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1
Attractors of Distribution
Generalized Central limit theorem and Stable distribution
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
1
2
Outline
Normal distributionmost prominent probability distribution in simple system
Central limit theoremWhy normal distribution is so normal
Power Law most prominent probability distribution in complex system
Generalized central limit theoremStable distribution Attractor family of distributions
2
NORMAL DISTRIBUTION
Part 1
3
4
Probability density function
5
Moment and variance
Normal distribution
where
parameter μ
is the mean
or
expectation
(location of
the peak)
and σthinsp2 is the
variance the
mean of the
squared
deviation
6
3-sigma rule
about 997 are within three standard
deviations7
CENTRAL LIMIT THEOREM
Part 2
8
9
Central limit theorem
10
Central limit theorem
The central limit
theorem states that the
sum of a number of
independent and
identically distributed
(iid) random
variables with finite
variances will tend to a
normal distribution as
the number of
variables grows
CchaosTalklevyIllustratingTheCentralLimitThe
httpdemonstrationswolframcomIllustratingTheCentralLimitTheoremWith
SumsOfUniformAndExpone
Other distributions can be
approximated by the normal
The binomial distribution B(nthinspp) is
approximately normal N(npthinspnp(1 minus p)) for
large n and for p not too close to zero or one
The Poisson(λ) distribution is approximately
normal N(λthinspλ) for large values of λ
The chi-squared distribution χ2(k) is
approximately normal N(kthinsp2k) for large ks
The Students t-distribution t(ν) is
approximately normal N(0thinsp1) when ν is large 11
Galton Board
If the probability of bouncing right on a pin is p (which equals 05 on an unbiased machine) the probability that the ball ends up in the kth bin equals
According to the central limit theorem the binomial distribution approximates the normal distribution provided that n the number of rows of pins in the machine is large
httpwwwyoutubecomwatchv=xDIyAOBa_yU
CchaosTalklevyIllustratingTheCentralLimitTheoremWithSumsOfBernoulliRandomVcdf
12
Principle of maximum entropy
According to the principle of maximum
entropy if nothing is known about a
distribution except that it belongs to a certain
class then the distribution with the largest
entropy should be chosen as the default
13
Another viewpoint
For a given mean and variance the
corresponding normal distribution is the
continuous distribution with the maximum
entropy
Therefore the assumption of normality
imposes the minimal prior structural
constraint beyond these moments14
Summary of normal
distribution
First the normal distribution is very tractable
analytically that is a large number of results
involving this distribution can be derived in
explicit form
15
Summary of normal
distribution
Second the normal distribution arises as the
outcome of the central limit theorem which
states that under mild conditions the sum of a
large number of random variables is
distributed approximately normally
Finally the bell shape of the normal
distribution makes it a convenient choice for
modelling a large variety of random variables
encountered in practice
16
17
POWER LAW
Part 3
18
Income distribution
19
20
Normal vs Power law
You can hardly find a person twice as tall as
you
Fair enoughhellip
This is normal distribution
But you can easily find a person 10000 times
richer than youhellip
Extremely unfairhellip
This is power law distribution
21
Examples of power laws
a Word frequency Estoup
b Citations of scientific papers Price
c Web hits Adamic and Huberman
d Copies of books sold
e Diameter of moon craters Neukum amp Ivanov
f Intensity of solar flares Lu and Hamilton
g Intensity of wars Small and Singer
h Wealth of the richest people
i Frequencies of family names eg US amp Japan not Korea
j Populations of cities
23
The Power Law Phenomenon
Most nodes have the same number of links
No highly connected nodes
Many nodes with few links
A few nodes with many links
of links (k) of links (k)
No
of
node
s w
ith
k li
nks
No
of
node
s w
ith
k li
nks
Bell CurvePower Law Distribution
GENERALIZED CENTRAL LIMIT THEOREM
Part 4
25
Attraction basin of Gaussian
The central limit theorem states that the sum
of a number of independent and identically
distributed (iid) random variables with finite
variances will tend to a normal distribution as
the number of variables grows
All distribution with finite variance form the
attraction basin of Gaunssian
26
what about the distribution having infinite variance
27
Characteristic function
28
Gaussian pdf and its
characteristic function
CchaosTalklevy01FourierTransformPairs
cdf
29
Characteristic function as a
moment generating function
30
CauchyndashLorentz distribution
CauchyndashLorentz distribution
Characteristic function
Observe that the characteristic function is not
differentiable at the origin So the Cauchy
distribution does not have an expected value
or Variance31
Generalized central limit
theorem
A generalization due to Gnedenko and
Kolmogorov states that the sum of a number
of random variables with power-law tail
distributions decreasing as 1 | x | α + 1 where
0 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
32
Stable distribution
In probability theory a random variable is
said to be stable (or to have a stable
distribution) if it has the property that a linear
combination of two independent copies of the
variable has the same distribution up to
location and scale parameters
The stable distribution family is also
sometimes referred to as the Leacutevy alpha-
stable distribution
33
Such distributions form a four-parameter
family of continuous probability distributions
parametrized by location and scale
parameters μ and c respectively and two
shape parameters β and α roughly
corresponding to measures of asymmetry
and concentration respectively (see the
figures)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
35
36
Symmetric α-stable distributions
with unit scale factor
37
Skewed centered stable
distributions with different β
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ the
skewness parameter β has no effect
The asymptotic behavior is described for α lt 2
Unified normal and power law
38
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
2
Outline
Normal distributionmost prominent probability distribution in simple system
Central limit theoremWhy normal distribution is so normal
Power Law most prominent probability distribution in complex system
Generalized central limit theoremStable distribution Attractor family of distributions
2
NORMAL DISTRIBUTION
Part 1
3
4
Probability density function
5
Moment and variance
Normal distribution
where
parameter μ
is the mean
or
expectation
(location of
the peak)
and σthinsp2 is the
variance the
mean of the
squared
deviation
6
3-sigma rule
about 997 are within three standard
deviations7
CENTRAL LIMIT THEOREM
Part 2
8
9
Central limit theorem
10
Central limit theorem
The central limit
theorem states that the
sum of a number of
independent and
identically distributed
(iid) random
variables with finite
variances will tend to a
normal distribution as
the number of
variables grows
CchaosTalklevyIllustratingTheCentralLimitThe
httpdemonstrationswolframcomIllustratingTheCentralLimitTheoremWith
SumsOfUniformAndExpone
Other distributions can be
approximated by the normal
The binomial distribution B(nthinspp) is
approximately normal N(npthinspnp(1 minus p)) for
large n and for p not too close to zero or one
The Poisson(λ) distribution is approximately
normal N(λthinspλ) for large values of λ
The chi-squared distribution χ2(k) is
approximately normal N(kthinsp2k) for large ks
The Students t-distribution t(ν) is
approximately normal N(0thinsp1) when ν is large 11
Galton Board
If the probability of bouncing right on a pin is p (which equals 05 on an unbiased machine) the probability that the ball ends up in the kth bin equals
According to the central limit theorem the binomial distribution approximates the normal distribution provided that n the number of rows of pins in the machine is large
httpwwwyoutubecomwatchv=xDIyAOBa_yU
CchaosTalklevyIllustratingTheCentralLimitTheoremWithSumsOfBernoulliRandomVcdf
12
Principle of maximum entropy
According to the principle of maximum
entropy if nothing is known about a
distribution except that it belongs to a certain
class then the distribution with the largest
entropy should be chosen as the default
13
Another viewpoint
For a given mean and variance the
corresponding normal distribution is the
continuous distribution with the maximum
entropy
Therefore the assumption of normality
imposes the minimal prior structural
constraint beyond these moments14
Summary of normal
distribution
First the normal distribution is very tractable
analytically that is a large number of results
involving this distribution can be derived in
explicit form
15
Summary of normal
distribution
Second the normal distribution arises as the
outcome of the central limit theorem which
states that under mild conditions the sum of a
large number of random variables is
distributed approximately normally
Finally the bell shape of the normal
distribution makes it a convenient choice for
modelling a large variety of random variables
encountered in practice
16
17
POWER LAW
Part 3
18
Income distribution
19
20
Normal vs Power law
You can hardly find a person twice as tall as
you
Fair enoughhellip
This is normal distribution
But you can easily find a person 10000 times
richer than youhellip
Extremely unfairhellip
This is power law distribution
21
Examples of power laws
a Word frequency Estoup
b Citations of scientific papers Price
c Web hits Adamic and Huberman
d Copies of books sold
e Diameter of moon craters Neukum amp Ivanov
f Intensity of solar flares Lu and Hamilton
g Intensity of wars Small and Singer
h Wealth of the richest people
i Frequencies of family names eg US amp Japan not Korea
j Populations of cities
23
The Power Law Phenomenon
Most nodes have the same number of links
No highly connected nodes
Many nodes with few links
A few nodes with many links
of links (k) of links (k)
No
of
node
s w
ith
k li
nks
No
of
node
s w
ith
k li
nks
Bell CurvePower Law Distribution
GENERALIZED CENTRAL LIMIT THEOREM
Part 4
25
Attraction basin of Gaussian
The central limit theorem states that the sum
of a number of independent and identically
distributed (iid) random variables with finite
variances will tend to a normal distribution as
the number of variables grows
All distribution with finite variance form the
attraction basin of Gaunssian
26
what about the distribution having infinite variance
27
Characteristic function
28
Gaussian pdf and its
characteristic function
CchaosTalklevy01FourierTransformPairs
cdf
29
Characteristic function as a
moment generating function
30
CauchyndashLorentz distribution
CauchyndashLorentz distribution
Characteristic function
Observe that the characteristic function is not
differentiable at the origin So the Cauchy
distribution does not have an expected value
or Variance31
Generalized central limit
theorem
A generalization due to Gnedenko and
Kolmogorov states that the sum of a number
of random variables with power-law tail
distributions decreasing as 1 | x | α + 1 where
0 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
32
Stable distribution
In probability theory a random variable is
said to be stable (or to have a stable
distribution) if it has the property that a linear
combination of two independent copies of the
variable has the same distribution up to
location and scale parameters
The stable distribution family is also
sometimes referred to as the Leacutevy alpha-
stable distribution
33
Such distributions form a four-parameter
family of continuous probability distributions
parametrized by location and scale
parameters μ and c respectively and two
shape parameters β and α roughly
corresponding to measures of asymmetry
and concentration respectively (see the
figures)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
35
36
Symmetric α-stable distributions
with unit scale factor
37
Skewed centered stable
distributions with different β
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ the
skewness parameter β has no effect
The asymptotic behavior is described for α lt 2
Unified normal and power law
38
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
NORMAL DISTRIBUTION
Part 1
3
4
Probability density function
5
Moment and variance
Normal distribution
where
parameter μ
is the mean
or
expectation
(location of
the peak)
and σthinsp2 is the
variance the
mean of the
squared
deviation
6
3-sigma rule
about 997 are within three standard
deviations7
CENTRAL LIMIT THEOREM
Part 2
8
9
Central limit theorem
10
Central limit theorem
The central limit
theorem states that the
sum of a number of
independent and
identically distributed
(iid) random
variables with finite
variances will tend to a
normal distribution as
the number of
variables grows
CchaosTalklevyIllustratingTheCentralLimitThe
httpdemonstrationswolframcomIllustratingTheCentralLimitTheoremWith
SumsOfUniformAndExpone
Other distributions can be
approximated by the normal
The binomial distribution B(nthinspp) is
approximately normal N(npthinspnp(1 minus p)) for
large n and for p not too close to zero or one
The Poisson(λ) distribution is approximately
normal N(λthinspλ) for large values of λ
The chi-squared distribution χ2(k) is
approximately normal N(kthinsp2k) for large ks
The Students t-distribution t(ν) is
approximately normal N(0thinsp1) when ν is large 11
Galton Board
If the probability of bouncing right on a pin is p (which equals 05 on an unbiased machine) the probability that the ball ends up in the kth bin equals
According to the central limit theorem the binomial distribution approximates the normal distribution provided that n the number of rows of pins in the machine is large
httpwwwyoutubecomwatchv=xDIyAOBa_yU
CchaosTalklevyIllustratingTheCentralLimitTheoremWithSumsOfBernoulliRandomVcdf
12
Principle of maximum entropy
According to the principle of maximum
entropy if nothing is known about a
distribution except that it belongs to a certain
class then the distribution with the largest
entropy should be chosen as the default
13
Another viewpoint
For a given mean and variance the
corresponding normal distribution is the
continuous distribution with the maximum
entropy
Therefore the assumption of normality
imposes the minimal prior structural
constraint beyond these moments14
Summary of normal
distribution
First the normal distribution is very tractable
analytically that is a large number of results
involving this distribution can be derived in
explicit form
15
Summary of normal
distribution
Second the normal distribution arises as the
outcome of the central limit theorem which
states that under mild conditions the sum of a
large number of random variables is
distributed approximately normally
Finally the bell shape of the normal
distribution makes it a convenient choice for
modelling a large variety of random variables
encountered in practice
16
17
POWER LAW
Part 3
18
Income distribution
19
20
Normal vs Power law
You can hardly find a person twice as tall as
you
Fair enoughhellip
This is normal distribution
But you can easily find a person 10000 times
richer than youhellip
Extremely unfairhellip
This is power law distribution
