Power laws and scale-free networks
10.11.2016
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Degree distribution
I
Hubs: well-connected vertices
Source: M. E. J. Newman, The structure and function of complex networks, 2003.
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Power-law distribution
Many real-world networks have very skewed degree distributions:
I
I
Power-law degree distribution
Linear on log-log plot
Random network: Poisson degree distribution
Will show that later
Source: Source: M. E. J. Newman, The structure and function of complex networks, 2003.
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Power-law distribution
A probability distribution pk on the natural numbers k ∈ N is
called power-law distribution if
pk ∝ k −α
with (tail) exponent α > 0
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Power-law distribution
A probability distribution pk on the natural numbers k ∈ N is
called power-law distribution if
pk ∝ k −α
with (tail) exponent α > 0
P
Normalization requires that ∞
k=0 pk = 1 and thus
(
0
for k = 0
pk =
k −α
ζ(α) for k > 0
where ζ(α)
Riemann zeta function defined as
P∞is the
−α
ζ(α) = k=1 k
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Power-law distribution
Continuous power-law distribution is often more convenient:
p(x) ∝ x −α
I
Minimum value xmin
I
Normalization given by
Z ∞
Z=
x −α dx =
xmin
for x ≥ xmin
∞
1
1
−α+1
−α+1 x
=
xmin
−α + 1
α
−
1
xmin
when α > 1
1
α−1
p(x) = x −α =
Z
xmin
x
−α
xmin
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Power-law distribution
Properties of power-law distributions:
I Linear on log-log plot
log pk = log ζ(α) − α log k
1e−01
pk
Distribution
Poisson
1e−03
Power law
1e−05
1
10
100
k
I
I
I
Note: A power-law distribution has heavy tails as the
probability decays much slower than exponential for k → ∞
Cumulative distribution function
Moments
Scale-free
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Power-law distribution
Properties of power-law distributions:
I
Linear on log-log plot
I
Cumulative distribution function
Z
1
1 ∞ 0−α 0
x dx =
x −(α−1)
P(X ≥ x) =
Z x
Z (α − 1)
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Moments
I
Scale-free
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Power-law distribution
Properties of power-law distributions:
I
Linear on log-log plot
I
Cumulative distribution function
I
Moments
Z
m
∞
x m p(x)dx
hx i =
xmin
=
=
Mean
Variance
I
1
Z
Z
∞
x m−α dx
xmin
1
x m−α+1
−Z (m − α + 1) min
hxi
hx 2 i − hxi2
hx m i
α>2
α>3
α>m+1
Scale-free
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Power-law distribution
Properties of power-law distributions:
I
Linear on log-log plot
I
Cumulative distribution function
I
Moments
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Scale-free
p(sx) =
I
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1
(sx)−α = s −α p(x) ∝ p(x)
Z
Distribution is self-similar
Data exhibits no typical scale
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Power-law fitting
How to detect power-laws in empirical data?
Log-log plot of empirical frequencies, i.e. pk ≈
nk
n
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pk
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1000
k
Note: Bad resolution in tail due to discrete counts
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Power-law fitting
How to detect power-laws in empirical data?
Log-log plot of empirical frequencies, i.e. pk ≈
1e+00
nk
n
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1e−02
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pk
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1e−04
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1e−06
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10
1000
k
Logarithmic binning gives better estimates of tail probabilities
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Power-law fitting
How to detect power-laws in empirical data?
nk
n
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Log-log plot of empirical frequencies, i.e. pk ≈
I
Log-log plot of empirical cumulative distribution:
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cum prob.
0.100
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10
1000
k
Note: Equivalent to rank-order plot as
P(X ≥ ki ) =
ri
n
where ri denotes the rank of data point ki
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Power-law fitting
Determine the (tail) exponent α:
I NOT recommended: Fit linear model to log-log data
I
I
Bad tail resolution of empirical counts
Ranks/cumulative probabilities are not independent
I
Empirical moments are always finite
True moments might not exist . . .
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Maximum likelihood estimation:
Continuous data
Discrete data
Statistical error: σ =
α̂ = 1 + N
hP
N
α̂ ≈ 1 + N
PN
α−1
√
N
xi
i=1 ln xmin
i=1 ln
i−1
ki
kmin − 12
−1
+ O( N1 )
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Power-law fitting
Determine the (tail) exponent α:
I NOT recommended: Fit linear model to log-log data
I
I
Bad tail resolution of empirical counts
Ranks/cumulative probabilities are not independent
I
Empirical moments are always finite
True moments might not exist . . .
I
Maximum likelihood estimation:
Continuous data
Discrete data
α̂ = 1 + N
hP
N
α̂ ≈ 1 + N
PN
xi
i=1 ln xmin
i=1 ln
i−1
ki
kmin − 12
−1
Often only tail, i.e. above kmin is fitted by power-law:
I
Difficult to determine best lower cut-off kmin
A. Clauset at al., Power-law distributions in empirical data, 2009.
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Scale-free networks
Networks with power-law degree distribution are commonly called
scale-free networks
I Exponents of real-world networks: 2 < α < 3
I Fraction w of edges connected to fraction p of highest degree
vertices
w = p (α−2)/(α−1)
Lorenz curves
100%
Fraction of edges w
75%
alpha = 2.1
alpha = 2.2
50%
alpha = 2.4
alpha = 2.7
alpha = 3.5
25%
0%
0%
25%
50%
75%
100%
Fraction of vertices p
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Scale-free networks
Networks with power-law degree distribution are commonly called
scale-free networks
I
Exponents of real-world networks: 2 < α < 3
I
Fraction w of edges connected to fraction p of highest degree
vertices
w = p (α−2)/(α−1)
α < 2 No such graph exists!
Note: w > 1, i.e. highest degree vertex needs
degree kmax > n
α > 3 hk 2 i is finite:
Many properties similar to random network
(Poisson distribution)
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