 2000.
6
100
102
104
wor
d fr
eque
ncy
100
102
104
100
102
104
cita
tions
100
102
104
106
100
102
104
web
hits
100
102
104
106
107
book
s sol
d
110100
100
102
104
106
tele
phon
e ca
lls re
ceiv
ed
100
103
106
23
45
67
earth
quak
e m
agni
tude
102
103
104
0.01
0.1
1cr
ater
dia
met
er in
km
10-4
10-2
100
102
102
103
104
105
peak
inte
nsity
101
102
103
104
110
100
inte
nsity
110100
109
1010
net w
orth
in U
S do
llars
110100
104
105
106
nam
e fr
eque
ncy
100
102
104
103
105
107
popu
latio
n of
city
100
102
104
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)(k
)(l)
FIG
.4
Cum
ula
tive
distr
ibutionsor
“ra
nk/fr
equen
cyplo
ts”
oftw
elve
quantities
repute
dto
follow
pow
erla
ws.
The
distr
ibutions
wer
eco
mpute
das
des
crib
edin
Appen
dix
A.
Data
inth
esh
aded
regio
ns
wer
eex
cluded
from
the
calc
ula
tions
ofth
eex
ponen
tsin
Table
I.Sourc
ere
fere
nce
sfo
rth
edata
are
giv
enin
the
text.
(a)
Num
ber
sofocc
urr
ence
sofw
ord
sin
the
nov
elM
oby
Dic
kby
Her
mann
Mel
ville
.(b
)N
um
ber
sof
cita
tions
tosc
ientific
paper
spublish
edin
1981,
from
tim
eof
publica
tion
until
June
1997.
(c)
Num
ber
sofhits
on
web
site
sby
60
000
use
rsofth
eA
mer
ica
Online
Inte
rnet
serv
ice
for
the
day
of1
Dec
ember
1997.
(d)
Num
ber
sof
copie
sof
bes
tsel
ling
books
sold
inth
eU
Sbet
wee
n1895
and
1965.
(e)
Num
ber
of
calls
rece
ived
by
AT
&T
tele
phone
cust
om
ersin
the
US
fora
single
day
.(f
)M
agnitude
ofea
rthquakes
inC
alifo
rnia
bet
wee
nJanuary
1910
and
May
1992.
Magnitude
ispro
port
ionalto
the
logarith
mofth
em
axim
um
am
plitu
de
ofth
eea
rthquake,
and
hen
ceth
edistr
ibution
obey
sa
pow
erla
wev
enth
ough
the
horizo
nta
laxis
islinea
r.(g
)D
iam
eter
ofcr
ate
rson
the
moon.
Ver
tica
laxis
ism
easu
red
per
square
kilom
etre
.(h
)Pea
kgam
ma-r
ayin
tensity
of
sola
rflare
sin
counts
per
seco
nd,
mea
sure
dfr
om
Eart
horb
itbet
wee
nFeb
ruary
1980
and
Nov
ember
1989.
(i)
Inte
nsity
ofw
ars
from
1816
to1980,m
easu
red
as
batt
ledea
ths
per
10000
ofth
epopula
tion
ofth
epart
icip
ating
countr
ies.
(j)
Aggre
gate
net
wort
hin
dollars
ofth
erich
est
indiv
iduals
inth
eU
Sin
Oct
ober
2003.
(k)
Fre
quen
cyofocc
urr
ence
offa
mily
nam
esin
the
US
inth
eyea
r1990.
(l)
Popula
tions
ofU
Sci
ties
inth
eyea
r2000.
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
21 of 43
Size distributions:
Examples:I Earthquake magnitude (Gutenberg-Richter
law ()): [6, 2] P(M) ∝ M−2
I Number of war deaths: [11] P(d) ∝ d−1.8
I Sizes of forest fires [5]
I Sizes of cities: [12] P(n) ∝ n−2.1
I Number of links to and from websites [3]
I See in part Simon [12] and M.E.J. Newman [7] “Powerlaws, Pareto distributions and Zipf’s law” for more.