21
Examples of power laws
a Word frequency Estoup
b Citations of scientific papers Price
c Web hits Adamic and Huberman
d Copies of books sold
e Diameter of moon craters Neukum amp Ivanov
f Intensity of solar flares Lu and Hamilton
g Intensity of wars Small and Singer
h Wealth of the richest people
i Frequencies of family names eg US amp Japan not Korea
j Populations of cities
23
The Power Law Phenomenon
Most nodes have the same number of links
No highly connected nodes
Many nodes with few links
A few nodes with many links
of links (k) of links (k)
No
of
node
s w
ith
k li
nks
No
of
node
s w
ith
k li
nks
Bell CurvePower Law Distribution
GENERALIZED CENTRAL LIMIT THEOREM
Part 4
25
Attraction basin of Gaussian
The central limit theorem states that the sum
of a number of independent and identically
distributed (iid) random variables with finite
variances will tend to a normal distribution as
the number of variables grows
All distribution with finite variance form the
attraction basin of Gaunssian
26
what about the distribution having infinite variance
27
Characteristic function
28
Gaussian pdf and its
characteristic function
CchaosTalklevy01FourierTransformPairs
cdf
29
Characteristic function as a
moment generating function
30
CauchyndashLorentz distribution
CauchyndashLorentz distribution
Characteristic function
Observe that the characteristic function is not
differentiable at the origin So the Cauchy
distribution does not have an expected value
or Variance31
Generalized central limit
theorem
A generalization due to Gnedenko and
Kolmogorov states that the sum of a number
of random variables with power-law tail
distributions decreasing as 1 | x | α + 1 where
0 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
32
Stable distribution
In probability theory a random variable is
said to be stable (or to have a stable
distribution) if it has the property that a linear
combination of two independent copies of the
variable has the same distribution up to
location and scale parameters
The stable distribution family is also
sometimes referred to as the Leacutevy alpha-
stable distribution
33
Such distributions form a four-parameter
family of continuous probability distributions
parametrized by location and scale
parameters μ and c respectively and two
shape parameters β and α roughly
corresponding to measures of asymmetry
and concentration respectively (see the
figures)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
35
36
Symmetric α-stable distributions
with unit scale factor
37
Skewed centered stable
distributions with different β
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ the
skewness parameter β has no effect
The asymptotic behavior is described for α lt 2
Unified normal and power law
38
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
4
Probability density function
5
Moment and variance
Normal distribution
where
parameter μ
is the mean
or
expectation
(location of
the peak)
and σthinsp2 is the
variance the
mean of the
squared
deviation
6
3-sigma rule
about 997 are within three standard
deviations7
CENTRAL LIMIT THEOREM
Part 2
8
9
Central limit theorem
10
Central limit theorem
The central limit
theorem states that the
sum of a number of
independent and
identically distributed
(iid) random
variables with finite
variances will tend to a
normal distribution as
the number of
variables grows
CchaosTalklevyIllustratingTheCentralLimitThe
httpdemonstrationswolframcomIllustratingTheCentralLimitTheoremWith
SumsOfUniformAndExpone
Other distributions can be
approximated by the normal
The binomial distribution B(nthinspp) is
approximately normal N(npthinspnp(1 minus p)) for
large n and for p not too close to zero or one
The Poisson(λ) distribution is approximately
normal N(λthinspλ) for large values of λ
The chi-squared distribution χ2(k) is
approximately normal N(kthinsp2k) for large ks
The Students t-distribution t(ν) is
approximately normal N(0thinsp1) when ν is large 11
Galton Board
If the probability of bouncing right on a pin is p (which equals 05 on an unbiased machine) the probability that the ball ends up in the kth bin equals
According to the central limit theorem the binomial distribution approximates the normal distribution provided that n the number of rows of pins in the machine is large
httpwwwyoutubecomwatchv=xDIyAOBa_yU
CchaosTalklevyIllustratingTheCentralLimitTheoremWithSumsOfBernoulliRandomVcdf
12
Principle of maximum entropy
According to the principle of maximum
entropy if nothing is known about a
distribution except that it belongs to a certain
class then the distribution with the largest
entropy should be chosen as the default
13
Another viewpoint
For a given mean and variance the
corresponding normal distribution is the
continuous distribution with the maximum
entropy
Therefore the assumption of normality
imposes the minimal prior structural
constraint beyond these moments14
Summary of normal
distribution
First the normal distribution is very tractable
analytically that is a large number of results
involving this distribution can be derived in
explicit form
15
Summary of normal
distribution
Second the normal distribution arises as the
outcome of the central limit theorem which
states that under mild conditions the sum of a
large number of random variables is
distributed approximately normally
Finally the bell shape of the normal
distribution makes it a convenient choice for
modelling a large variety of random variables
encountered in practice
16
17
POWER LAW
Part 3
18
Income distribution
19
20
Normal vs Power law
You can hardly find a person twice as tall as
you
Fair enoughhellip
This is normal distribution
But you can easily find a person 10000 times
richer than youhellip
Extremely unfairhellip
This is power law distribution
21
Examples of power laws
a Word frequency Estoup
b Citations of scientific papers Price
c Web hits Adamic and Huberman
d Copies of books sold
e Diameter of moon craters Neukum amp Ivanov
f Intensity of solar flares Lu and Hamilton
g Intensity of wars Small and Singer
h Wealth of the richest people
i Frequencies of family names eg US amp Japan not Korea
j Populations of cities
23
The Power Law Phenomenon
Most nodes have the same number of links
No highly connected nodes
Many nodes with few links
A few nodes with many links
of links (k) of links (k)
No
of
node
s w
ith
k li
nks
No
of
node
s w
ith
k li
nks
Bell CurvePower Law Distribution
GENERALIZED CENTRAL LIMIT THEOREM
Part 4
25
Attraction basin of Gaussian
The central limit theorem states that the sum
of a number of independent and identically
distributed (iid) random variables with finite
variances will tend to a normal distribution as
the number of variables grows
All distribution with finite variance form the
attraction basin of Gaunssian
26
what about the distribution having infinite variance
27
Characteristic function
28
Gaussian pdf and its
characteristic function
CchaosTalklevy01FourierTransformPairs
cdf
29
Characteristic function as a
moment generating function
30
CauchyndashLorentz distribution
CauchyndashLorentz distribution
Characteristic function
Observe that the characteristic function is not
differentiable at the origin So the Cauchy
distribution does not have an expected value
or Variance31
Generalized central limit
theorem
A generalization due to Gnedenko and
Kolmogorov states that the sum of a number
of random variables with power-law tail
distributions decreasing as 1 | x | α + 1 where
0 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
32
Stable distribution
In probability theory a random variable is
said to be stable (or to have a stable
distribution) if it has the property that a linear
combination of two independent copies of the
variable has the same distribution up to
location and scale parameters
The stable distribution family is also
sometimes referred to as the Leacutevy alpha-
stable distribution
33
Such distributions form a four-parameter
family of continuous probability distributions
parametrized by location and scale
parameters μ and c respectively and two
shape parameters β and α roughly
corresponding to measures of asymmetry
and concentration respectively (see the
figures)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
35
36
Symmetric α-stable distributions
with unit scale factor
37
Skewed centered stable
distributions with different β
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ the
skewness parameter β has no effect
The asymptotic behavior is described for α lt 2
Unified normal and power law
38
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
5
Moment and variance
Normal distribution
where
parameter μ
is the mean
or
expectation
(location of
the peak)
and σthinsp2 is the
variance the
mean of the
squared
deviation
6
3-sigma rule
about 997 are within three standard
deviations7
CENTRAL LIMIT THEOREM
Part 2
8
9
Central limit theorem
10
Central limit theorem
The central limit
theorem states that the
sum of a number of
independent and
identically distributed
(iid) random
variables with finite
variances will tend to a
normal distribution as
the number of
variables grows
CchaosTalklevyIllustratingTheCentralLimitThe
httpdemonstrationswolframcomIllustratingTheCentralLimitTheoremWith
SumsOfUniformAndExpone
Other distributions can be
approximated by the normal
The binomial distribution B(nthinspp) is
approximately normal N(npthinspnp(1 minus p)) for
large n and for p not too close to zero or one
The Poisson(λ) distribution is approximately
normal N(λthinspλ) for large values of λ
The chi-squared distribution χ2(k) is
approximately normal N(kthinsp2k) for large ks
The Students t-distribution t(ν) is
approximately normal N(0thinsp1) when ν is large 11
Galton Board
If the probability of bouncing right on a pin is p (which equals 05 on an unbiased machine) the probability that the ball ends up in the kth bin equals
According to the central limit theorem the binomial distribution approximates the normal distribution provided that n the number of rows of pins in the machine is large
httpwwwyoutubecomwatchv=xDIyAOBa_yU
CchaosTalklevyIllustratingTheCentralLimitTheoremWithSumsOfBernoulliRandomVcdf
12
Principle of maximum entropy
According to the principle of maximum
entropy if nothing is known about a
distribution except that it belongs to a certain
class then the distribution with the largest
entropy should be chosen as the default
13
Another viewpoint
For a given mean and variance the
corresponding normal distribution is the
continuous distribution with the maximum
entropy
Therefore the assumption of normality
imposes the minimal prior structural
constraint beyond these moments14
Summary of normal
distribution
First the normal distribution is very tractable
analytically that is a large number of results
involving this distribution can be derived in
explicit form
15
Summary of normal
distribution
Second the normal distribution arises as the
outcome of the central limit theorem which
states that under mild conditions the sum of a
large number of random variables is
distributed approximately normally
Finally the bell shape of the normal
distribution makes it a convenient choice for
modelling a large variety of random variables
encountered in practice
16
17
POWER LAW
Part 3
18
Income distribution
19
20
Normal vs Power law
You can hardly find a person twice as tall as
you
Fair enoughhellip
This is normal distribution
But you can easily find a person 10000 times
richer than youhellip
Extremely unfairhellip
This is power law distribution
21
Examples of power laws
a Word frequency Estoup
b Citations of scientific papers Price
c Web hits Adamic and Huberman
d Copies of books sold
e Diameter of moon craters Neukum amp Ivanov
f Intensity of solar flares Lu and Hamilton
g Intensity of wars Small and Singer
h Wealth of the richest people
i Frequencies of family names eg US amp Japan not Korea
j Populations of cities
23
The Power Law Phenomenon
Most nodes have the same number of links
No highly connected nodes
Many nodes with few links
A few nodes with many links
of links (k) of links (k)
No
of
node
s w
ith
k li
nks
No
of
node
s w
ith
k li
nks
Bell CurvePower Law Distribution
GENERALIZED CENTRAL LIMIT THEOREM
Part 4
25
Attraction basin of Gaussian
The central limit theorem states that the sum
of a number of independent and identically
distributed (iid) random variables with finite
variances will tend to a normal distribution as
the number of variables grows
All distribution with finite variance form the
attraction basin of Gaunssian
26
what about the distribution having infinite variance
27
Characteristic function
28
Gaussian pdf and its
characteristic function
CchaosTalklevy01FourierTransformPairs
cdf
29
Characteristic function as a
moment generating function
30
CauchyndashLorentz distribution
CauchyndashLorentz distribution
Characteristic function
Observe that the characteristic function is not
differentiable at the origin So the Cauchy
distribution does not have an expected value
or Variance31
Generalized central limit
theorem
A generalization due to Gnedenko and
Kolmogorov states that the sum of a number
of random variables with power-law tail
distributions decreasing as 1 | x | α + 1 where
0 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
32
Stable distribution
In probability theory a random variable is
said to be stable (or to have a stable
distribution) if it has the property that a linear
combination of two independent copies of the
variable has the same distribution up to
location and scale parameters
The stable distribution family is also
sometimes referred to as the Leacutevy alpha-
stable distribution
33
Such distributions form a four-parameter
family of continuous probability distributions
parametrized by location and scale
parameters μ and c respectively and two
shape parameters β and α roughly
corresponding to measures of asymmetry
and concentration respectively (see the
figures)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
35
36
Symmetric α-stable distributions
with unit scale factor
37
Skewed centered stable
distributions with different β
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ the
skewness parameter β has no effect
The asymptotic behavior is described for α lt 2
Unified normal and power law
38
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
Normal distribution
where
parameter μ
is the mean
or
expectation
(location of
the peak)
and σthinsp2 is the
variance the
mean of the
squared
deviation
6
3-sigma rule
about 997 are within three standard
deviations7
CENTRAL LIMIT THEOREM
Part 2
8
9
Central limit theorem
10
Central limit theorem
The central limit
theorem states that the
sum of a number of
independent and
identically distributed
(iid) random
variables with finite
variances will tend to a
normal distribution as
the number of
variables grows
CchaosTalklevyIllustratingTheCentralLimitThe
httpdemonstrationswolframcomIllustratingTheCentralLimitTheoremWith
SumsOfUniformAndExpone
Other distributions can be
approximated by the normal
The binomial distribution B(nthinspp) is
approximately normal N(npthinspnp(1 minus p)) for
large n and for p not too close to zero or one
The Poisson(λ) distribution is approximately
normal N(λthinspλ) for large values of λ
The chi-squared distribution χ2(k) is
approximately normal N(kthinsp2k) for large ks
The Students t-distribution t(ν) is
