Lognormals and friends Lognormals and friends Lognormals Empirical Confusability Principles of Complex Systems Random Multiplicative Growth Model Course 300, Fall, 2008 Random Growth with Variable Lifespan References Prof. Peter Dodds Department of Mathematics & Statistics University of Vermont Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License . Frame 1/23
Lognormals and Outline friends Lognormals Empirical Confusability Random Multiplicative Growth Model Random Growth with Variable Lifespan References Lognormals Empirical Confusability Random Multiplicative Growth Model Random Growth with Variable Lifespan References Frame 2/23
Lognormals and Alternative distributions friends Lognormals Empirical Confusability Random Multiplicative Growth Model Random Growth with Variable Lifespan References There are other heavy-tailed distributions: 1. Lognormal 2. Stretched exponential (Weibull) 3. ... (Gamma) Frame 4/23
Lognormals and lognormals friends Lognormals Empirical Confusability Random Multiplicative Growth Model Random Growth with The lognormal distribution: Variable Lifespan References − ( ln x − µ ) 2 1 � � P ( x ) = √ exp 2 σ 2 x 2 πσ ◮ 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. Frame 5/23
Lognormals and lognormals friends Lognormals Empirical Confusability Standard form reveals the mean µ and variance σ 2 of the Random Multiplicative Growth Model Random Growth with underlying normal distribution: Variable Lifespan References − ( ln x − µ ) 2 � � 1 P ( x ) = √ exp 2 σ 2 x 2 πσ For lognormals: µ lognormal = e µ + 1 2 σ 2 , median lognormal = e µ , σ lognormal = ( e σ 2 − 1 ) e 2 µ + σ 2 , mode lognormal = e µ − σ 2 . All moments of lognormals are finite. Frame 6/23
Lognormals and Derivation from a normal distribution friends Take Y as distributed normally: Lognormals Empirical Confusability Random Multiplicative ◮ Growth Model − ( y − µ ) 2 1 � � Random Growth with P ( y ) d y = √ Variable Lifespan d y exp 2 σ 2 2 πσ References Set Y = ln X : ◮ Transform according to P ( x ) d x = P ( y ) d y : ◮ d y d x = 1 / x ⇒ d y = d x / x ◮ − ( ln x − µ ) 2 1 � � √ ⇒ P ( x ) d x = exp d x 2 σ 2 x 2 πσ Frame 7/23
Lognormals and Confusion between lognormals and pure friends power laws Lognormals Empirical Confusability Random Multiplicative Growth Model Random Growth with Variable Lifespan −2 References −4 log 10 P(x) Near agreement −6 over four orders −8 of magnitude! −10 0 2 4 6 8 10 log 10 x ◮ For lognormal (blue), µ = 0 and σ = 10. ◮ For power law (red), α = 1 and c = 0 . 03. Frame 8/23
Lognormals and Confusion friends What’s happening: Lognormals Empirical Confusability Random Multiplicative Growth Model ◮ Random Growth with − ( ln x − µ ) 2 � 1 � �� Variable Lifespan ln P ( x ) = ln √ exp References 2 σ 2 2 πσ x ◮ √ 2 π − ( ln x − µ ) 2 = − ln x − ln 2 σ 2 ◮ � µ √ 2 π − µ 2 = − 1 2 σ 2 ( ln x ) 2 + � σ 2 − 1 ln x − ln 2 σ 2 . ◮ ⇒ If σ 2 ≫ 1 and µ , ◮ ln P ( x ) ∼ − ln x + const. Frame 9/23
Lognormals and Confusion friends Lognormals Empirical Confusability ◮ Expect -1 scaling to hold until ( ln x ) 2 term becomes Random Multiplicative Growth Model significant compared to ( ln x ) . Random Growth with Variable Lifespan ◮ This happens when (roughly) References ◮ � µ − 1 2 σ 2 ( ln x ) 2 ≃ 0 . 05 � σ 2 − 1 ln x ◮ ⇒ log 10 x � 0 . 05 × 2 ( σ 2 − µ ) log 10 e ◮ ≃ 0 . 05 ( σ 2 − µ ) ◮ ⇒ If you find a -1 exponent, you may have a lognormal distribution... Frame 10/23
Lognormals and Generating lognormals: friends Lognormals Empirical Confusability Random multiplicative growth: Random Multiplicative Growth Model Random Growth with Variable Lifespan ◮ References x n + 1 = rx n where r > 0 is a random growth variable ◮ (Shrinkage is allowed) ◮ In log space, growth is by addition: ln x n + 1 = ln r + ln x n ◮ ⇒ ln x n is normally distributed ◮ ⇒ x n is lognormally distributed Frame 12/23
