Lecture 2: Power Laws and Dragon-kingsgeza.kzoo.edu/~erdi/powerlaw/sornlec3.pdf · D-MTEC Chair of...

D-MTEC Chair of Entrepreneurial Risks

Prof. Dr. Didier Sornette www.er.ethz.ch

Entrepreneurial RisksLecture 2:

Power Laws and Dragon-kingsHeavy Tails and Long Tails

-normal versus power laws-calculation tools-new business model: the long tail-many examples of power laws in nature and society-scale invariance, fractal and multifractals-a few (among many) mechanisms for power laws-beyond power laws: black swans vs dragon-kings



Entrepreneurial RisksLecture 2:

Power Laws vs Long Tails

- Chapters 5, 14 and 15 ofD. Sornette, Critical Phenomena in Natural Sciences, Chaos, Fractals, Self-organization and Disorder: Concepts and Tools, 2nd ed. (Springer Series in Synergetics, Heidelberg, 2004)

-Wolfgang Amann, Cuno Pümpin and Didier Sornette, How power laws may shape the face of corporate strategy, Critical Eye 17, 21-26 (2008)

-D. Sornette, Dragon-Kings, Black Swans and the Prediction of Crises, International Journal of Terraspace Science and Engineering 2(1), 1-18 (2009) (http://arXiv.org/abs/0907.4290)

-D. Sornette and G. Ouillon, Dragon-kings: mechanisms, statistical methods and empirical evidence, to appear in European Physical Journal Special Topics (2012) (special issue on power laws and dragon-kings)



Readings

Risk

• Loss x Chance

• Not a number, but a curve

• Not just a single curve

• Not necessarily quantitative

Chance of a severe hurricane in 2010 is 20%



Definitions



Intuition on PDFs



Normal distribution



Development of Normal Distribution

• Jacob Bernoulli (ca. 1700) First experiments with distributions

• De Moivre (1733): Tossing coins• Laplace (1749 – 1828) / Poisson (ca. 1800): Law

of large numbers• Gauss (1807)• Quetelet (1847): considers normal distribution as

a fundamental law of humanity



Normal distribution and CLT

(CLT: Central Limit Theorem)



Key Concepts in Asset Management

• MPT: Markowitz Portfolio Theory (1952)– risk = std dev of return

• CAPM: Capital Asset Pricing Model (Sharpe 1964)– std risk remunerated only when not diversifiable (market risk)

• Efficient Market Hypothesis (Fama 1966...)• Black-Scholes-Merton: option theory (1973)• Value at Risk (first applications before WW II)

All concepts developed between 1950 - 1975

All concepts use standard deviation.



Capital Allocation Line (1945 – 2001)

Quelle: Morgan Stanley, Ibbotson Associates

Annual Returns

Volatility: std



Normal laws have “typical scales”



“What is the probability that someone has twice your height ?Essentially zero! The height, weight and many other variables are distributed with ʻmildʼ probability distribution functions with a well defined typical value and relatively small variations around it.

Mild forms and wild forms of societal self-organization

Didier Sornette (2004), 2nd ed., Critical Phenomena in Natural Sciences. Chaos, Fractals, Self-organization and Disorder: Concepts and Tools, Springer, Heidelberg

Crow J F Journal of Heredity 2004;95:365-374 © 2004 The American Genetic Association

(Top) A living histogram from the Connecticut State Agricultural College (J. Heredity 5:511–518, 1914). (Bottom) A modern version from the same university, arranged by Linda Strausbaugh (Genetics 147:5, 1997). The mean height of males in 1914 was 67.3 inches and 70.1 in 1997. In 1997 the height of females, shown in white, was 64.8 inches

Source: http://jhered.oxfordjournals.org/content/95/5/365.full



“What is the probability that someone has twice your height ?Essentially zero! The height, weight and many other variables are distributed with ʻmildʼ probability distribution functions with a well defined typical value and relatively small variations around it.