approximately normal N(0thinsp1) when ν is large 11
Galton Board
If the probability of bouncing right on a pin is p (which equals 05 on an unbiased machine) the probability that the ball ends up in the kth bin equals
According to the central limit theorem the binomial distribution approximates the normal distribution provided that n the number of rows of pins in the machine is large
httpwwwyoutubecomwatchv=xDIyAOBa_yU
CchaosTalklevyIllustratingTheCentralLimitTheoremWithSumsOfBernoulliRandomVcdf
12
Principle of maximum entropy
According to the principle of maximum
entropy if nothing is known about a
distribution except that it belongs to a certain
class then the distribution with the largest
entropy should be chosen as the default
13
Another viewpoint
For a given mean and variance the
corresponding normal distribution is the
continuous distribution with the maximum
entropy
Therefore the assumption of normality
imposes the minimal prior structural
constraint beyond these moments14
Summary of normal
distribution
First the normal distribution is very tractable
analytically that is a large number of results
involving this distribution can be derived in
explicit form
15
Summary of normal
distribution
Second the normal distribution arises as the
outcome of the central limit theorem which
states that under mild conditions the sum of a
large number of random variables is
distributed approximately normally
Finally the bell shape of the normal
distribution makes it a convenient choice for
modelling a large variety of random variables
encountered in practice
16
17
POWER LAW
Part 3
18
Income distribution
19
20
Normal vs Power law
You can hardly find a person twice as tall as
you
Fair enoughhellip
This is normal distribution
But you can easily find a person 10000 times
richer than youhellip
Extremely unfairhellip
This is power law distribution
21
Examples of power laws
a Word frequency Estoup
b Citations of scientific papers Price
c Web hits Adamic and Huberman
d Copies of books sold
e Diameter of moon craters Neukum amp Ivanov
f Intensity of solar flares Lu and Hamilton
g Intensity of wars Small and Singer
h Wealth of the richest people
i Frequencies of family names eg US amp Japan not Korea
j Populations of cities
23
The Power Law Phenomenon
Most nodes have the same number of links
No highly connected nodes
Many nodes with few links
A few nodes with many links
of links (k) of links (k)
No
of
node
s w
ith
k li
nks
No
of
node
s w
ith
k li
nks
Bell CurvePower Law Distribution
GENERALIZED CENTRAL LIMIT THEOREM
Part 4
25
Attraction basin of Gaussian
The central limit theorem states that the sum
of a number of independent and identically
distributed (iid) random variables with finite
variances will tend to a normal distribution as
the number of variables grows
All distribution with finite variance form the
attraction basin of Gaunssian
26
what about the distribution having infinite variance
27
Characteristic function
28
Gaussian pdf and its
characteristic function
CchaosTalklevy01FourierTransformPairs
cdf
29
Characteristic function as a
moment generating function
30
CauchyndashLorentz distribution
CauchyndashLorentz distribution
Characteristic function
Observe that the characteristic function is not
differentiable at the origin So the Cauchy
distribution does not have an expected value
or Variance31
Generalized central limit
theorem
A generalization due to Gnedenko and
Kolmogorov states that the sum of a number
of random variables with power-law tail
distributions decreasing as 1 | x | α + 1 where
0 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
32
Stable distribution
In probability theory a random variable is
said to be stable (or to have a stable
distribution) if it has the property that a linear
combination of two independent copies of the
variable has the same distribution up to
location and scale parameters
The stable distribution family is also
sometimes referred to as the Leacutevy alpha-
stable distribution
33
Such distributions form a four-parameter
family of continuous probability distributions
parametrized by location and scale
parameters μ and c respectively and two
shape parameters β and α roughly
corresponding to measures of asymmetry
and concentration respectively (see the
figures)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
35
36
Symmetric α-stable distributions
with unit scale factor
37
Skewed centered stable
distributions with different β
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ the
skewness parameter β has no effect
The asymptotic behavior is described for α lt 2
Unified normal and power law
38
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
3-sigma rule
about 997 are within three standard
deviations7
CENTRAL LIMIT THEOREM
Part 2
8
9
Central limit theorem
10
Central limit theorem
The central limit
theorem states that the
sum of a number of
independent and
identically distributed
(iid) random
variables with finite
variances will tend to a
normal distribution as
the number of
variables grows
CchaosTalklevyIllustratingTheCentralLimitThe
httpdemonstrationswolframcomIllustratingTheCentralLimitTheoremWith
SumsOfUniformAndExpone
Other distributions can be
approximated by the normal
The binomial distribution B(nthinspp) is
approximately normal N(npthinspnp(1 minus p)) for
large n and for p not too close to zero or one
The Poisson(λ) distribution is approximately
normal N(λthinspλ) for large values of λ
The chi-squared distribution χ2(k) is
approximately normal N(kthinsp2k) for large ks
The Students t-distribution t(ν) is
approximately normal N(0thinsp1) when ν is large 11
Galton Board
If the probability of bouncing right on a pin is p (which equals 05 on an unbiased machine) the probability that the ball ends up in the kth bin equals
According to the central limit theorem the binomial distribution approximates the normal distribution provided that n the number of rows of pins in the machine is large
httpwwwyoutubecomwatchv=xDIyAOBa_yU
CchaosTalklevyIllustratingTheCentralLimitTheoremWithSumsOfBernoulliRandomVcdf
12
Principle of maximum entropy
According to the principle of maximum
entropy if nothing is known about a
distribution except that it belongs to a certain
class then the distribution with the largest
entropy should be chosen as the default
13
Another viewpoint
For a given mean and variance the
corresponding normal distribution is the
continuous distribution with the maximum
entropy
Therefore the assumption of normality
imposes the minimal prior structural
constraint beyond these moments14
Summary of normal
distribution
First the normal distribution is very tractable
analytically that is a large number of results
involving this distribution can be derived in
explicit form
15
Summary of normal
distribution
Second the normal distribution arises as the
outcome of the central limit theorem which
states that under mild conditions the sum of a
large number of random variables is
distributed approximately normally
Finally the bell shape of the normal
distribution makes it a convenient choice for
modelling a large variety of random variables
encountered in practice
16
17
POWER LAW
Part 3
18
Income distribution
19
20
Normal vs Power law
You can hardly find a person twice as tall as
you
Fair enoughhellip
This is normal distribution
But you can easily find a person 10000 times
richer than youhellip
Extremely unfairhellip
This is power law distribution
21
Examples of power laws
a Word frequency Estoup
b Citations of scientific papers Price
c Web hits Adamic and Huberman
d Copies of books sold
e Diameter of moon craters Neukum amp Ivanov
f Intensity of solar flares Lu and Hamilton
g Intensity of wars Small and Singer
h Wealth of the richest people
i Frequencies of family names eg US amp Japan not Korea
j Populations of cities
23
The Power Law Phenomenon
Most nodes have the same number of links
No highly connected nodes
Many nodes with few links
A few nodes with many links
of links (k) of links (k)
No
of
node
s w
ith
k li
nks
No
of
node
s w
ith
k li
nks
Bell CurvePower Law Distribution
GENERALIZED CENTRAL LIMIT THEOREM
Part 4
25
Attraction basin of Gaussian
The central limit theorem states that the sum
of a number of independent and identically
distributed (iid) random variables with finite
variances will tend to a normal distribution as
the number of variables grows
All distribution with finite variance form the
attraction basin of Gaunssian
26
what about the distribution having infinite variance
27
Characteristic function
28
Gaussian pdf and its
characteristic function
CchaosTalklevy01FourierTransformPairs
cdf
29
Characteristic function as a
moment generating function
30
CauchyndashLorentz distribution
CauchyndashLorentz distribution
Characteristic function
Observe that the characteristic function is not
differentiable at the origin So the Cauchy
distribution does not have an expected value
or Variance31
Generalized central limit
theorem
A generalization due to Gnedenko and
Kolmogorov states that the sum of a number
of random variables with power-law tail
distributions decreasing as 1 | x | α + 1 where
0 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
32
Stable distribution
In probability theory a random variable is
said to be stable (or to have a stable
distribution) if it has the property that a linear
combination of two independent copies of the
variable has the same distribution up to
location and scale parameters
The stable distribution family is also
sometimes referred to as the Leacutevy alpha-
stable distribution
33
Such distributions form a four-parameter
family of continuous probability distributions
parametrized by location and scale
parameters μ and c respectively and two
shape parameters β and α roughly
corresponding to measures of asymmetry
and concentration respectively (see the
figures)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
35
36
Symmetric α-stable distributions
with unit scale factor
37
Skewed centered stable
distributions with different β
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ the
skewness parameter β has no effect
The asymptotic behavior is described for α lt 2
Unified normal and power law
38
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
CENTRAL LIMIT THEOREM
Part 2
8
9
Central limit theorem
10
Central limit theorem
The central limit
theorem states that the
sum of a number of
independent and
identically distributed
(iid) random
variables with finite
variances will tend to a
normal distribution as
the number of
variables grows
CchaosTalklevyIllustratingTheCentralLimitThe
httpdemonstrationswolframcomIllustratingTheCentralLimitTheoremWith
SumsOfUniformAndExpone
Other distributions can be
approximated by the normal
The binomial distribution B(nthinspp) is
approximately normal N(npthinspnp(1 minus p)) for
large n and for p not too close to zero or one
The Poisson(λ) distribution is approximately
normal N(λthinspλ) for large values of λ
The chi-squared distribution χ2(k) is
approximately normal N(kthinsp2k) for large ks
The Students t-distribution t(ν) is
approximately normal N(0thinsp1) when ν is large 11
Galton Board
If the probability of bouncing right on a pin is p (which equals 05 on an unbiased machine) the probability that the ball ends up in the kth bin equals
According to the central limit theorem the binomial distribution approximates the normal distribution provided that n the number of rows of pins in the machine is large
httpwwwyoutubecomwatchv=xDIyAOBa_yU
CchaosTalklevyIllustratingTheCentralLimitTheoremWithSumsOfBernoulliRandomVcdf
12
Principle of maximum entropy
According to the principle of maximum
entropy if nothing is known about a
distribution except that it belongs to a certain
class then the distribution with the largest
entropy should be chosen as the default
13
Another viewpoint
For a given mean and variance the
corresponding normal distribution is the
continuous distribution with the maximum
entropy
Therefore the assumption of normality
imposes the minimal prior structural
constraint beyond these moments14
Summary of normal
distribution
First the normal distribution is very tractable
analytically that is a large number of results
involving this distribution can be derived in
explicit form
15
Summary of normal
distribution
Second the normal distribution arises as the
outcome of the central limit theorem which
states that under mild conditions the sum of a
large number of random variables is
distributed approximately normally
Finally the bell shape of the normal
distribution makes it a convenient choice for
modelling a large variety of random variables
encountered in practice
16
17
POWER LAW
Part 3
18
Income distribution
19
20
Normal vs Power law
You can hardly find a person twice as tall as
you
Fair enoughhellip
This is normal distribution
But you can easily find a person 10000 times
richer than youhellip
Extremely unfairhellip
This is power law distribution
21
Examples of power laws
a Word frequency Estoup
b Citations of scientific papers Price
c Web hits Adamic and Huberman
d Copies of books sold
e Diameter of moon craters Neukum amp Ivanov
f Intensity of solar flares Lu and Hamilton
g Intensity of wars Small and Singer
h Wealth of the richest people
i Frequencies of family names eg US amp Japan not Korea
j Populations of cities
23
The Power Law Phenomenon
Most nodes have the same number of links
No highly connected nodes
Many nodes with few links
A few nodes with many links
of links (k) of links (k)
No
of
node
s w
ith
k li
nks
No
of
node
s w
ith
k li
nks
Bell CurvePower Law Distribution
GENERALIZED CENTRAL LIMIT THEOREM
Part 4
25
Attraction basin of Gaussian
The central limit theorem states that the sum
of a number of independent and identically
distributed (iid) random variables with finite
variances will tend to a normal distribution as
the number of variables grows
All distribution with finite variance form the
attraction basin of Gaunssian
26
what about the distribution having infinite variance
27
Characteristic function
28
Gaussian pdf and its
characteristic function
CchaosTalklevy01FourierTransformPairs
cdf
29
Characteristic function as a
moment generating function
30
CauchyndashLorentz distribution
CauchyndashLorentz distribution
Characteristic function
Observe that the characteristic function is not
differentiable at the origin So the Cauchy
distribution does not have an expected value
or Variance31
Generalized central limit
theorem
A generalization due to Gnedenko and
Kolmogorov states that the sum of a number
of random variables with power-law tail
distributions decreasing as 1 | x | α + 1 where
0 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
32
Stable distribution
In probability theory a random variable is
said to be stable (or to have a stable
distribution) if it has the property that a linear
combination of two independent copies of the
variable has the same distribution up to
location and scale parameters
The stable distribution family is also
sometimes referred to as the Leacutevy alpha-
stable distribution
33
Such distributions form a four-parameter
family of continuous probability distributions
parametrized by location and scale
parameters μ and c respectively and two
shape parameters β and α roughly
corresponding to measures of asymmetry
and concentration respectively (see the
figures)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