Lognormals and Lognormals or power laws? friends ◮ Gibrat [2] (1931) uses this argument to explain Lognormals lognormal distribution of firm sizes Empirical Confusability Random Multiplicative Growth Model ◮ Robert Axtell (2001) shows power law fits the data Random Growth with Variable Lifespan very well [1] γ ≃ 2 References Frame 13/23 ◮
Lognormals and An explanation friends Lognormals Empirical Confusability ◮ Axtel (mis)cites Malcai et al.’s (1999) argument [6] for Random Multiplicative Growth Model Random Growth with why power laws appear with exponent γ ≃ 1 Variable Lifespan References ◮ The set up: N entities with size x i ( t ) ◮ Generally: x i ( t + 1 ) = rx i ( t ) where r is drawn from some happy distribution ◮ Same as for lognormal but one extra piece: ◮ Each x i cannot drop too low with respect to the other sizes: x i ( t + 1 ) = max ( rx i ( t ) , c � x i � ) Frame 14/23
Lognormals and An explanation friends Some math later... Lognormals Empirical Confusability ◮ Find Random Multiplicative Growth Model P ( x ) ∼ x − γ Random Growth with Variable Lifespan References where ◮ ( c / N ) γ − 1 − 1 � � N = ( γ − 2 ) ( c / N ) γ − 1 − ( c / N ) ( γ − 1 ) ◮ Now, if c / N ≪ 1, N = ( γ − 2 ) � − 1 � ( γ − 1 ) − ( c / N ) ◮ Which gives 1 γ ∼ 1 + 1 − c Frame 15/23 ◮ Groovy... c small ⇒ γ ≃ 2
Lognormals and The second tweak friends Lognormals Empirical Confusability Random Multiplicative Ages of firms/people/... may not be the same Growth Model Random Growth with Variable Lifespan ◮ Allow the number of updates for each size x i to vary References ◮ Example: P ( t ) d t = ae − at d t ◮ Back to no bottom limit: each x i follows a lognormal ◮ Sizes are distributed as [7] � ∞ − ( ln x − µ ) 2 1 � � ae − at √ P ( x ) = exp d t 2 t x 2 π t t = 0 (Assume for this example that σ ∼ t and µ = ln m ) ◮ Now averaging different lognormal distributions. Frame 17/23
Lognormals and Averaging lognormals friends ◮ Lognormals � ∞ − ( ln x / m ) 2 Empirical Confusability 1 � � ae − at Random Multiplicative P ( x ) = √ exp d t Growth Model 2 t x 2 π t Random Growth with t = 0 Variable Lifespan References ◮ Substitute t = u 2 : � ∞ 2 λ � − λ u 2 − ( ln x / m ) 2 / 2 u 2 � √ P ( x ) = exp d u 2 π x u = 0 ◮ We can (lazily) look this up: [3] � ∞ √ � π d u = 1 � − au 2 − b / u 2 � exp a exp ( − 2 ab ) 2 0 ◮ We have a = λ and b = ( ln x / m ) 2 / 2: P ( x ) ∝ x − 1 e − √ 2 λ ( ln x / m ) 2 Frame 18/23
Lognormals and The second tweak friends Lognormals Empirical Confusability P ( x ) ∝ x − 1 e − √ ◮ Random Multiplicative 2 λ ( ln x / m ) 2 Growth Model Random Growth with Variable Lifespan ◮ Depends on sign of ln x / m , i.e., whether x / m > 1 or References x / m < 1. ◮ √ � x − 1 + 2 λ if x / m < 1 P ( x ) ∝ √ x − 1 − 2 λ 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 Frame 19/23
Lognormals and Quick summary of these exciting friends developments Lognormals Empirical Confusability Random Multiplicative Growth Model Random Growth with Variable Lifespan References ◮ 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... Frame 20/23
Lognormals and References I friends Lognormals R. Axtell. Empirical Confusability Random Multiplicative Growth Model Zipf distribution of U.S. firm sizes. Random Growth with Variable Lifespan Science , 293(5536):1818–1820, 2001. pdf ( ⊞ ) References 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. Frame 21/23
Lognormals and References II friends Lognormals B. A. Huberman and L. A. Adamic. Empirical Confusability Random Multiplicative Growth Model The nature of markets in the World Wide Web. Random Growth with Variable Lifespan Quarterly Journal of Economic Commerce , 1:5–12, References 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 ( ⊞ ) Frame 22/23
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