I Note: Exponents range in error
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
22 of 43
Size distributions:
Examples:I Number of citations to papers: [9, 10] P(k) ∝ k−3.I Individual wealth (maybe): P(W ) ∝W−2.I Distributions of tree trunk diameters: P(d) ∝ d−2.I The gravitational force at a random point in the
universe: [1] P(F ) ∝ F−5/2. (see the Holtsmarkdistribution () and stable distributions ()
I Diameter of moon craters: [7] P(d) ∝ d−3.I Word frequency: [12] e.g., P(k) ∝ k−2.2 (variable)
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
23 of 43
power-law distributions
Gaussians versus power-law distributions:I Mediocristan versus ExtremistanI Mild versus Wild (Mandelbrot)I Example: Height versus wealth.
I See “The Black Swan” by NassimTaleb. [13]
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
24 of 43
Turkeys...
From “The Black Swan” [13]
Power-Law SizeDistributions
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Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
25 of 43
Taleb’s table [13]
Mediocristan/ExtremistanI Most typical member is mediocre/Most typical is either
giant or tiny
I Winners get a small segment/Winner take almost alleffects
I When you observe for a while, you know what’s goingon/It takes a very long time to figure out what’s going on
I Prediction is easy/Prediction is hard
I History crawls/History makes jumps
I Tyranny of the collective/Tyranny of the rare andaccidental
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
26 of 43
Size distributions:
Power-law size distributions aresometimes calledPareto distributions () after Italianscholar Vilfredo Pareto. ()
I Pareto noted wealth in Italy wasdistributed unevenly (80–20 rule;misleading).
I Term used especially bypractitioners of the DismalScience ().
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
27 of 43
Devilish power-law size distribution details:
Exhibit A:I Given P(x) = cx−γ with 0 < xmin < x < xmax,
the mean is (γ 6= 2):
〈x〉 = c2− γ
(x2−γ
max − x2−γmin
).
I Mean ‘blows up’ with upper cutoff if γ < 2.I Mean depends on lower cutoff if γ > 2.I γ < 2: Typical sample is large.I γ > 2: Typical sample is small.
Insert question from assignment 1 ()
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
28 of 43
And in general...
Moments:I All moments depend only on cutoffs.I No internal scale that dominates/matters.I Compare to a Gaussian, exponential, etc.
For many real size distributions: 2 < γ < 3I mean is finite (depends on lower cutoff)I σ2 = variance is ‘infinite’ (depends on upper cutoff)I Width of distribution is ‘infinite’I If γ > 3, distribution is less terrifying and may be
easily confused with other kinds of distributions.
Insert question from assignment 1 ()
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
29 of 43
Moments
Standard deviation is a mathematical convenience:I Variance is nice analytically...I Another measure of distribution width:
Mean average deviation (MAD) = 〈|x − 〈x〉|〉
I For a pure power law with 2 < γ < 3:
〈|x − 〈x〉|〉 is finite.
I But MAD is mildly unpleasant analytically...I We still speak of infinite ‘width’ if γ < 3.
Insert question from assignment 2 ()
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
30 of 43
How sample sizes grow...
Given P(x) ∼ cx−γ:I We can show that after n samples, we expect the
largest sample to be
x1 & c′n1/(γ−1)
I Sampling from a finite-variance distribution gives amuch slower growth with n.
I e.g., for P(x) = λe−λx , we find
x1 &1λ
ln n.
Insert question from assignment 2 ()
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
31 of 43
Complementary Cumulative Distribution Function:
CCDF:I
P≥(x) = P(x ′ ≥ x) = 1− P(x ′ < x)
I
=
∫ ∞x ′=x
P(x ′)dx ′
I
∝∫ ∞
x ′=x(x ′)−γdx ′
I
=1
−γ + 1(x ′)−γ+1
∣∣∣∣∞x ′=x
I
∝ x−γ+1
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
32 of 43
Complementary Cumulative Distribution Function:
CCDF:I
P≥(x) ∝ x−γ+1
I Use when tail of P follows a power law.I Increases exponent by one.I Useful in cleaning up data.
PDF:
−3 −2 −1 0 10
1
2
3
4
log10
q
log 10
Nq
CCDF:
−3 −2 −1 0 10
1
2
3
4
log10
q
log 10
N>
q
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
33 of 43
Complementary Cumulative Distribution Function:
I Discrete variables:
P≥(k) = P(k ′ ≥ k)
=∞∑
k ′=k
P(k)
∝ k−γ+1
I Use integrals to approximate sums.
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
34 of 43
Zipfian rank-frequency plots
George Kingsley Zipf:I Noted various rank distributions
have power-law tails, often with exponent -1(word frequency, city sizes...)