What is the probability that someone has twice your wealth? The answer of course depends somewhat on your wealth but in general there is a non-vanishing fraction of the population twice, ten times, or even one hundred times wealthier as you are.”

Mild forms and wild forms of societal self-organization

Didier Sornette (2004), 2nd ed., Critical Phenomena in Natural Sciences. Chaos, Fractals, Self-organization and Disorder: Concepts and Tools, Springer, Heidelberg





October 1987 NYSE: 21.6 standard deviations

Return time=

44‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘00‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘000‘

000‘000‘000 years (= 44 * 1099)

(Age of Universe = 10 * 109 years)



Shocks are quite often 1994 interest rates schock

1996 Asia boom

1997 Asia crisis

1998 Russia crisis / LTCM

2000 Internet rally

2001 Bursting of internet bubble, 11.9.01

2002 Nemax - 68,8 %

2003 Irak crisis

2003 Stock exchange rally

2005 Japan, Arabian markets (Boom)

2005 / 2006 Februar: Crash of Arabian markets

You can often hear or read of reports of a 5, …, 9, … sigma event!



Few large earthquakes, many small ones

Few large cities, many small ones

Few frequently occurring words (“and”, “the”, “of”), manywhich rarely occur

Few extremely wealthy individuals, many poor

Few popular websites (Google, Yahoo, YouTube),many millions of others that are rarely visited.

Examples where “Normal” distribution does not apply



How do we represent this graphically?



“Long Tail” vs “Heavy Tail”



Jean Le Rond d'Alembert (November 16, 1717 – October 29, 1783)

Life duration, years

Paradox: Average life duration is 26 years, while the chances to die before 8 and live more than 8 years are the same. 26 yrs

Average lifeduration

8 yrs

Die before 8

Die after 8

Basic Paradox of Heavy Tails

mean median median

mean



The Power Law: a “Heavy Tailed” distribution

f(x) =C

x1+β



Heavy Tails



Probability density function f(x)

The distribution is called heavy tailed if it has infinite second moment:

For power-law distributions (Pareto distributions):

Note also that if(Infinite expectation)

Heavy Tails

1 (0,1)

n

iiX nm

Nnσ

=

−→

∑1

1/ ( )

n

i niX b

SC n β β=

−→

∑

Central Limit Theorem Generalized Central Limit Theorem

1

1 ( , ) ,n

ii

N nm nX N m mn n n

σ σ=

⎛ ⎞→ → →⎜ ⎟⎝ ⎠∑

1

1

( ) ,1 21 ( ) /, 1

n

i ni

bX CS n b n

n

ββ β β

ββ

−

=

< ∞ < <⎧→ + → ⎨ ∞ ≤⎩∑



Limit Theorems



Heavy Tails:how to determine the optimal price

�

S N p(x)dxαS

+∞

∫⎡

⎣ ⎢

⎤

⎦ ⎥ ∝ SN 1

x1+βdx ∝

αS

+∞

∫ NS1−β

Product price Number of customers who can afford the price

•β < 1: S⇑ (luxury goods)

•β > 1: S⇓ (mass product)

Total sales:

p(x): PDF of customers’ wealth (buying power)



Normal Distribution vs. Power Law

• Normal distributions:– uniform distribution, vertex, no fat tails– mostly in static systems with weak interactions

• Power Laws:– weak vertex, continuous decent from highest

point, strong fat tails– System elements are in long-range interaction – Systems grow in a dynamic / evolutionary way

→ Out-of-Equilibrium Systems



• Earthquakes• Avalanches• Wealth, rich get richer• Extinction of Dinosaurs• Forest Fires• Epidemics• Pulsars• Scientific theories• Size of towns• Revolutions• Wars• Board membership• Movie actors• Proteins• Financial markets

Examples of Heavy Tails



Cities

Left: histogram of the populations of all US cities with population of 10 000 or more. Right: another histogram of the same data, but plotted on logarithmic scales. The approximate straight-line form of the histogram in the right panel implies that the distribution follows a power law. Data from the 2000 US Census.

