zurich mechanisms for power laws - eth z · mitzenmacher m (2004) a brief history of generative...
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D-MTEC Chair of Entrepreneurial Risks
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Prof. Dr. Didier Sornette www.er.ethz.ch
MECHANISMS FOR POWER LAWSZurich
First edition2000
Second enlarged edition2004
Nov 2005
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Heavy tails in pdf of earthquakes
Heavy tails in ruptures
Heavy tails in pdf of seismic rates
Harvard catalog
(CNES, France)
Turcotte (1999)
Heavy tails in pdf of rock falls, Landslides, mountain collapses
SCEC, 1985-2003, m≥2, grid of 5x5 km, time step=1 day
(Saichev and Sornette, 2005)
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Heavy tails in pdf of Solar flares
Heavy tails in pdf of Hurricane losses
1000
104
105
1 10
Damage values for top 30 damaging hurricanes normalized to 1995 dollars by inflation, personal
property increases and coastal county population change
Normalized1925
Normalized1900
N
Da
ma
ge
(m
illi
on
19
95
do
lla
rs)
RANK
Y = M0*XM1
57911M0
-0.80871M1
0.97899R
(Newman, 2005)
Heavy tails in pdf of rain events
Peters et al. (2002)
Heavy tails in pdf of forest fires
Malamud et al., Science 281 (1998)
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40
200
400
600
800
1000
1200
1 2 3 4 5 6 7 8 9 10
After-tax present value in millions of 1990 dollars
DBC
1980-84 pharmaceuticals in groups of deciles
Exponential model 1dataExponential model 2
OUTLIERS
Heavy-tail of Pharmaceutical sales
Heavy-tail of movie sales
OUTLIERS
Heavy-tail ofcrash losses(drawdowns)
Heavy-tail of price financialreturns
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Heavy-tail of pdf of war sizes
Levy (1983); Turcotte (1999)
Heavy-tail of pdf of health care costs
Rupper et al. (2002)
Heavy-tail of pdf of book sales
Heavy-tail of pdf of terrorist intensityJohnson et al. (2006)
Survivor Cdf
Sales per day
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Numbers of occurrencesof words in the novelMoby Dick by HermannMelville.
Numbers of citationsto scientific paperspublished in 1981,from time ofpublication until June1997.
Numbers of hits on websites by 60 000 users ofthe America OnlineInternet service for theday of 1 December 1997.
Numbers of copiesof bestsellingbooks sold in theUS between 1895and 1965.
Number of callsreceived by AT&Ttelephonecustomers in the USfor a single day.
M. E. J. Newman, Power laws, Pareto distributions and Zipf’s law (2005)
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Peak gamma-ray intensity ofsolar flares in counts persecond, measured from Earthorbit between February1980 and November 1989.
Intensity of wars from 1816 to1980, measured as battledeaths per 10 000 of thepopulation of theparticipating countries.
Diameter of craterson the moon. Verticalaxis is measured persquarekilometre.
Frequencyof occurrence of familynames in the US in the year1990.
M. E. J. Newman, Power laws, Pareto distributions and Zipf’s law (2005)
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Probability density function f(x)
Pr{ } ( )
b
a
a x b f x dx! ! = "The distribution is called heavy tailed if it has infinite second moment:
2 ( )x f x dx
!
"!
= !#For power-law distributions (Pareto distributions):
( ) : Pr{ } 1 , 1F x x x x!" #
= $ = # %1( ) : ( ) / , 1f x dF x dx x x!! " "
= = #
2 2 1 2
1 1
( ) 22
x f x dx x x dx x! !!! !
!
"" "
# # #
#"
= = = "$ <#% %
1 1
1 1
11
x x dx x! !!
! !!
""
# # #$ % = = "#&Note also that if
(Infinite expectation)
Heavy Tails
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9D. Sornette, Probability Distributions in Complex Systems Encyclopedia of Complexity and System Science (Springer Science), 2007
1. percolation, fragmentation and other related processes,
2. directed percolation and its universality class of so-called “contact processes”,
3. cracking noise and avalanches resulting from the competition between frozen disorderand local interactions, as exemplified in the random field Ising model, whereavalanches result from hysteretic loops [34],
4. random walks and their properties associated with their first passage statistics [35]in homogenous as well as in random landscapes,5. flashing annihilation in Verhulst kinetics [36],
6. sweeping of a control parameter towards an instability [25, 37],
7. proportional growth by multiplicative noise with constraints (the Kesten process [38]and its generalization for instance in terms of generalized Lotka-Volterra processes[39], whose ancestry can be traced to Simon and Yule,
8. competition between multiplicative noise and birth-death processes [40],
MECHANISMS FOR POWER LAWS
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9. growth by preferential attachment [32],
10. exponential deterministic growth with random times of observations (which gives theZipf law) [41],
11. constrained optimization with power law constraints (HOT for highly optimized tolerant),
12. control algorithms, which employ optimal parameter estimation based on pastobservations, have been shown to generate broad power law distributions of fluctuationsand of their corresponding corrections in the control process [42, 43],
13. on-off intermittency as a mechanism for power law pdf of laminar phases [44, 45],
14. self-organized criticality which comes in many flavors:• cellular automata sandpiles with and without conservation laws,• systems made of coupled elements with threshold dynamics,• critical de-synchronization of coupled oscillators of relaxation,• nonlinear feedback of the order parameter onto the control parameter• generic scale invariance,• mapping onto a critical point,• extremal dynamics.
