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Lognormals and friends Lognormals

Empirical Confusability Random Multiplicative Growth Model Random Growth with Variable Lifespan

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Lognormals and friends

Principles of Complex Systems Course 300, Fall, 2008

• Prof. Peter Dodds

Department of Mathematics & Statistics University of Vermont

Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.

Lognormals and friends Lognormals

Empirical Confusability Random Multiplicative Growth Model Random Growth with Variable Lifespan

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Outline

Lognormals Empirical Confusability Random Multiplicative Growth Model Random Growth with Variable Lifespan References

Lognormals and friends Lognormals

Empirical Confusability Random Multiplicative Growth Model Random Growth with Variable Lifespan

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Alternative distributions

There are other heavy-tailed distributions:

• 1. Lognormal
• 2. Stretched exponential (Weibull)
• 3. ... (Gamma)

Lognormals and friends Lognormals

Empirical Confusability Random Multiplicative Growth Model Random Growth with Variable Lifespan

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lognormals

The lognormal distribution:

P(x) = 1 x √ 2πσ exp

• −(ln x − µ)2

2σ2

• ◮ ln x is distributed according to a normal distribution

with mean µ and variance σ.

◮ Appears in economics and biology where growth

increments are distributed normally.

Lognormals and friends Lognormals

Empirical Confusability Random Multiplicative Growth Model Random Growth with Variable Lifespan

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lognormals

Standard form reveals the mean µ and variance σ2 of the underlying normal distribution: P(x) = 1 x √ 2πσ exp

• −(ln x − µ)2

2σ2

• For lognormals:

µlognormal = eµ+ 1

2σ2,

medianlognormal = eµ, σlognormal = (eσ2 − 1)e2µ+σ2, modelognormal = eµ−σ2. All moments of lognormals are finite.

Lognormals and friends Lognormals

Empirical Confusability Random Multiplicative Growth Model Random Growth with Variable Lifespan

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Derivation from a normal distribution

Take Y as distributed normally:

◮

P(y)dy = 1 √ 2πσ dy exp

• −(y − µ)2

2σ2

• Set Y = ln X:

◮ Transform according to P(x)dx = P(y)dy : ◮

dy dx = 1/x ⇒ dy = dx /x

◮

⇒ P(x)dx = 1 x √ 2πσ exp

• −(ln x − µ)2

2σ2

• dx

Lognormals and friends Lognormals

Empirical Confusability Random Multiplicative Growth Model Random Growth with Variable Lifespan

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Confusion between lognormals and pure power laws

2 4 6 8 10 −10 −8 −6 −4 −2

log10 x log10 P(x)

Near agreement

• ver four orders
• f magnitude!

◮ For lognormal (blue), µ = 0 and σ = 10. ◮ For power law (red), α = 1 and c = 0.03.

Lognormals and friends Lognormals

Empirical Confusability Random Multiplicative Growth Model Random Growth with Variable Lifespan

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Confusion

What’s happening:

◮

ln P(x) = ln

• 1

x √ 2πσ exp

• −(ln x − µ)2

2σ2

• ◮

= − ln x − ln √ 2π − (ln x − µ)2 2σ2

◮

= − 1 2σ2 (ln x)2 + µ σ2 − 1

• ln x − ln

√ 2π − µ2 2σ2 .

◮ ⇒ If σ2 ≫ 1 and µ, ◮

ln P(x) ∼ − ln x + const.

Lognormals and friends Lognormals

Empirical Confusability Random Multiplicative Growth Model Random Growth with Variable Lifespan

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Confusion

◮ Expect -1 scaling to hold until (ln x)2 term becomes

significant compared to (ln x).

◮ This happens when (roughly) ◮

− 1 2σ2 (ln x)2 ≃ 0.05 µ σ2 − 1

• ln x

◮

⇒ log10 x 0.05 × 2(σ2 − µ) log10 e

◮

≃ 0.05(σ2 − µ)

◮ ⇒ If you find a -1 exponent,

you may have a lognormal distribution...

Lognormals and friends Lognormals

Empirical Confusability Random Multiplicative Growth Model Random Growth with Variable Lifespan

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Generating lognormals:

Random multiplicative growth:

◮

xn+1 = rxn where r > 0 is a random growth variable

◮ (Shrinkage is allowed) ◮ In log space, growth is by addition:

ln xn+1 = ln r + ln xn

◮ ⇒ ln xn is normally distributed ◮ ⇒ xn is lognormally distributed

Lognormals and friends Lognormals

Empirical Confusability Random Multiplicative Growth Model Random Growth with Variable Lifespan

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Lognormals or power laws?

◮ Gibrat [2] (1931) uses this argument to explain

lognormal distribution of firm sizes

◮ Robert Axtell (2001) shows power law fits the data

very well [1] γ ≃ 2

◮

Lognormals and friends Lognormals

Empirical Confusability Random Multiplicative Growth Model Random Growth with Variable Lifespan

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An explanation

◮ Axtel (mis)cites Malcai et al.’s (1999) argument [6] for

why power laws appear with exponent γ ≃ 1

◮ The set up: N entities with size xi(t) ◮ Generally:

xi(t + 1) = rxi(t) where r is drawn from some happy distribution

◮ Same as for lognormal but one extra piece: ◮ Each xi cannot drop too low with respect to the other

sizes: xi(t + 1) = max(rxi(t), c xi)

Lognormals and friends Lognormals

Empirical Confusability Random Multiplicative Growth Model Random Growth with Variable Lifespan

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An explanation

Some math later...