31

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)

32

Heavy tails in cdf of Solar flares

Heavy tails in cdf 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

Normalized1925Normalized1900N

Dam

age

(mill

ion

1995

dol

lars

)

RANK

Y = M0*XM1

57911M0-0.80871M10.97899R

(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)

33

OUTLIERS OUTLIERS

Heavy-tail of movie sales

Heavy-tail of price financialreturns

Firm sizes (Zipf’s law)

City sizes (Zipf’s law)

34

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 cdf of book sales

Heavy-tail of cdf of terrorist intensityJohnson et al. (2006)

Survivor Cdf

Sales per day

Heavy-tail of cdf of cyber risks

b=0.7

ID Thefts

Heavy-tail of YouTube view counts

0

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

Heavy-tail of Pharmaceutical sales

Number

waiting time

Software vulnerabilities

views



(Axtell, Science, 2001)

Heavy tails in pdf of firm sizes

37

Numbers of occurrences of words in the novel Moby Dick by Hermann Melville.

Numbers of citations to scientific papers published in 1981, from time of publication until June 1997.

Numbers of hits on web sites by 60 000 users of the America Online Internet service for the day of 1 December 1997.

Numbers of copies of bestselling books sold in the US between 1895 and 1965.

Number of calls received by AT&Ttelephone customers in the US for a single day.

M. E. J. Newman, Power laws, Pareto distributions and Zipf’s law (2005)

38

Peak gamma-ray intensity of solar flares in counts per second, measured from Earth orbit between February1980 and November 1989.

Intensity of wars from 1816 to 1980, measured as battle deaths per 10 000 of the population of theparticipating countries.

Diameter of craters on the moon. Vertical axis is measured per squarekilometre.

Frequencyof occurrence of family names in the US in the year 1990.

M. E. J. Newman, Power laws, Pareto distributions and Zipf’s law (2005)

39

Not heavy tailed ⇒ most probablyWILDLY UNDERESTIMATED

Frequency of fatalities due to man-caused events



What do these heavy tails mean?

Why do we see power laws everywhere?



Coherent-noise mechanism

List of mechanisms for power laws I

(Self-organized criticality)



List of mechanisms for power laws II

• Sweeping of an instability

•Avalanches in hysteretic loops

(Coupling of sub-critical bifurcations)

43

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.

D. Sornette (2004) Probability Distributions in Complex Systems, Encyclopedia of Complexity and System Science (Springer Science), 2004

D. Sornette (2006) Critical Phenomena in Natural Sciences,Chaos, Fractals, Self-organization and Disorder: Concepts and Tools,2nd ed., 2nd print, pp.528, 102 figs. , 4 tabs (Springer Series in Synergetics, Heidelberg)



Power laws, Scale Invariance and Fractals

Chapter 5Didier Sornette (2004), 2nd ed., Critical Phenomena in Natural Sciences. Chaos, Fractals, Self-organization and Disorder: Concepts and Tools, Springer, Heidelberg

Under a change of scale of the control parameter

Functional equation whose solution is

Power laws are the hallmark of scale invariance:



Scale invariance



Scale invariance



Reflection symmetry



Rotation symmetry



Translation symmetry



Magnification symmetry => fractals













Romanesco broccoli











Illustration of scale invariance

Non-Fractal

Fractal

Non - Fractal

Size of Features1 cm

1 characteristic scale

FractalSize of Features

2 cm1 cm1/2 cm

1/4 cm

many different scales

Self-Similarity

Water

Land

Water

Land

Water

Land

ScalingThe value measured for a property depends on the resolution at which it is measured.

Allometric scaling

http://online.itp.ucsb.edu/online/pattern_i03/west/

Allometric scaling for cities

here, the exponent ~1.1 > 1 ! => positive feedbacks / synergies

Bettencourt, l. M. a., lobo, J., Helbing, d., Kuhnert, C. & West, G. B. Proc. Natl Acad. Sci. USA 104, 7301–7306 (2007).