35
36
Symmetric α-stable distributions
with unit scale factor
37
Skewed centered stable
distributions with different β
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ the
skewness parameter β has no effect
The asymptotic behavior is described for α lt 2
Unified normal and power law
38
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
9
Central limit theorem
10
Central limit theorem
The central limit
theorem states that the
sum of a number of
independent and
identically distributed
(iid) random
variables with finite
variances will tend to a
normal distribution as
the number of
variables grows
CchaosTalklevyIllustratingTheCentralLimitThe
httpdemonstrationswolframcomIllustratingTheCentralLimitTheoremWith
SumsOfUniformAndExpone
Other distributions can be
approximated by the normal
The binomial distribution B(nthinspp) is
approximately normal N(npthinspnp(1 minus p)) for
large n and for p not too close to zero or one
The Poisson(λ) distribution is approximately
normal N(λthinspλ) for large values of λ
The chi-squared distribution χ2(k) is
approximately normal N(kthinsp2k) for large ks
The Students t-distribution t(ν) is
approximately normal N(0thinsp1) when ν is large 11
Galton Board
If the probability of bouncing right on a pin is p (which equals 05 on an unbiased machine) the probability that the ball ends up in the kth bin equals
According to the central limit theorem the binomial distribution approximates the normal distribution provided that n the number of rows of pins in the machine is large
httpwwwyoutubecomwatchv=xDIyAOBa_yU
CchaosTalklevyIllustratingTheCentralLimitTheoremWithSumsOfBernoulliRandomVcdf
12
Principle of maximum entropy
According to the principle of maximum
entropy if nothing is known about a
distribution except that it belongs to a certain
class then the distribution with the largest
entropy should be chosen as the default
13
Another viewpoint
For a given mean and variance the
corresponding normal distribution is the
continuous distribution with the maximum
entropy
Therefore the assumption of normality
imposes the minimal prior structural
constraint beyond these moments14
Summary of normal
distribution
First the normal distribution is very tractable
analytically that is a large number of results
involving this distribution can be derived in
explicit form
15
Summary of normal
distribution
Second the normal distribution arises as the
outcome of the central limit theorem which
states that under mild conditions the sum of a
large number of random variables is
distributed approximately normally
Finally the bell shape of the normal
distribution makes it a convenient choice for
modelling a large variety of random variables
encountered in practice
16
17
POWER LAW
Part 3
18
Income distribution
19
20
Normal vs Power law
You can hardly find a person twice as tall as
you
Fair enoughhellip
This is normal distribution
But you can easily find a person 10000 times
richer than youhellip
Extremely unfairhellip
This is power law distribution
21
Examples of power laws
a Word frequency Estoup
b Citations of scientific papers Price
c Web hits Adamic and Huberman
d Copies of books sold
e Diameter of moon craters Neukum amp Ivanov
f Intensity of solar flares Lu and Hamilton
g Intensity of wars Small and Singer
h Wealth of the richest people
i Frequencies of family names eg US amp Japan not Korea
j Populations of cities
23
The Power Law Phenomenon
Most nodes have the same number of links
No highly connected nodes
Many nodes with few links
A few nodes with many links
of links (k) of links (k)
No
of
node
s w
ith
k li
nks
No
of
node
s w
ith
k li
nks
Bell CurvePower Law Distribution
GENERALIZED CENTRAL LIMIT THEOREM
Part 4
25
Attraction basin of Gaussian
The central limit theorem states that the sum
of a number of independent and identically
distributed (iid) random variables with finite
variances will tend to a normal distribution as
the number of variables grows
All distribution with finite variance form the
attraction basin of Gaunssian
26
what about the distribution having infinite variance
27
Characteristic function
28
Gaussian pdf and its
characteristic function
CchaosTalklevy01FourierTransformPairs
cdf
29
Characteristic function as a
moment generating function
30
CauchyndashLorentz distribution
CauchyndashLorentz distribution
Characteristic function
Observe that the characteristic function is not
differentiable at the origin So the Cauchy
distribution does not have an expected value
or Variance31
Generalized central limit
theorem
A generalization due to Gnedenko and
Kolmogorov states that the sum of a number
of random variables with power-law tail
distributions decreasing as 1 | x | α + 1 where
0 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
32
Stable distribution
In probability theory a random variable is
said to be stable (or to have a stable
distribution) if it has the property that a linear
combination of two independent copies of the
variable has the same distribution up to
location and scale parameters
The stable distribution family is also
sometimes referred to as the Leacutevy alpha-
stable distribution
33
Such distributions form a four-parameter
family of continuous probability distributions
parametrized by location and scale
parameters μ and c respectively and two
shape parameters β and α roughly
corresponding to measures of asymmetry
and concentration respectively (see the
figures)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
35
36
Symmetric α-stable distributions
with unit scale factor
37
Skewed centered stable
distributions with different β
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ the
skewness parameter β has no effect
The asymptotic behavior is described for α lt 2
Unified normal and power law
38
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
10
Central limit theorem
The central limit
theorem states that the
sum of a number of
independent and
identically distributed
(iid) random
variables with finite
variances will tend to a
normal distribution as
the number of
variables grows
CchaosTalklevyIllustratingTheCentralLimitThe
httpdemonstrationswolframcomIllustratingTheCentralLimitTheoremWith
SumsOfUniformAndExpone
Other distributions can be
approximated by the normal
The binomial distribution B(nthinspp) is
approximately normal N(npthinspnp(1 minus p)) for
large n and for p not too close to zero or one
The Poisson(λ) distribution is approximately
normal N(λthinspλ) for large values of λ
The chi-squared distribution χ2(k) is
approximately normal N(kthinsp2k) for large ks
The Students t-distribution t(ν) is
approximately normal N(0thinsp1) when ν is large 11
Galton Board
If the probability of bouncing right on a pin is p (which equals 05 on an unbiased machine) the probability that the ball ends up in the kth bin equals
According to the central limit theorem the binomial distribution approximates the normal distribution provided that n the number of rows of pins in the machine is large
httpwwwyoutubecomwatchv=xDIyAOBa_yU
CchaosTalklevyIllustratingTheCentralLimitTheoremWithSumsOfBernoulliRandomVcdf
12
Principle of maximum entropy
According to the principle of maximum
entropy if nothing is known about a
distribution except that it belongs to a certain
class then the distribution with the largest
entropy should be chosen as the default
13
Another viewpoint
For a given mean and variance the
corresponding normal distribution is the
continuous distribution with the maximum
entropy
Therefore the assumption of normality
imposes the minimal prior structural
constraint beyond these moments14
Summary of normal
distribution
First the normal distribution is very tractable
analytically that is a large number of results
involving this distribution can be derived in
explicit form
15
Summary of normal
distribution
Second the normal distribution arises as the
outcome of the central limit theorem which
states that under mild conditions the sum of a
large number of random variables is
distributed approximately normally
Finally the bell shape of the normal
distribution makes it a convenient choice for
modelling a large variety of random variables
encountered in practice
16
17
POWER LAW
Part 3
18
Income distribution
19
20
Normal vs Power law
You can hardly find a person twice as tall as
you
Fair enoughhellip
This is normal distribution
But you can easily find a person 10000 times
richer than youhellip
Extremely unfairhellip
This is power law distribution
21
Examples of power laws
a Word frequency Estoup
b Citations of scientific papers Price
c Web hits Adamic and Huberman
d Copies of books sold
e Diameter of moon craters Neukum amp Ivanov
f Intensity of solar flares Lu and Hamilton
g Intensity of wars Small and Singer
h Wealth of the richest people
i Frequencies of family names eg US amp Japan not Korea
j Populations of cities
23
The Power Law Phenomenon
Most nodes have the same number of links
No highly connected nodes
Many nodes with few links
A few nodes with many links
of links (k) of links (k)
No
of
node
s w
ith
k li
nks
No
of
node
s w
ith
k li
nks
Bell CurvePower Law Distribution
GENERALIZED CENTRAL LIMIT THEOREM
Part 4
25
Attraction basin of Gaussian
The central limit theorem states that the sum
of a number of independent and identically
distributed (iid) random variables with finite
variances will tend to a normal distribution as
the number of variables grows
All distribution with finite variance form the
attraction basin of Gaunssian
26
what about the distribution having infinite variance
27
Characteristic function
28
Gaussian pdf and its
characteristic function
CchaosTalklevy01FourierTransformPairs
cdf
29
Characteristic function as a
moment generating function
30
CauchyndashLorentz distribution
CauchyndashLorentz distribution
Characteristic function
Observe that the characteristic function is not
differentiable at the origin So the Cauchy
distribution does not have an expected value
or Variance31
Generalized central limit
theorem
A generalization due to Gnedenko and
Kolmogorov states that the sum of a number
of random variables with power-law tail
distributions decreasing as 1 | x | α + 1 where
0 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
32
Stable distribution
In probability theory a random variable is
said to be stable (or to have a stable
distribution) if it has the property that a linear
combination of two independent copies of the
variable has the same distribution up to
location and scale parameters
The stable distribution family is also
sometimes referred to as the Leacutevy alpha-
stable distribution
33
Such distributions form a four-parameter
family of continuous probability distributions
parametrized by location and scale
parameters μ and c respectively and two
shape parameters β and α roughly
corresponding to measures of asymmetry
and concentration respectively (see the
figures)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
35
36
Symmetric α-stable distributions
with unit scale factor
37
Skewed centered stable
distributions with different β
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ the
skewness parameter β has no effect
The asymptotic behavior is described for α lt 2
Unified normal and power law
38
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
Other distributions can be
approximated by the normal
The binomial distribution B(nthinspp) is
approximately normal N(npthinspnp(1 minus p)) for
large n and for p not too close to zero or one
The Poisson(λ) distribution is approximately
normal N(λthinspλ) for large values of λ
The chi-squared distribution χ2(k) is
approximately normal N(kthinsp2k) for large ks
The Students t-distribution t(ν) is
approximately normal N(0thinsp1) when ν is large 11
Galton Board
If the probability of bouncing right on a pin is p (which equals 05 on an unbiased machine) the probability that the ball ends up in the kth bin equals
According to the central limit theorem the binomial distribution approximates the normal distribution provided that n the number of rows of pins in the machine is large
httpwwwyoutubecomwatchv=xDIyAOBa_yU
CchaosTalklevyIllustratingTheCentralLimitTheoremWithSumsOfBernoulliRandomVcdf
12
Principle of maximum entropy
According to the principle of maximum
entropy if nothing is known about a
distribution except that it belongs to a certain
class then the distribution with the largest
entropy should be chosen as the default
13
Another viewpoint
For a given mean and variance the
corresponding normal distribution is the
continuous distribution with the maximum
entropy
Therefore the assumption of normality
imposes the minimal prior structural
constraint beyond these moments14
Summary of normal
distribution
First the normal distribution is very tractable
analytically that is a large number of results
involving this distribution can be derived in
explicit form
15
Summary of normal
distribution
Second the normal distribution arises as the
outcome of the central limit theorem which
states that under mild conditions the sum of a
large number of random variables is
distributed approximately normally
Finally the bell shape of the normal
distribution makes it a convenient choice for
modelling a large variety of random variables
encountered in practice
16
17
POWER LAW
Part 3
18
Income distribution
19
20
Normal vs Power law
You can hardly find a person twice as tall as
you
Fair enoughhellip
This is normal distribution
But you can easily find a person 10000 times
richer than youhellip
Extremely unfairhellip
This is power law distribution
21
Examples of power laws
a Word frequency Estoup
b Citations of scientific papers Price
c Web hits Adamic and Huberman
d Copies of books sold
e Diameter of moon craters Neukum amp Ivanov
f Intensity of solar flares Lu and Hamilton
g Intensity of wars Small and Singer
h Wealth of the richest people
i Frequencies of family names eg US amp Japan not Korea
j Populations of cities
23
The Power Law Phenomenon
Most nodes have the same number of links
No highly connected nodes
Many nodes with few links
A few nodes with many links
of links (k) of links (k)
No
of
node
s w
ith
k li
nks
No
of
node
s w
ith
k li
nks
Bell CurvePower Law Distribution
GENERALIZED CENTRAL LIMIT THEOREM
Part 4
25
Attraction basin of Gaussian
The central limit theorem states that the sum
of a number of independent and identically
distributed (iid) random variables with finite
variances will tend to a normal distribution as
the number of variables grows
All distribution with finite variance form the
attraction basin of Gaunssian
26
what about the distribution having infinite variance
27
Characteristic function
28
Gaussian pdf and its
characteristic function
CchaosTalklevy01FourierTransformPairs
cdf
29
Characteristic function as a
moment generating function
30
CauchyndashLorentz distribution
CauchyndashLorentz distribution
Characteristic function
Observe that the characteristic function is not
differentiable at the origin So the Cauchy
distribution does not have an expected value
or Variance31
Generalized central limit
theorem
A generalization due to Gnedenko and
Kolmogorov states that the sum of a number
of random variables with power-law tail
distributions decreasing as 1 | x | α + 1 where
0 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
32
Stable distribution
In probability theory a random variable is
said to be stable (or to have a stable