◮ Find

P(x) ∼ x−γ where

◮

N = (γ − 2) (γ − 1)

• (c/N)γ−1 − 1

(c/N)γ−1 − (c/N)

• ◮ Now, if c/N ≪ 1,

N = (γ − 2) (γ − 1)

• −1

−(c/N)

• ◮ Which gives

γ ∼ 1 + 1 1 − c

◮ Groovy... c small ⇒ γ ≃ 2

Lognormals and friends Lognormals

Empirical Confusability Random Multiplicative Growth Model Random Growth with Variable Lifespan

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The second tweak

Ages of firms/people/... may not be the same

◮ Allow the number of updates for each size xi to vary ◮ Example: P(t)dt = ae−atdt ◮ Back to no bottom limit: each xi follows a lognormal ◮ Sizes are distributed as [7]

P(x) = ∞

t=0

ae−at 1 x √ 2πt exp

• −(ln x − µ)2

2t

• dt

(Assume for this example that σ ∼ t and µ = ln m)

◮ Now averaging different lognormal distributions.

Lognormals and friends Lognormals

Empirical Confusability Random Multiplicative Growth Model Random Growth with Variable Lifespan

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Averaging lognormals

◮

P(x) = ∞

t=0

ae−at 1 x √ 2πt exp

• −(ln x/m)2

2t

• dt

◮ Substitute t = u2:

P(x) = 2λ √ 2πx ∞

u=0

exp

• −λu2 − (ln x/m)2/2u2

du

◮ We can (lazily) look this up: [3]

∞ exp

• −au2 − b/u2

du = 1 2 π a exp(−2 √ ab)

◮ We have a = λ and b = (ln x/m)2/2:

P(x) ∝ x−1e−√

2λ(ln x/m)2

Lognormals and friends Lognormals

Empirical Confusability Random Multiplicative Growth Model Random Growth with Variable Lifespan

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The second tweak

◮

P(x) ∝ x−1e−√

2λ(ln x/m)2 ◮ Depends on sign of ln x/m, i.e., whether x/m > 1 or

x/m < 1.

◮

P(x) ∝

• x−1+

√ 2λ

if x/m < 1 x−1−

√ 2λ

if x/m > 1

◮ ‘Break’ in scaling (not uncommon) ◮ Double-Pareto distribution ◮ First noticed by Montroll and Shlesinger [8, 9] ◮ Later: Huberman and Adamic [4, 5]: Number of pages

per website

Lognormals and friends Lognormals

Empirical Confusability Random Multiplicative Growth Model Random Growth with Variable Lifespan

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Quick summary of these exciting developments

◮ Lognormals and power laws can be awfully similar ◮ Random Multiplicative Growth leads to lognormal

distributions

◮ Enforcing a minimum size leads to a power law tail ◮ With no minimum size but a distribution of lifetimes,

double Pareto distribution appear

◮ Take home message: Be careful out there...

Lognormals and friends Lognormals

Empirical Confusability Random Multiplicative Growth Model Random Growth with Variable Lifespan

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References I

• R. Axtell.

Zipf distribution of U.S. firm sizes. Science, 293(5536):1818–1820, 2001. pdf (⊞)

• R. Gibrat.

Les inégalités économiques. Librairie du Recueil Sirey, Paris, France, 1931.

• I. Gradshteyn and I. Ryzhik.

Table of Integrals, Series, and Products. Academic Press, San Diego, fifth edition, 1994.

• B. A. Huberman and L. A. Adamic.

Evolutionary dynamics of the World Wide Web. Technical report, Xerox Palo Alto Research Center, 1999.

Lognormals and friends Lognormals

Empirical Confusability Random Multiplicative Growth Model Random Growth with Variable Lifespan

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References II

• B. A. Huberman and L. A. Adamic.

The nature of markets in the World Wide Web. Quarterly Journal of Economic Commerce, 1:5–12, 2000.

• O. Malcai, O. Biham, and S. Solomon.

Power-law distributions and lévy-stable intermittent fluctuations in stochastic systems of many autocatalytic elements.

• Phys. Rev. E, 60(2):1299–1303, Aug 1999. pdf (⊞)
• M. Mitzenmacher.

A brief history of generative models for power law and lognormal distributions. Internet Mathematics, 1:226–251, 2003. pdf (⊞)

Lognormals and friends Lognormals

Empirical Confusability Random Multiplicative Growth Model Random Growth with Variable Lifespan

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References III

• E. W. Montroll and M. W. Shlesinger.

On 1/f noise aned other distributions with long tails.

• Proc. Natl. Acad. Sci., 79:3380–3383, 1982.
• E. W. Montroll and M. W. Shlesinger.

Maximum entropy formalism, fractals, scaling phenomena, and 1/f noise: a tale of tails.

• J. Stat. Phys., 32:209–230, 1983.