Allometric scaling for open source softwares

Number of lines changed as a function of the number of developers active per time bins for the Apache HTTP server (left) and the GNU C compiler (right)

Maillart et al. (2012)

Exponent ~2expresses superlinear positive feedbacks



Size Number1 11/3 21/9 41/27 8

1/3^n 2^n

Self-similarity and power law



Size Number1 11/3 21/9 41/27 8

1/3^n 2^n

Fractal dimension:

N(r) =1

rD

2n =1

[(1/3)n]D





6

B.B. Mandelbrot, The Fractal Geometry Of Nature, 1983, W.H.Freeman

G.A. Edgar, Measure, Topology and Fractal Geometry,1990, Springer-Verlag

M. Barnsley, Fractals Everywhere, 1988, Academic Press

J. Feder, Fractals, 1988, Plenum

Books About Fractals

Movie: courtesy of Nicola Pestalozzi (ETH Zurich)



Coherent-noise mechanism

List of mechanisms for power laws I

(Self-organized criticality)



List of mechanisms for power laws II

• Sweeping of an instability

•Avalanches in hysteretic loops

(Coupling of sub-critical bifurcations)



Preferential attachment



Source: Lada A. Adamic

Growth with Preferential Attachment



MECHANISMS FOR POWER LAWSComplex system approach

• One general idea: System elements are interconnected • Network with hubs and knots• Hubs with many connections have a information

advantage.

Chapters 14 and 15Didier Sornette (2004), 2nd ed., Critical Phenomena in Natural Sciences. Chaos, Fractals, Self-organization and Disorder: Concepts and Tools, Springer, Heidelberg

Saint Matthew effect



: number of pages with in-degree j and total number t of pages

Probability that increases is


this page.



Recurrence equation

For large j:

Yule (1925); Simon (1955); Lokta, Zipf, …., Barabasi

Steady state



Stochastic Recurrence Equations

with ‘a’ stochastic

Law of proportional growth:







whose solution is

with

Note relationship with mechanism of “competition between exponentials”

Xt+1 = atXt + bt





Model of financial bubbles (Rational Expectation bubbles)



Why do we care?

Very general and ubiquitous mechanism: multiplicative process with a source or minimum size

Insight into the dynamics of firms and other economic entities

For instance, for firms, recent research show that the Zipf law of firms’sizes is mainly due to risks while returns are subdominant: risk/luck dominates! (Malevergne, Saichev, Sornette, 2007)



Source: Lada A. Adamic


2009

Cumulative distributions of market share of digital cameras measured for 4 snapshots in Japan

(Ryohei Hisano, Didier Sornette, Takayuki Mizuno, 2011)

Test of Gibrat’s law of proportional growth for market share of products until January 31 2008


The panel depicts the distribution of lifetimes of digital cameras using the definition mentioned in the text. Green circles denote the distribution of lifetimes of products which died during 2005, blue squares correspond to 2006, red triangles to 2007, and black crosses mixing all of these years together.



Comparison between the theoretical predicted power law exponents μ(TH) and the empirical exponents μ(MLE).

95

Self-organized criticality

Earthquakes Cannot Be PredictedRobert J. Geller, David D. Jackson, Yan Y. Kagan, Francesco MulargiaScience 275, 1616-1617 (1997)Turcotte (1999)

(Bak, Tang, Wiesenfeld, 1987)



Sandpile model as paradigm of Self-Organization

Bak, P., Tang, C. and Wiesenfeld, K. (1987). "Self-organized criticality: an explanation of 1 / f noise". Physical Review Letters 59 (4): 381–384.