distribution) if it has the property that a linear
combination of two independent copies of the
variable has the same distribution up to
location and scale parameters
The stable distribution family is also
sometimes referred to as the Leacutevy alpha-
stable distribution
33
Such distributions form a four-parameter
family of continuous probability distributions
parametrized by location and scale
parameters μ and c respectively and two
shape parameters β and α roughly
corresponding to measures of asymmetry
and concentration respectively (see the
figures)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
35
36
Symmetric α-stable distributions
with unit scale factor
37
Skewed centered stable
distributions with different β
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ the
skewness parameter β has no effect
The asymptotic behavior is described for α lt 2
Unified normal and power law
38
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
Galton Board
If the probability of bouncing right on a pin is p (which equals 05 on an unbiased machine) the probability that the ball ends up in the kth bin equals
According to the central limit theorem the binomial distribution approximates the normal distribution provided that n the number of rows of pins in the machine is large
httpwwwyoutubecomwatchv=xDIyAOBa_yU
CchaosTalklevyIllustratingTheCentralLimitTheoremWithSumsOfBernoulliRandomVcdf
12
Principle of maximum entropy
According to the principle of maximum
entropy if nothing is known about a
distribution except that it belongs to a certain
class then the distribution with the largest
entropy should be chosen as the default
13
Another viewpoint
For a given mean and variance the
corresponding normal distribution is the
continuous distribution with the maximum
entropy
Therefore the assumption of normality
imposes the minimal prior structural
constraint beyond these moments14
Summary of normal
distribution
First the normal distribution is very tractable
analytically that is a large number of results
involving this distribution can be derived in
explicit form
15
Summary of normal
distribution
Second the normal distribution arises as the
outcome of the central limit theorem which
states that under mild conditions the sum of a
large number of random variables is
distributed approximately normally
Finally the bell shape of the normal
distribution makes it a convenient choice for
modelling a large variety of random variables
encountered in practice
16
17
POWER LAW
Part 3
18
Income distribution
19
20
Normal vs Power law
You can hardly find a person twice as tall as
you
Fair enoughhellip
This is normal distribution
But you can easily find a person 10000 times
richer than youhellip
Extremely unfairhellip
This is power law distribution
21
Examples of power laws
a Word frequency Estoup
b Citations of scientific papers Price
c Web hits Adamic and Huberman
d Copies of books sold
e Diameter of moon craters Neukum amp Ivanov
f Intensity of solar flares Lu and Hamilton
g Intensity of wars Small and Singer
h Wealth of the richest people
i Frequencies of family names eg US amp Japan not Korea
j Populations of cities
23
The Power Law Phenomenon
Most nodes have the same number of links
No highly connected nodes
Many nodes with few links
A few nodes with many links
of links (k) of links (k)
No
of
node
s w
ith
k li
nks
No
of
node
s w
ith
k li
nks
Bell CurvePower Law Distribution
GENERALIZED CENTRAL LIMIT THEOREM
Part 4
25
Attraction basin of Gaussian
The central limit theorem states that the sum
of a number of independent and identically
distributed (iid) random variables with finite
variances will tend to a normal distribution as
the number of variables grows
All distribution with finite variance form the
attraction basin of Gaunssian
26
what about the distribution having infinite variance
27
Characteristic function
28
Gaussian pdf and its
characteristic function
CchaosTalklevy01FourierTransformPairs
cdf
29
Characteristic function as a
moment generating function
30
CauchyndashLorentz distribution
CauchyndashLorentz distribution
Characteristic function
Observe that the characteristic function is not
differentiable at the origin So the Cauchy
distribution does not have an expected value
or Variance31
Generalized central limit
theorem
A generalization due to Gnedenko and
Kolmogorov states that the sum of a number
of random variables with power-law tail
distributions decreasing as 1 | x | α + 1 where
0 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
32
Stable distribution
In probability theory a random variable is
said to be stable (or to have a stable
distribution) if it has the property that a linear
combination of two independent copies of the
variable has the same distribution up to
location and scale parameters
The stable distribution family is also
sometimes referred to as the Leacutevy alpha-
stable distribution
33
Such distributions form a four-parameter
family of continuous probability distributions
parametrized by location and scale
parameters μ and c respectively and two
shape parameters β and α roughly
corresponding to measures of asymmetry
and concentration respectively (see the
figures)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
35
36
Symmetric α-stable distributions
with unit scale factor
37
Skewed centered stable
distributions with different β
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ the
skewness parameter β has no effect
The asymptotic behavior is described for α lt 2
Unified normal and power law
38
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
Principle of maximum entropy
According to the principle of maximum
entropy if nothing is known about a
distribution except that it belongs to a certain
class then the distribution with the largest
entropy should be chosen as the default
13
Another viewpoint
For a given mean and variance the
corresponding normal distribution is the
continuous distribution with the maximum
entropy
Therefore the assumption of normality
imposes the minimal prior structural
constraint beyond these moments14
Summary of normal
distribution
First the normal distribution is very tractable
analytically that is a large number of results
involving this distribution can be derived in
explicit form
15
Summary of normal
distribution
Second the normal distribution arises as the
outcome of the central limit theorem which
states that under mild conditions the sum of a
large number of random variables is
distributed approximately normally
Finally the bell shape of the normal
distribution makes it a convenient choice for
modelling a large variety of random variables
encountered in practice
16
17
POWER LAW
Part 3
18
Income distribution
19
20
Normal vs Power law
You can hardly find a person twice as tall as
you
Fair enoughhellip
This is normal distribution
But you can easily find a person 10000 times
richer than youhellip
Extremely unfairhellip
This is power law distribution
21
Examples of power laws
a Word frequency Estoup
b Citations of scientific papers Price
c Web hits Adamic and Huberman
d Copies of books sold
e Diameter of moon craters Neukum amp Ivanov
f Intensity of solar flares Lu and Hamilton
g Intensity of wars Small and Singer
h Wealth of the richest people
i Frequencies of family names eg US amp Japan not Korea
j Populations of cities
23
The Power Law Phenomenon
Most nodes have the same number of links
No highly connected nodes
Many nodes with few links
A few nodes with many links
of links (k) of links (k)
No
of
node
s w
ith
k li
nks
No
of
node
s w
ith
k li
nks
Bell CurvePower Law Distribution
GENERALIZED CENTRAL LIMIT THEOREM
Part 4
25
Attraction basin of Gaussian
The central limit theorem states that the sum
of a number of independent and identically
distributed (iid) random variables with finite
variances will tend to a normal distribution as
the number of variables grows
All distribution with finite variance form the
attraction basin of Gaunssian
26
what about the distribution having infinite variance
27
Characteristic function
28
Gaussian pdf and its
characteristic function
CchaosTalklevy01FourierTransformPairs
cdf
29
Characteristic function as a
moment generating function
30
CauchyndashLorentz distribution
CauchyndashLorentz distribution
Characteristic function
Observe that the characteristic function is not
differentiable at the origin So the Cauchy
distribution does not have an expected value
or Variance31
Generalized central limit
theorem
A generalization due to Gnedenko and
Kolmogorov states that the sum of a number
of random variables with power-law tail
distributions decreasing as 1 | x | α + 1 where
0 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
32
Stable distribution
In probability theory a random variable is
said to be stable (or to have a stable
distribution) if it has the property that a linear
combination of two independent copies of the
variable has the same distribution up to
location and scale parameters
The stable distribution family is also
sometimes referred to as the Leacutevy alpha-
stable distribution
33
Such distributions form a four-parameter
family of continuous probability distributions
parametrized by location and scale
parameters μ and c respectively and two
shape parameters β and α roughly
corresponding to measures of asymmetry
and concentration respectively (see the
figures)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
35
36
Symmetric α-stable distributions
with unit scale factor
37
Skewed centered stable
distributions with different β
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ the
skewness parameter β has no effect
The asymptotic behavior is described for α lt 2
Unified normal and power law
38
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
Another viewpoint
For a given mean and variance the
corresponding normal distribution is the
continuous distribution with the maximum
entropy
Therefore the assumption of normality
imposes the minimal prior structural
constraint beyond these moments14
Summary of normal
distribution
First the normal distribution is very tractable
analytically that is a large number of results
involving this distribution can be derived in
explicit form
15
Summary of normal
distribution
Second the normal distribution arises as the
outcome of the central limit theorem which
states that under mild conditions the sum of a
large number of random variables is
distributed approximately normally
Finally the bell shape of the normal
distribution makes it a convenient choice for
modelling a large variety of random variables
encountered in practice
16
17
POWER LAW
Part 3
18
Income distribution
19
20
Normal vs Power law
You can hardly find a person twice as tall as
you
Fair enoughhellip
This is normal distribution
But you can easily find a person 10000 times
richer than youhellip
Extremely unfairhellip
This is power law distribution
21
Examples of power laws
a Word frequency Estoup
b Citations of scientific papers Price
c Web hits Adamic and Huberman
d Copies of books sold
e Diameter of moon craters Neukum amp Ivanov
f Intensity of solar flares Lu and Hamilton
g Intensity of wars Small and Singer
h Wealth of the richest people
i Frequencies of family names eg US amp Japan not Korea
j Populations of cities
23
The Power Law Phenomenon
Most nodes have the same number of links
No highly connected nodes
Many nodes with few links
A few nodes with many links
of links (k) of links (k)
No
of
node
s w
ith
k li
nks
No
of
node
s w
ith
k li
nks
Bell CurvePower Law Distribution
GENERALIZED CENTRAL LIMIT THEOREM
Part 4
25
Attraction basin of Gaussian
The central limit theorem states that the sum
of a number of independent and identically
distributed (iid) random variables with finite
variances will tend to a normal distribution as
the number of variables grows
All distribution with finite variance form the
attraction basin of Gaunssian
26
what about the distribution having infinite variance
27
Characteristic function
28
Gaussian pdf and its
characteristic function
CchaosTalklevy01FourierTransformPairs
cdf
29
Characteristic function as a
moment generating function
30
CauchyndashLorentz distribution
CauchyndashLorentz distribution
Characteristic function
Observe that the characteristic function is not
differentiable at the origin So the Cauchy
distribution does not have an expected value
or Variance31
Generalized central limit
theorem
A generalization due to Gnedenko and
Kolmogorov states that the sum of a number
of random variables with power-law tail
distributions decreasing as 1 | x | α + 1 where
0 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
32
Stable distribution
In probability theory a random variable is
said to be stable (or to have a stable
distribution) if it has the property that a linear
combination of two independent copies of the
variable has the same distribution up to
location and scale parameters
The stable distribution family is also
sometimes referred to as the Leacutevy alpha-
stable distribution
33
Such distributions form a four-parameter
family of continuous probability distributions
parametrized by location and scale
parameters μ and c respectively and two
shape parameters β and α roughly
corresponding to measures of asymmetry
and concentration respectively (see the
figures)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
35
36
Symmetric α-stable distributions
with unit scale factor
37
Skewed centered stable
distributions with different β
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ the
skewness parameter β has no effect
The asymptotic behavior is described for α lt 2
Unified normal and power law
38
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
Summary of normal
distribution
First the normal distribution is very tractable
analytically that is a large number of results
involving this distribution can be derived in
explicit form
15
Summary of normal
distribution
Second the normal distribution arises as the
outcome of the central limit theorem which
states that under mild conditions the sum of a
large number of random variables is
distributed approximately normally
Finally the bell shape of the normal
distribution makes it a convenient choice for
modelling a large variety of random variables
encountered in practice
16
17
POWER LAW
Part 3
18
Income distribution
19
20
Normal vs Power law
You can hardly find a person twice as tall as
you
Fair enoughhellip
This is normal distribution
But you can easily find a person 10000 times
richer than youhellip
Extremely unfairhellip
This is power law distribution
21
Examples of power laws
a Word frequency Estoup
b Citations of scientific papers Price
c Web hits Adamic and Huberman
d Copies of books sold
e Diameter of moon craters Neukum amp Ivanov
f Intensity of solar flares Lu and Hamilton
g Intensity of wars Small and Singer
h Wealth of the richest people
i Frequencies of family names eg US amp Japan not Korea
j Populations of cities
23
The Power Law Phenomenon
Most nodes have the same number of links
No highly connected nodes
Many nodes with few links
A few nodes with many links
of links (k) of links (k)
No
of
node
s w
ith
k li
nks
No
of
node
s w
ith
k li
nks
Bell CurvePower Law Distribution
GENERALIZED CENTRAL LIMIT THEOREM
Part 4
25
Attraction basin of Gaussian
The central limit theorem states that the sum
of a number of independent and identically