I Zipf’s 1949 Magnum Opus ():
I We’ll study Zipf’s law in depth...
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
35 of 43
Zipfian rank-frequency plots
Zipf’s way:I Given a collection of entities, rank them by size,
largest to smallest.I xr = the size of the r th ranked entity.I r = 1 corresponds to the largest size.I Example: x1 could be the frequency of occurrence of
the most common word in a text.I Zipf’s observation:
xr ∝ r−α
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
36 of 43
Size distributions:
Brown Corpus (1,015,945 words):
CCDF:
−3 −2 −1 0 10
1
2
3
4
log10
q
log 10
N>
q
Zipf:
0 1 2 3 4−3
−2
−1
0
1
log10
rank i
log 10
qi
I The, of, and, to, a, ... = ‘objects’I ‘Size’ = word frequencyI Beep: (Important) CCDF and Zipf plots are related...
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
37 of 43
Size distributions:
Brown Corpus (1,015,945 words):CCDF:
−3 −2 −1 0 10
1
2
3
4
log10
q
log 10
N>
q
Zipf:
−3 −2 −1 0 10
1
2
3
4
log10
qi
log 10
ran
k i
I The, of, and, to, a, ... = ‘objects’I ‘Size’ = word frequencyI Beep: (Important) CCDF and Zipf plots are related...
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
38 of 43
Observe:I NP≥(x) = the number of objects with size at least x
where N = total number of objects.I If an object has size xr , then NP≥(xr ) is its rank r .I So
xr ∝ r−α = (NP≥(xr ))−α
∝ x (−γ+1)(−α)r since P≥(x) ∼ x−γ+1.
We therefore have 1 = (−γ + 1)(−α) or:
α =1
γ − 1
I A rank distribution exponent of α = 1 corresponds toa size distribution exponent γ = 2.
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
39 of 43
The Don. ()Extreme deviations in test cricket:
1000 10 20 30 9040 50 60 70 80
I Don Bradman’s batting average ()= 166% next best.
I That’s pretty solid.I Later in the course: Understanding success—
is the Mona Lisa like Don Bradman?
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
40 of 43
References I
[1]
[2] P. Bak, K. Christensen, L. Danon, and T. Scanlon.Unified scaling law for earthquakes.Phys. Rev. Lett., 88:178501, 2002. pdf ()
[3] A.-L. Barabási and R. Albert.Emergence of scaling in random networks.Science, 286:509–511, 1999. pdf ()
[4] K. Christensen, L. Danon, T. Scanlon, and P. Bak.Unified scaling law for earthquakes.Proc. Natl. Acad. Sci., 99:2509–2513, 2002. pdf ()
[5] P. Grassberger.Critical behaviour of the Drossel-Schwabl forest firemodel.New Journal of Physics, 4:17.1–17.15, 2002. pdf ()
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
41 of 43
References II
[6] B. Gutenberg and C. F. Richter.Earthquake magnitude, intensity, energy, andacceleration.Bull. Seism. Soc. Am., 499:105–145, 1942. pdf ()
[7] M. E. J. Newman.Power laws, pareto doistributions and zipf’s law.Contemporary Physics, pages 323–351, 2005.pdf ()
[8] M. I. Norton and D. Ariely.Building a better America—One wealth quintile at atime.Perspectives on Psychological Science, 6:9–12,2011. pdf ()
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
42 of 43
References III
[9] D. J. d. S. Price.Networks of scientific papers.Science, 149:510–515, 1965. pdf ()
[10] D. J. d. S. Price.A general theory of bibliometric and other cumulativeadvantage processes.J. Amer. Soc. Inform. Sci., 27:292–306, 1976.
[11] L. F. Richardson.Variation of the frequency of fatal quarrels withmagnitude.J. Amer. Stat. Assoc., 43:523–546, 1949. pdf ()
[12] H. A. Simon.On a class of skew distribution functions.Biometrika, 42:425–440, 1955. pdf ()
Power-Law SizeDistributions
Our Intuition
Definition
Examples
Wild vs. Mild
CCDFs
Zipf’s law
Zipf ⇔ CCDF
References
43 of 43
References IV
[13] N. N. Taleb.The Black Swan.Random House, New York, 2007.
[14] G. K. Zipf.Human Behaviour and the Principle of Least-Effort.Addison-Wesley, Cambridge, MA, 1949.