Artwork by Elaine Wiesenfeld (from Bak, How Nature Works, 1996)

1996



Aggregate Fluctuations from Independent Sectoral Shocks: Self-Organized Criticality in a Model of Production and Inventory Dynamics

Per Bak, Kan Chen (1992)

J.A. Scheinkman and M. Woodford, Self-organized criticality and economic fluctuations,American Economic Review 84 (2), 417-421 (1994)



PDF of avalanche sizesPDF of avalanche durations

Self-organization + power laws = Self-organized Criticality





Stress field immediately before (a) and after (b) a mainshock. The stress change due to the mainshock in shownin (c). The elements that broke during the avalanche are shown in dark in (c) (stress decrease) and were mostly close to the rupture threshold before the mainshock [light gray in (a)]. The upper panels show the whole grid of size L=1024 and the lower plots represent a subset of thegrid delineated by the square in the upper plot.

(Helmsetter et al., Phys. Rev. E 70, 046120, 2004)



Forest Fire Model (FFM)

During each time step the system is updated according to the rules:

The Forest Fire Model is a stochastic 3-state cellular automaton defined on a d-dimensional lattice with Ld sites.

Each site is occupied by a tree, a burning tree, or is empty.

• empty site → tree with the growth rate probability p • tree → burning tree with the lightning rate probability f, if

no nearest neighbour is burning • tree → burning tree with the probability 1-g, if at least

one nearest neighbour is burning, where g defines immunity.

• burning tree → empty site SOC in the limit f << p <<1



Michael Biggs, Strikes as Forest Fires: Chicago and Paris in the Late Nineteenth CenturyAmerican Journal of Sociology, volume 110 (2005), pages 1684–1714

•Strikes

•Power grid dynamics

•Network of firms and credit risk dependences

•Economic policy implications

•Spatial SIR epidemic with recurrence

Forest Fire Model (FFM)



A typical simulation

http://www.shodor.org/interactivate/activities/ABetterFire/?version=1.5.0_13&browser=Mozilla&vendor=Apple_Inc.



Power laws and large risks

• Power laws are ubiquitous• They express scale invariance• Large and extreme events

-example of height vs wealth

• Gaussian approach inappropriate: underestimation of the real risks

– Markowitz mean-variance portfolio– Black-Scholes option pricing and hedging– Asset valuation (CAPM, APT, factor models)– Financial crashes

TWO PROBLEMS What tail? What risks?



What model(s) for the Distributions (of Returns)?



Implications of the two models

Practical consequences :•Extreme risk assessment,•Multi-moment asset pricing methods.

“Long Tail” vs “Heavy Tail”



The Long Tail defined

In Mathematics The name for a statistical

distribution curve based on a high amplitude population followed by a low frequency population which gradually tails off (Wikipedia)

Any Industry where there is demand / availability for specialized products can and will be influenced by the long tail as accessible selection increases

and transaction friction decreases

In Business Products that are in low demand

that can collectively make up a market share that rivals the relatively few current bestsellers when distributed over such a big channel as the Internet (Chris Anderson)



Power law dynamics in the online worldCompanies and consumers discover the long-tail

Another way to describe the long tail



The diversity of the ecosystem makes it a fertile environment for small players. You don’t have to dominate the food chain to get by in the Web world; you can find a productive niche and thrive…

-Steven Johnson, “Web 2.0 Arrives”

Power law dynamics in the online worldCompanies and consumers discover the long-tail

Analogy with the power of bacteria on Earth...



Power law dynamics in the online world

Different products and different strategies emerge

Bestsellers (Head):• Highly visible• Marketed to us• Long sales cycle

Long Tail Products: Rarely visible Word-of-mouth Sales, not cycles



Amazon Barnes & Noble

130000

2300000

Total Inventory (books)

Rhapsody Walmart

39000

735000

Total Inventory (songs)

Netflix Blockbuster

3000

25000

Total Inventory (movies)

Power law dynamics in the online worldSome examples

22.00

78.00

Rhapsody % Sales from Long Tail

57.00

43.00

Amazon % Sales from Long Tail

20.00

80.00

Netflix % Sales from Long Tail

Source: jotspot

Movie sales example

Didier Sornette and Daniel Zajdenweber, The economic return of research\,: the Pareto law and its implications, European Physical Journal B, 8 (4), 653-664 (1999).