distributed (iid) random variables with finite
variances will tend to a normal distribution as
the number of variables grows
All distribution with finite variance form the
attraction basin of Gaunssian
26
what about the distribution having infinite variance
27
Characteristic function
28
Gaussian pdf and its
characteristic function
CchaosTalklevy01FourierTransformPairs
cdf
29
Characteristic function as a
moment generating function
30
CauchyndashLorentz distribution
CauchyndashLorentz distribution
Characteristic function
Observe that the characteristic function is not
differentiable at the origin So the Cauchy
distribution does not have an expected value
or Variance31
Generalized central limit
theorem
A generalization due to Gnedenko and
Kolmogorov states that the sum of a number
of random variables with power-law tail
distributions decreasing as 1 | x | α + 1 where
0 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
32
Stable distribution
In probability theory a random variable is
said to be stable (or to have a stable
distribution) if it has the property that a linear
combination of two independent copies of the
variable has the same distribution up to
location and scale parameters
The stable distribution family is also
sometimes referred to as the Leacutevy alpha-
stable distribution
33
Such distributions form a four-parameter
family of continuous probability distributions
parametrized by location and scale
parameters μ and c respectively and two
shape parameters β and α roughly
corresponding to measures of asymmetry
and concentration respectively (see the
figures)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
35
36
Symmetric α-stable distributions
with unit scale factor
37
Skewed centered stable
distributions with different β
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ the
skewness parameter β has no effect
The asymptotic behavior is described for α lt 2
Unified normal and power law
38
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
Summary of normal
distribution
Second the normal distribution arises as the
outcome of the central limit theorem which
states that under mild conditions the sum of a
large number of random variables is
distributed approximately normally
Finally the bell shape of the normal
distribution makes it a convenient choice for
modelling a large variety of random variables
encountered in practice
16
17
POWER LAW
Part 3
18
Income distribution
19
20
Normal vs Power law
You can hardly find a person twice as tall as
you
Fair enoughhellip
This is normal distribution
But you can easily find a person 10000 times
richer than youhellip
Extremely unfairhellip
This is power law distribution
21
Examples of power laws
a Word frequency Estoup
b Citations of scientific papers Price
c Web hits Adamic and Huberman
d Copies of books sold
e Diameter of moon craters Neukum amp Ivanov
f Intensity of solar flares Lu and Hamilton
g Intensity of wars Small and Singer
h Wealth of the richest people
i Frequencies of family names eg US amp Japan not Korea
j Populations of cities
23
The Power Law Phenomenon
Most nodes have the same number of links
No highly connected nodes
Many nodes with few links
A few nodes with many links
of links (k) of links (k)
No
of
node
s w
ith
k li
nks
No
of
node
s w
ith
k li
nks
Bell CurvePower Law Distribution
GENERALIZED CENTRAL LIMIT THEOREM
Part 4
25
Attraction basin of Gaussian
The central limit theorem states that the sum
of a number of independent and identically
distributed (iid) random variables with finite
variances will tend to a normal distribution as
the number of variables grows
All distribution with finite variance form the
attraction basin of Gaunssian
26
what about the distribution having infinite variance
27
Characteristic function
28
Gaussian pdf and its
characteristic function
CchaosTalklevy01FourierTransformPairs
cdf
29
Characteristic function as a
moment generating function
30
CauchyndashLorentz distribution
CauchyndashLorentz distribution
Characteristic function
Observe that the characteristic function is not
differentiable at the origin So the Cauchy
distribution does not have an expected value
or Variance31
Generalized central limit
theorem
A generalization due to Gnedenko and
Kolmogorov states that the sum of a number
of random variables with power-law tail
distributions decreasing as 1 | x | α + 1 where
0 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
32
Stable distribution
In probability theory a random variable is
said to be stable (or to have a stable
distribution) if it has the property that a linear
combination of two independent copies of the
variable has the same distribution up to
location and scale parameters
The stable distribution family is also
sometimes referred to as the Leacutevy alpha-
stable distribution
33
Such distributions form a four-parameter
family of continuous probability distributions
parametrized by location and scale
parameters μ and c respectively and two
shape parameters β and α roughly
corresponding to measures of asymmetry
and concentration respectively (see the
figures)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
35
36
Symmetric α-stable distributions
with unit scale factor
37
Skewed centered stable
distributions with different β
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ the
skewness parameter β has no effect
The asymptotic behavior is described for α lt 2
Unified normal and power law
38
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
17
POWER LAW
Part 3
18
Income distribution
19
20
Normal vs Power law
You can hardly find a person twice as tall as
you
Fair enoughhellip
This is normal distribution
But you can easily find a person 10000 times
richer than youhellip
Extremely unfairhellip
This is power law distribution
21
Examples of power laws
a Word frequency Estoup
b Citations of scientific papers Price
c Web hits Adamic and Huberman
d Copies of books sold
e Diameter of moon craters Neukum amp Ivanov
f Intensity of solar flares Lu and Hamilton
g Intensity of wars Small and Singer
h Wealth of the richest people
i Frequencies of family names eg US amp Japan not Korea
j Populations of cities
23
The Power Law Phenomenon
Most nodes have the same number of links
No highly connected nodes
Many nodes with few links
A few nodes with many links
of links (k) of links (k)
No
of
node
s w
ith
k li
nks
No
of
node
s w
ith
k li
nks
Bell CurvePower Law Distribution
GENERALIZED CENTRAL LIMIT THEOREM
Part 4
25
Attraction basin of Gaussian
The central limit theorem states that the sum
of a number of independent and identically
distributed (iid) random variables with finite
variances will tend to a normal distribution as
the number of variables grows
All distribution with finite variance form the
attraction basin of Gaunssian
26
what about the distribution having infinite variance
27
Characteristic function
28
Gaussian pdf and its
characteristic function
CchaosTalklevy01FourierTransformPairs
cdf
29
Characteristic function as a
moment generating function
30
CauchyndashLorentz distribution
CauchyndashLorentz distribution
Characteristic function
Observe that the characteristic function is not
differentiable at the origin So the Cauchy
distribution does not have an expected value
or Variance31
Generalized central limit
theorem
A generalization due to Gnedenko and
Kolmogorov states that the sum of a number
of random variables with power-law tail
distributions decreasing as 1 | x | α + 1 where
0 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
32
Stable distribution
In probability theory a random variable is
said to be stable (or to have a stable
distribution) if it has the property that a linear
combination of two independent copies of the
variable has the same distribution up to
location and scale parameters
The stable distribution family is also
sometimes referred to as the Leacutevy alpha-
stable distribution
33
Such distributions form a four-parameter
family of continuous probability distributions
parametrized by location and scale
parameters μ and c respectively and two
shape parameters β and α roughly
corresponding to measures of asymmetry
and concentration respectively (see the
figures)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
35
36
Symmetric α-stable distributions
with unit scale factor
37
Skewed centered stable
distributions with different β
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ the
skewness parameter β has no effect
The asymptotic behavior is described for α lt 2
Unified normal and power law
38
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
POWER LAW
Part 3
18
Income distribution
19
20
Normal vs Power law
You can hardly find a person twice as tall as
you
Fair enoughhellip
This is normal distribution
But you can easily find a person 10000 times
richer than youhellip
Extremely unfairhellip
This is power law distribution
21
Examples of power laws
a Word frequency Estoup
b Citations of scientific papers Price
c Web hits Adamic and Huberman
d Copies of books sold
e Diameter of moon craters Neukum amp Ivanov
f Intensity of solar flares Lu and Hamilton
g Intensity of wars Small and Singer
h Wealth of the richest people
i Frequencies of family names eg US amp Japan not Korea
j Populations of cities
23
The Power Law Phenomenon
Most nodes have the same number of links
No highly connected nodes
Many nodes with few links
A few nodes with many links
of links (k) of links (k)
No
of
node
s w
ith
k li
nks
No
of
node
s w
ith
k li
nks
Bell CurvePower Law Distribution
GENERALIZED CENTRAL LIMIT THEOREM
Part 4
25
Attraction basin of Gaussian
The central limit theorem states that the sum
of a number of independent and identically
distributed (iid) random variables with finite
variances will tend to a normal distribution as
the number of variables grows
All distribution with finite variance form the
attraction basin of Gaunssian
26
what about the distribution having infinite variance
27
Characteristic function
28
Gaussian pdf and its
characteristic function
CchaosTalklevy01FourierTransformPairs
cdf
29
Characteristic function as a
moment generating function
30
CauchyndashLorentz distribution
CauchyndashLorentz distribution
Characteristic function
Observe that the characteristic function is not
differentiable at the origin So the Cauchy
distribution does not have an expected value
or Variance31
Generalized central limit
theorem
A generalization due to Gnedenko and
Kolmogorov states that the sum of a number
of random variables with power-law tail
distributions decreasing as 1 | x | α + 1 where
0 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
32
Stable distribution
In probability theory a random variable is
said to be stable (or to have a stable
distribution) if it has the property that a linear
combination of two independent copies of the
variable has the same distribution up to
location and scale parameters
The stable distribution family is also
sometimes referred to as the Leacutevy alpha-
stable distribution
33
Such distributions form a four-parameter
family of continuous probability distributions
parametrized by location and scale
parameters μ and c respectively and two
shape parameters β and α roughly
corresponding to measures of asymmetry
and concentration respectively (see the
figures)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
35
36
Symmetric α-stable distributions
with unit scale factor
37
Skewed centered stable
distributions with different β
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ the
skewness parameter β has no effect
The asymptotic behavior is described for α lt 2
Unified normal and power law
38
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
Income distribution
19
20
Normal vs Power law
You can hardly find a person twice as tall as
you
Fair enoughhellip
This is normal distribution
But you can easily find a person 10000 times
richer than youhellip
Extremely unfairhellip
This is power law distribution
21
Examples of power laws
a Word frequency Estoup
b Citations of scientific papers Price
c Web hits Adamic and Huberman
d Copies of books sold
e Diameter of moon craters Neukum amp Ivanov
f Intensity of solar flares Lu and Hamilton
g Intensity of wars Small and Singer
h Wealth of the richest people
i Frequencies of family names eg US amp Japan not Korea
j Populations of cities
23
The Power Law Phenomenon
Most nodes have the same number of links
No highly connected nodes
Many nodes with few links
A few nodes with many links
of links (k) of links (k)
No
of
node
s w
ith
k li
nks
No
of
node
s w
ith
k li
nks
Bell CurvePower Law Distribution
GENERALIZED CENTRAL LIMIT THEOREM
Part 4
25
Attraction basin of Gaussian
The central limit theorem states that the sum
of a number of independent and identically
distributed (iid) random variables with finite
variances will tend to a normal distribution as
the number of variables grows
All distribution with finite variance form the
attraction basin of Gaunssian
26
what about the distribution having infinite variance
27
Characteristic function
28
Gaussian pdf and its
characteristic function
CchaosTalklevy01FourierTransformPairs
cdf
29
Characteristic function as a
moment generating function
30
CauchyndashLorentz distribution
CauchyndashLorentz distribution
Characteristic function
Observe that the characteristic function is not
differentiable at the origin So the Cauchy
distribution does not have an expected value
or Variance31
Generalized central limit
theorem
A generalization due to Gnedenko and
Kolmogorov states that the sum of a number
of random variables with power-law tail
distributions decreasing as 1 | x | α + 1 where
0 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
32
Stable distribution
In probability theory a random variable is
said to be stable (or to have a stable
distribution) if it has the property that a linear
combination of two independent copies of the
variable has the same distribution up to
location and scale parameters
The stable distribution family is also
sometimes referred to as the Leacutevy alpha-
stable distribution
33
Such distributions form a four-parameter
family of continuous probability distributions
parametrized by location and scale
parameters μ and c respectively and two
shape parameters β and α roughly
corresponding to measures of asymmetry
and concentration respectively (see the
figures)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
35
36
Symmetric α-stable distributions
with unit scale factor
37
Skewed centered stable
distributions with different β
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ the
skewness parameter β has no effect
The asymptotic behavior is described for α lt 2
Unified normal and power law
38
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
20
Normal vs Power law
You can hardly find a person twice as tall as
you
Fair enoughhellip
This is normal distribution
But you can easily find a person 10000 times
richer than youhellip
Extremely unfairhellip
This is power law distribution
21
Examples of power laws
a Word frequency Estoup
b Citations of scientific papers Price
c Web hits Adamic and Huberman
d Copies of books sold
e Diameter of moon craters Neukum amp Ivanov
f Intensity of solar flares Lu and Hamilton
g Intensity of wars Small and Singer
h Wealth of the richest people
i Frequencies of family names eg US amp Japan not Korea
j Populations of cities
23
The Power Law Phenomenon