Mitzenmacher M (2004) A brief history of generative models for power law and lognormal distributions, Internet Mathematics 1, 226-251.
Newman MEJ (2005) Power laws, Pareto distributions and Zipf’s law, Contemporary Physics 46, 323-351.
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M
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STUDENTDISTRIBUTION
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Yule's process
Nb of species of a given genus;Size of a given firm
Growth of the number of genus;growth of the number of firms
Pdf(S) ~ 1/S1+γ
Zipf’s law γ=1 for D=c
D eDt
D
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SUPERPOSITIONOF DISTRIBUTIONS
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31Hemmer and Hansen (1992)
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Mean field theory:
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37(Sethna et al.)
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: number of pages with in-degree j at time t
Probability that increases is
This is nothing but Simon (1956)’s model for firms translated for the Internet
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Steady state
Recurrence equation
For large j:
Yule (1925); Simon (1955); Lokta, Zipf, ….
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Kesten processstochastic recurrence equations
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D. Sornette and R. Cont
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P. Jogi, D. Sornette and M. Blank, Fine structure and complex exponents in power law distributions from random maps, Phys. Rev. E 57 120-134 (1998)
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Immigration; Investment growth with capitalization
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Generalization
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Multi-dimensional Stochastic Recurrence Equations
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(stationarity)
(strong fluctuations)
(ergodicity)
(no traps)
(tail not controlledby A or B)
(Aperiodicity)
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L. Gil and D.Sornette“Landau-Ginzburgtheory of self-organizedcriticality”, Phys.Rev.Lett. 76,3991-3994 (1996)
Mechanism:Negative effectiveDiffusion coefficient
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Coherent-noise mechanism
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Securitization of credit risks:is it the next “systemic collapse”?
• Securitization of credit risks leads tosmaller risks
• But more inter-connected⇒ global risk?
CDS and CDO: form of insurance contracts linked to underlying debt that protects thebuyer in case of default.
The market has almost doubled in size every year for the past five years, reaching $20trillion in notional amounts outstanding last June 2007, according to the Bank forInternational Settlements.
Bundling of indexes of CDSs together and slicing them into tranches, based on riskinessand return. The most toxic tranche at the bottom exposes the holder to the first 3%of losses but also gives him a large portion of the returns. At the top, the risks and returnsare much smaller-unless there is a systemic failure.
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Securitization leads to larger inter-connectivitySeparation of financial and credit risks
pdfpdf
risks risks
Coupling strength increases
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Traditional emphasis onDaily returns do not revealany anomalous events
THE CONCEPT OF “Kings”
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A. Johansen and D. Sornette, Stock market crashes are outliers,European Physical Journal B 1, 141-143 (1998)
A. Johansen and D. Sornette, Large Stock Market Price Drawdowns Are Outliers,Journal of Risk 4(2), 69-110, Winter 2001/02
Distribution of drawdowns
OUTLIERSOUTLIERS
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OUTLIERS
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Thirty Major US companies
OUTLIERSOUTLIERS
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D. Sornette and A. JohansenSignificance of log-periodic precursors to financial crashes,Quantitative Finance 1 (4), 452-471 (2001)
A. Johansen and D. Sornette,Endogenous versus Exogenous Crashes in Financial Markets,in press in ``Contemporary Issues in International Finance''(Nova Science Publishers, 2004)(http://arXiv.org/abs/cond-mat/0210509)
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Outliers, Kings, “Black swans”(require special mechanism and may be more predictable)
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Beyond power laws: five examples of “kings”
Material failure and rupture processes.
Gutenberg-Richter law and characteristic earthquakes.
Extreme king events in the pdf of turbulent velocity fluctuations.
Outliers and kings in the distribution of financial drawdowns.
Paris as the king in the Zipf distribution of French city sizes.
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2004
1997Second edition