µ � 1.5

How does the long tail arise?

Mechanism for the long tail

•“order” versus “disorder”

•“social” versus “individual”

How do we exploit it?

Is there “life” beyond power laws?

127

2008 FINANCIAL CRISIS

128

March 2009

2008 FINANCIAL CRISIS

Crises are not

but

“Dragon-kings”

Black Swans versus Dragons-Kings

Black Swan (Cygnus atratus)

131

Beyond power laws: 7 examples of “Dragons”

Material science: failure and rupture processes.

Geophysics: Gutenberg-Richter law and characteristic earthquakes.

Hydrodynamics: Extreme dragon events in the pdf of turbulent velocity fluctuations.

Financial economics: Outliers and dragons in the distribution of financial drawdowns.

Population geography: Paris (London) as the dragon-king in the Zipf distribution of French (English) city sizes.

Brain medicine: Epileptic seizures

Metastable states in complex optimization problems: Self-organized critical random directed “polymers”

132

Traditional emphasis onDaily returns do not revealany anomalous events

Crashes as “Black swans”?

“Black swans”

Better risk measure: drawdowns

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

“Dragons” of financial risks

“Dragons” of financial risks(require special mechanism and may be more predictable)

10% daily drop on Nasdaq : 1/1000 probability

1 in 1000 days => 1 day in 4 years

30% drop in three consecutive days?

(1/1000)*(1/1000)*(1/1000) = (1/1000’000’000)

=> one event in 4 millions years!

137






Population geography: Paris as the dragon-king in the Zipf distribution of French city sizes.



138

Paris as a dragon-king

Jean Laherrere and Didier Sornette, Stretched exponential distributions in Nature and Economy: ``Fat tails''with characteristic scales, European Physical Journal B 2, 525-539 (1998)

2009

(Size)c

139









Energy distribution for the [+-62] specimen #4 at different times, for 5 time windows with 3400events each. The average time (in seconds) of events in each window is given in the caption.

H. Nechad, A. Helmstetter, R. El Guerjouma and D. Sornette, Andrade and Critical Time-to-Failure Laws in Fiber-Matrix Composites: Experiments and Model, Journal of Mechanics and Physics of Solids (JMPS) 53, 1099-1127 (2005)

...

time-to-failure analysisS.G. Sammis and D. Sornette, Positive Feedback, Memory and the Predictability of Earthquakes, Proceedings of the National Academy of Sciences USA, V99 SUPP1:2501-2508 (2002 FEB 19)

142









Mathematical Geophysics Conference Extreme Earth EventsVillefranche-sur-Mer, 18-23 June 2000

L'vov, V.S., Pomyalov, A. and Procaccia, I. (2001) Outliers, Extreme Events and Multiscaling,Physical Review E 6305 (5), 6118, U158-U166.

Pdf of the square of theVelocity as in the previous figure but for a much longertime series, so that the tailof the distributions for large Fluctuations is much betterconstrained. The hypothesisthat there are no outliers is tested here by collapsing the distributions for the three shown layers. While this is a success for small fluctuations, the tails of the distributions for large events are very different, indicating that extreme fluctuations belong to a different class of their own and hence are outliers.

L'vov, V.S., Pomyalov, A. and Procaccia, I. (2001) Outliers, Extreme Events and Multiscaling,Physical Review E 6305 (5), 6118, U158-U166.

Collapse ~of positions and amplitudes! for five intensivepeaks belonging to the 20th shell.