Most nodes have the same number of links
No highly connected nodes
Many nodes with few links
A few nodes with many links
of links (k) of links (k)
No
of
node
s w
ith
k li
nks
No
of
node
s w
ith
k li
nks
Bell CurvePower Law Distribution
GENERALIZED CENTRAL LIMIT THEOREM
Part 4
25
Attraction basin of Gaussian
The central limit theorem states that the sum
of a number of independent and identically
distributed (iid) random variables with finite
variances will tend to a normal distribution as
the number of variables grows
All distribution with finite variance form the
attraction basin of Gaunssian
26
what about the distribution having infinite variance
27
Characteristic function
28
Gaussian pdf and its
characteristic function
CchaosTalklevy01FourierTransformPairs
cdf
29
Characteristic function as a
moment generating function
30
CauchyndashLorentz distribution
CauchyndashLorentz distribution
Characteristic function
Observe that the characteristic function is not
differentiable at the origin So the Cauchy
distribution does not have an expected value
or Variance31
Generalized central limit
theorem
A generalization due to Gnedenko and
Kolmogorov states that the sum of a number
of random variables with power-law tail
distributions decreasing as 1 | x | α + 1 where
0 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
32
Stable distribution
In probability theory a random variable is
said to be stable (or to have a stable
distribution) if it has the property that a linear
combination of two independent copies of the
variable has the same distribution up to
location and scale parameters
The stable distribution family is also
sometimes referred to as the Leacutevy alpha-
stable distribution
33
Such distributions form a four-parameter
family of continuous probability distributions
parametrized by location and scale
parameters μ and c respectively and two
shape parameters β and α roughly
corresponding to measures of asymmetry
and concentration respectively (see the
figures)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
35
36
Symmetric α-stable distributions
with unit scale factor
37
Skewed centered stable
distributions with different β
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ the
skewness parameter β has no effect
The asymptotic behavior is described for α lt 2
Unified normal and power law
38
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
Normal vs Power law
You can hardly find a person twice as tall as
you
Fair enoughhellip
This is normal distribution
But you can easily find a person 10000 times
richer than youhellip
Extremely unfairhellip
This is power law distribution
21
Examples of power laws
a Word frequency Estoup
b Citations of scientific papers Price
c Web hits Adamic and Huberman
d Copies of books sold
e Diameter of moon craters Neukum amp Ivanov
f Intensity of solar flares Lu and Hamilton
g Intensity of wars Small and Singer
h Wealth of the richest people
i Frequencies of family names eg US amp Japan not Korea
j Populations of cities
23
The Power Law Phenomenon
Most nodes have the same number of links
No highly connected nodes
Many nodes with few links
A few nodes with many links
of links (k) of links (k)
No
of
node
s w
ith
k li
nks
No
of
node
s w
ith
k li
nks
Bell CurvePower Law Distribution
GENERALIZED CENTRAL LIMIT THEOREM
Part 4
25
Attraction basin of Gaussian
The central limit theorem states that the sum
of a number of independent and identically
distributed (iid) random variables with finite
variances will tend to a normal distribution as
the number of variables grows
All distribution with finite variance form the
attraction basin of Gaunssian
26
what about the distribution having infinite variance
27
Characteristic function
28
Gaussian pdf and its
characteristic function
CchaosTalklevy01FourierTransformPairs
cdf
29
Characteristic function as a
moment generating function
30
CauchyndashLorentz distribution
CauchyndashLorentz distribution
Characteristic function
Observe that the characteristic function is not
differentiable at the origin So the Cauchy
distribution does not have an expected value
or Variance31
Generalized central limit
theorem
A generalization due to Gnedenko and
Kolmogorov states that the sum of a number
of random variables with power-law tail
distributions decreasing as 1 | x | α + 1 where
0 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
32
Stable distribution
In probability theory a random variable is
said to be stable (or to have a stable
distribution) if it has the property that a linear
combination of two independent copies of the
variable has the same distribution up to
location and scale parameters
The stable distribution family is also
sometimes referred to as the Leacutevy alpha-
stable distribution
33
Such distributions form a four-parameter
family of continuous probability distributions
parametrized by location and scale
parameters μ and c respectively and two
shape parameters β and α roughly
corresponding to measures of asymmetry
and concentration respectively (see the
figures)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
35
36
Symmetric α-stable distributions
with unit scale factor
37
Skewed centered stable
distributions with different β
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ the
skewness parameter β has no effect
The asymptotic behavior is described for α lt 2
Unified normal and power law
38
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
Examples of power laws
a Word frequency Estoup
b Citations of scientific papers Price
c Web hits Adamic and Huberman
d Copies of books sold
e Diameter of moon craters Neukum amp Ivanov
f Intensity of solar flares Lu and Hamilton
g Intensity of wars Small and Singer
h Wealth of the richest people
i Frequencies of family names eg US amp Japan not Korea
j Populations of cities
23
The Power Law Phenomenon
Most nodes have the same number of links
No highly connected nodes
Many nodes with few links
A few nodes with many links
of links (k) of links (k)
No
of
node
s w
ith
k li
nks
No
of
node
s w
ith
k li
nks
Bell CurvePower Law Distribution
GENERALIZED CENTRAL LIMIT THEOREM
Part 4
25
Attraction basin of Gaussian
The central limit theorem states that the sum
of a number of independent and identically
distributed (iid) random variables with finite
variances will tend to a normal distribution as
the number of variables grows
All distribution with finite variance form the
attraction basin of Gaunssian
26
what about the distribution having infinite variance
27
Characteristic function
28
Gaussian pdf and its
characteristic function
CchaosTalklevy01FourierTransformPairs
cdf
29
Characteristic function as a
moment generating function
30
CauchyndashLorentz distribution
CauchyndashLorentz distribution
Characteristic function
Observe that the characteristic function is not
differentiable at the origin So the Cauchy
distribution does not have an expected value
or Variance31
Generalized central limit
theorem
A generalization due to Gnedenko and
Kolmogorov states that the sum of a number
of random variables with power-law tail
distributions decreasing as 1 | x | α + 1 where
0 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
32
Stable distribution
In probability theory a random variable is
said to be stable (or to have a stable
distribution) if it has the property that a linear
combination of two independent copies of the
variable has the same distribution up to
location and scale parameters
The stable distribution family is also
sometimes referred to as the Leacutevy alpha-
stable distribution
33
Such distributions form a four-parameter
family of continuous probability distributions
parametrized by location and scale
parameters μ and c respectively and two
shape parameters β and α roughly
corresponding to measures of asymmetry
and concentration respectively (see the
figures)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
35
36
Symmetric α-stable distributions
with unit scale factor
37
Skewed centered stable
distributions with different β
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ the
skewness parameter β has no effect
The asymptotic behavior is described for α lt 2
Unified normal and power law
38
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
23
The Power Law Phenomenon
Most nodes have the same number of links
No highly connected nodes
Many nodes with few links
A few nodes with many links
of links (k) of links (k)
No
of
node
s w
ith
k li
nks
No
of
node
s w
ith
k li
nks
Bell CurvePower Law Distribution
GENERALIZED CENTRAL LIMIT THEOREM
Part 4
25
Attraction basin of Gaussian
The central limit theorem states that the sum
of a number of independent and identically
distributed (iid) random variables with finite
variances will tend to a normal distribution as
the number of variables grows
All distribution with finite variance form the
attraction basin of Gaunssian
26
what about the distribution having infinite variance
27
Characteristic function
28
Gaussian pdf and its
characteristic function
CchaosTalklevy01FourierTransformPairs
cdf
29
Characteristic function as a
moment generating function
30
CauchyndashLorentz distribution
CauchyndashLorentz distribution
Characteristic function
Observe that the characteristic function is not
differentiable at the origin So the Cauchy
distribution does not have an expected value
or Variance31
Generalized central limit
theorem
A generalization due to Gnedenko and
Kolmogorov states that the sum of a number
of random variables with power-law tail
distributions decreasing as 1 | x | α + 1 where
0 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
32
Stable distribution
In probability theory a random variable is
said to be stable (or to have a stable
distribution) if it has the property that a linear
combination of two independent copies of the
variable has the same distribution up to
location and scale parameters
The stable distribution family is also
sometimes referred to as the Leacutevy alpha-
stable distribution
33
Such distributions form a four-parameter
family of continuous probability distributions
parametrized by location and scale
parameters μ and c respectively and two
shape parameters β and α roughly
corresponding to measures of asymmetry
and concentration respectively (see the
figures)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
35
36
Symmetric α-stable distributions
with unit scale factor
37
Skewed centered stable
distributions with different β
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ the
skewness parameter β has no effect
The asymptotic behavior is described for α lt 2
Unified normal and power law
38
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
The Power Law Phenomenon
Most nodes have the same number of links
No highly connected nodes
Many nodes with few links
A few nodes with many links
of links (k) of links (k)
No
of
node
s w
ith
k li
nks
No
of
node
s w
ith
k li
nks
Bell CurvePower Law Distribution
GENERALIZED CENTRAL LIMIT THEOREM
Part 4
25
Attraction basin of Gaussian
The central limit theorem states that the sum
of a number of independent and identically
distributed (iid) random variables with finite
variances will tend to a normal distribution as
the number of variables grows
All distribution with finite variance form the
attraction basin of Gaunssian
26
what about the distribution having infinite variance
27
Characteristic function
28
Gaussian pdf and its
characteristic function
CchaosTalklevy01FourierTransformPairs
cdf
29
Characteristic function as a
moment generating function
30
CauchyndashLorentz distribution
CauchyndashLorentz distribution
Characteristic function
Observe that the characteristic function is not
differentiable at the origin So the Cauchy
distribution does not have an expected value
or Variance31
Generalized central limit
theorem
A generalization due to Gnedenko and
Kolmogorov states that the sum of a number
of random variables with power-law tail
distributions decreasing as 1 | x | α + 1 where
0 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
32
Stable distribution
In probability theory a random variable is
said to be stable (or to have a stable
distribution) if it has the property that a linear
combination of two independent copies of the
variable has the same distribution up to
location and scale parameters
The stable distribution family is also
sometimes referred to as the Leacutevy alpha-
stable distribution
33
Such distributions form a four-parameter
family of continuous probability distributions
parametrized by location and scale
parameters μ and c respectively and two
shape parameters β and α roughly
corresponding to measures of asymmetry
and concentration respectively (see the
figures)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
35
36
Symmetric α-stable distributions
with unit scale factor
37
Skewed centered stable
distributions with different β
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ the
skewness parameter β has no effect
The asymptotic behavior is described for α lt 2
Unified normal and power law
38
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
GENERALIZED CENTRAL LIMIT THEOREM
Part 4
25
Attraction basin of Gaussian
The central limit theorem states that the sum
of a number of independent and identically
distributed (iid) random variables with finite
variances will tend to a normal distribution as
the number of variables grows
All distribution with finite variance form the
attraction basin of Gaunssian
26
what about the distribution having infinite variance
27
Characteristic function
28
Gaussian pdf and its
characteristic function
CchaosTalklevy01FourierTransformPairs
cdf
29
Characteristic function as a
moment generating function
30
CauchyndashLorentz distribution
CauchyndashLorentz distribution
Characteristic function
Observe that the characteristic function is not
differentiable at the origin So the Cauchy
distribution does not have an expected value
or Variance31
Generalized central limit
theorem
A generalization due to Gnedenko and
Kolmogorov states that the sum of a number
of random variables with power-law tail
distributions decreasing as 1 | x | α + 1 where
0 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
32
Stable distribution
In probability theory a random variable is
said to be stable (or to have a stable
distribution) if it has the property that a linear
combination of two independent copies of the
variable has the same distribution up to
location and scale parameters
The stable distribution family is also
sometimes referred to as the Leacutevy alpha-
stable distribution
33
Such distributions form a four-parameter
family of continuous probability distributions
parametrized by location and scale
parameters μ and c respectively and two
shape parameters β and α roughly
corresponding to measures of asymmetry
and concentration respectively (see the
figures)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
35
36
Symmetric α-stable distributions
with unit scale factor
37
Skewed centered stable
distributions with different β
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ the
skewness parameter β has no effect
The asymptotic behavior is described for α lt 2
Unified normal and power law
38
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
Attraction basin of Gaussian
The central limit theorem states that the sum
of a number of independent and identically
distributed (iid) random variables with finite
variances will tend to a normal distribution as
the number of variables grows
All distribution with finite variance form the
attraction basin of Gaunssian
26
what about the distribution having infinite variance
27
Characteristic function
28
Gaussian pdf and its
characteristic function
CchaosTalklevy01FourierTransformPairs
cdf
29
Characteristic function as a
moment generating function
30
CauchyndashLorentz distribution
CauchyndashLorentz distribution
Characteristic function
Observe that the characteristic function is not
differentiable at the origin So the Cauchy
distribution does not have an expected value
or Variance31
Generalized central limit
theorem
A generalization due to Gnedenko and
Kolmogorov states that the sum of a number
of random variables with power-law tail
distributions decreasing as 1 | x | α + 1 where
0 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
32
Stable distribution
In probability theory a random variable is
said to be stable (or to have a stable
distribution) if it has the property that a linear
combination of two independent copies of the
variable has the same distribution up to
location and scale parameters
The stable distribution family is also
sometimes referred to as the Leacutevy alpha-
stable distribution
33
Such distributions form a four-parameter
family of continuous probability distributions
parametrized by location and scale
parameters μ and c respectively and two
shape parameters β and α roughly
corresponding to measures of asymmetry
and concentration respectively (see the
figures)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
35
36
Symmetric α-stable distributions
with unit scale factor
37
Skewed centered stable
distributions with different β
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ the
skewness parameter β has no effect
The asymptotic behavior is described for α lt 2
Unified normal and power law
38
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
27
Characteristic function
28
Gaussian pdf and its
characteristic function
CchaosTalklevy01FourierTransformPairs
cdf
29
Characteristic function as a
moment generating function
30
CauchyndashLorentz distribution
CauchyndashLorentz distribution
Characteristic function
Observe that the characteristic function is not
differentiable at the origin So the Cauchy
distribution does not have an expected value
or Variance31
Generalized central limit
theorem
A generalization due to Gnedenko and
Kolmogorov states that the sum of a number
of random variables with power-law tail
distributions decreasing as 1 | x | α + 1 where
0 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
32
Stable distribution
In probability theory a random variable is
said to be stable (or to have a stable
distribution) if it has the property that a linear
combination of two independent copies of the
variable has the same distribution up to
location and scale parameters
The stable distribution family is also
sometimes referred to as the Leacutevy alpha-
stable distribution
33
Such distributions form a four-parameter
family of continuous probability distributions
parametrized by location and scale
parameters μ and c respectively and two
shape parameters β and α roughly
corresponding to measures of asymmetry
and concentration respectively (see the
figures)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
35
36
Symmetric α-stable distributions
with unit scale factor
37
Skewed centered stable
distributions with different β
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ the
skewness parameter β has no effect
The asymptotic behavior is described for α lt 2
Unified normal and power law
38
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
28
Gaussian pdf and its
characteristic function
CchaosTalklevy01FourierTransformPairs
cdf
29
Characteristic function as a
moment generating function
30
CauchyndashLorentz distribution
CauchyndashLorentz distribution
Characteristic function
Observe that the characteristic function is not
differentiable at the origin So the Cauchy
distribution does not have an expected value
or Variance31
Generalized central limit
theorem
A generalization due to Gnedenko and
Kolmogorov states that the sum of a number
of random variables with power-law tail
distributions decreasing as 1 | x | α + 1 where
0 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
32
Stable distribution
In probability theory a random variable is
said to be stable (or to have a stable
distribution) if it has the property that a linear
combination of two independent copies of the
variable has the same distribution up to
location and scale parameters
The stable distribution family is also
sometimes referred to as the Leacutevy alpha-
stable distribution
33
Such distributions form a four-parameter
family of continuous probability distributions
parametrized by location and scale
parameters μ and c respectively and two
shape parameters β and α roughly
corresponding to measures of asymmetry
and concentration respectively (see the
figures)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
35
36
Symmetric α-stable distributions
with unit scale factor
37
Skewed centered stable
distributions with different β
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ the
skewness parameter β has no effect
The asymptotic behavior is described for α lt 2
Unified normal and power law
38
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
29
Characteristic function as a
moment generating function
30
CauchyndashLorentz distribution
CauchyndashLorentz distribution
Characteristic function
Observe that the characteristic function is not
differentiable at the origin So the Cauchy
distribution does not have an expected value
or Variance31
Generalized central limit
theorem
A generalization due to Gnedenko and
Kolmogorov states that the sum of a number
of random variables with power-law tail
distributions decreasing as 1 | x | α + 1 where
0 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
32
Stable distribution
In probability theory a random variable is
said to be stable (or to have a stable
distribution) if it has the property that a linear
combination of two independent copies of the
variable has the same distribution up to
location and scale parameters
The stable distribution family is also
sometimes referred to as the Leacutevy alpha-
stable distribution
33
Such distributions form a four-parameter
family of continuous probability distributions
parametrized by location and scale
parameters μ and c respectively and two
shape parameters β and α roughly
corresponding to measures of asymmetry
and concentration respectively (see the
figures)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
35
36
Symmetric α-stable distributions
with unit scale factor
37
Skewed centered stable
distributions with different β
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ the
skewness parameter β has no effect
The asymptotic behavior is described for α lt 2
Unified normal and power law
38
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
30
CauchyndashLorentz distribution
CauchyndashLorentz distribution
Characteristic function
Observe that the characteristic function is not
differentiable at the origin So the Cauchy
distribution does not have an expected value
or Variance31
Generalized central limit
theorem
A generalization due to Gnedenko and
Kolmogorov states that the sum of a number
of random variables with power-law tail
distributions decreasing as 1 | x | α + 1 where
0 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
32
Stable distribution
In probability theory a random variable is
said to be stable (or to have a stable
distribution) if it has the property that a linear
combination of two independent copies of the
variable has the same distribution up to
location and scale parameters
The stable distribution family is also
sometimes referred to as the Leacutevy alpha-
stable distribution
33
Such distributions form a four-parameter
family of continuous probability distributions
parametrized by location and scale
parameters μ and c respectively and two
shape parameters β and α roughly
corresponding to measures of asymmetry
and concentration respectively (see the
figures)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
35
36
Symmetric α-stable distributions
with unit scale factor
37
Skewed centered stable
distributions with different β
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ the
skewness parameter β has no effect
The asymptotic behavior is described for α lt 2
Unified normal and power law
38
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
CauchyndashLorentz distribution
Characteristic function
Observe that the characteristic function is not
differentiable at the origin So the Cauchy
distribution does not have an expected value
or Variance31
Generalized central limit
theorem
A generalization due to Gnedenko and
Kolmogorov states that the sum of a number
of random variables with power-law tail
distributions decreasing as 1 | x | α + 1 where
0 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
32
Stable distribution
In probability theory a random variable is
said to be stable (or to have a stable
distribution) if it has the property that a linear
combination of two independent copies of the
variable has the same distribution up to
location and scale parameters
The stable distribution family is also
sometimes referred to as the Leacutevy alpha-
stable distribution
33
Such distributions form a four-parameter
family of continuous probability distributions
parametrized by location and scale
parameters μ and c respectively and two
shape parameters β and α roughly
corresponding to measures of asymmetry
and concentration respectively (see the
figures)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
35
36
Symmetric α-stable distributions
with unit scale factor
37
Skewed centered stable
distributions with different β
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ the
skewness parameter β has no effect
The asymptotic behavior is described for α lt 2
Unified normal and power law
38
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
Generalized central limit
theorem
A generalization due to Gnedenko and
Kolmogorov states that the sum of a number
of random variables with power-law tail
distributions decreasing as 1 | x | α + 1 where
0 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
32
Stable distribution
In probability theory a random variable is
said to be stable (or to have a stable
distribution) if it has the property that a linear
combination of two independent copies of the
variable has the same distribution up to
location and scale parameters
The stable distribution family is also
sometimes referred to as the Leacutevy alpha-
stable distribution
33
Such distributions form a four-parameter
family of continuous probability distributions
parametrized by location and scale
parameters μ and c respectively and two
shape parameters β and α roughly
corresponding to measures of asymmetry
and concentration respectively (see the
figures)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
35
36
Symmetric α-stable distributions
with unit scale factor
37
Skewed centered stable
distributions with different β
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ the
skewness parameter β has no effect
The asymptotic behavior is described for α lt 2
Unified normal and power law
38
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
Stable distribution
In probability theory a random variable is
said to be stable (or to have a stable
distribution) if it has the property that a linear
combination of two independent copies of the
variable has the same distribution up to
location and scale parameters
The stable distribution family is also
sometimes referred to as the Leacutevy alpha-
stable distribution
33
Such distributions form a four-parameter
family of continuous probability distributions
parametrized by location and scale
parameters μ and c respectively and two
shape parameters β and α roughly
corresponding to measures of asymmetry
and concentration respectively (see the
figures)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
35
36
Symmetric α-stable distributions
with unit scale factor
37
Skewed centered stable
distributions with different β
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ the
skewness parameter β has no effect
The asymptotic behavior is described for α lt 2
Unified normal and power law
38
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
Such distributions form a four-parameter
family of continuous probability distributions
parametrized by location and scale
parameters μ and c respectively and two
shape parameters β and α roughly
corresponding to measures of asymmetry
and concentration respectively (see the
figures)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
35
36
Symmetric α-stable distributions
with unit scale factor
37
Skewed centered stable
distributions with different β
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ the
skewness parameter β has no effect
The asymptotic behavior is described for α lt 2
Unified normal and power law
38
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
35
36
Symmetric α-stable distributions
with unit scale factor
37
Skewed centered stable
distributions with different β
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ the
skewness parameter β has no effect
The asymptotic behavior is described for α lt 2
Unified normal and power law
38
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
36
Symmetric α-stable distributions
with unit scale factor
37
Skewed centered stable
distributions with different β
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ the
skewness parameter β has no effect
The asymptotic behavior is described for α lt 2
Unified normal and power law
38
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
37
Skewed centered stable
distributions with different β
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ the
skewness parameter β has no effect
The asymptotic behavior is described for α lt 2
Unified normal and power law
38
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ the
skewness parameter β has no effect
The asymptotic behavior is described for α lt 2
Unified normal and power law
38
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
4040
Concluding Remarks
The importance of stable probability
distributions is that they are attractors for
properly normed sums of independent and
identically-distributed (iid) random variables
The normal distribution is one family of stable
distributions
Without the finite variance assumption the limit
may be a stable distribution which has the
power law behavior for large x
40
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
Analogy
Chenrsquos attractor family
At first Lorenz attractor
was found
Then a family of
attractors were found
which unified Lorenz and
Chen attractor
Stable distribution family
At first normal
distribution was found
Then a family of attractor
of distribution were found
which unified normal
distribution and power
law
For a long time this was thought as the only storyhellipThen a question raised naturallyhellip
Could there be any extension
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
4242
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
42