146









LTAD 1-6(1-6)

LTMD 1-6(17-22)

LTPD 1-6(33-38)

RTAD 1-6(41-46)

RTMD 1-6(25-30)

RTAD 1-6(9-14)

RFD 1-8(57-64)

LFD 1-8(49-56)

Depth Needle Electrodes Contact Numbering: N … 3 2 1

Key: L=Left R=Right A=Anterior M=Mesial P=Posterior D=Depth T=Temporal F=Frontal

Focus

Epileptic Seizures – Quakes of the Brain?with Ivan Osorio – KUMC & FHS

Mark G. Frei - FHSJohn Milton -The Claremont Colleges

(arxiv.org/abs/0712.3929)

148

Bursts and Seizures

149

Gutenberg-Richter distribution of sizes Omori law: Direct and Inverse

pdf of inter-event waiting times The longer it has been since the last event, the longer it will be since the next one!

150

19 rats treated intravenously (2) with the convulsant 3-mercapto-proprionic acid (3-MPA)

SYNCHRONISATION AND COLLECTIVE EFFECTSIN EXTENDED STOCHASTIC SYSTEMS

Fireflies

Miltenberger et al. (1993)

Earthquake-fault model

(Prof. R.E. Amritkar)

Interaction (coupling) strength

Heterogeneity; level of compartmentalization

10

1

0.1

0.01

0.0010.001 0.01 0.1 1 10

SYNCHRONIZATIONEXTREME RISKS

SELF-ORGANIZED CRITICALITY

+

+

+

++

+

+

*

*

*

*

*

*

*

Coexistence of SOCand Synchronized behavior

INCOHERENT

Generic diagram for coupled threshold oscillators of relaxation

153

The pdf’s of the seizure energies and of the inter-seizure waiting times for subject 21.

Note the shoulder in each distribution, demonstrating the presence of a characteristic size and time scale, qualifying the periodic regime.

Some humans are like rats with large doses of

convulsant

154








Metastable states in random media: Self-organized critical random directed polymers

Singh, et. al., 1983, BSSA 73,

1779-1796

Southern California

Knopoff, 2000, PNAS 97, 11880-11884

Main, 1995, BSSA 85, 1299-1308

Complex magnitude distributions

Characteristic earthquakes?

Wesnousky, 1996, BSSA 86, 286-291

Probability distribution of size and power output of individual aurora region. (a) size distribution during quiet times. (b) size distribution during substorms. (c) power distribution during quiet times.(d) power distribution during substorms.

Lui et al, GEOPHYSICAL RESEARCH LETTERS, VOL. 27, NO. 7, PAGES 911-914, APRIL 1,2000

Chapman et al.,, GEOPHYSICAL RESEARCH LETTERS, VOL. 25, NO. 13, PAGES 2397-2400, JULY 1, 1998

A simple avalanche model as an analogue formagnetospheric activity

Global auroral energy deposition

Dynamic clustering in N balls in a billiardTwo balls are D-neighbors at epoch t if they collided during the time interval [t-D, t].Any connected component of this neighbor relation is called a D-cluster at epoch t .

Power-law cluster size distribution at critical instant tc for seven models with fixed N=5000 and rho=0.1, 0.01..., 0.00000001.

Critical mass of the largest cluster as a function of the billiard density rho at fixed N=5000.

Gabrielov, A., V. Keilis-Borok, Y. Sinai, and I. Zaliapin (2008) Statistical properties of the cluster dynamics of the systems of statistical mechanics. ESI Lecture Notes in Mathematics and Physics:Boltzmann's Legacy, European Mathematical Society, G. Gallavotti, W. Reiter and J. Yngvason (Eds.), 203-216.

Mechanisms for Dragon-kings

•Generalized correlated percolation

•Partial global synchronization

•A kind of condensation (a la Bose-Einstein)



•The ubiquity of “Heavy tails” (power laws)

•What are large and extremes events? (power law, stretched exponentials, dragon-king regime...)

•Promote outliers, dragon-kings … both at the individual and collective levels => “Social Bubble” hypothesis

•Entrepreneurs’ role to explore new scenarios, new horizons for wealth creation

•“Dragon-Kings vs black swans” view of the world

Highlights

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Lecture 2: Power Laws and Dragon-kingsgeza.kzoo.edu/~erdi/powerlaw/sornlec3.pdf · D-MTEC Chair of...

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