POWER LAWS

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POWER LAWS Bridges between microscopic and
macroscopic scales
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  Wolff's 1996 findings regarding the holdings
of the top 1, top 5, and top 10 of the U.S.
population in 1992 are reported in Table
Table 2
Pk denotes the percentage of total wealth held by
the top k percent
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Davis 1941 No. 6 of the Cowles Commission for
Research in Economics, 1941. No one however, has
yet exhibited a stable social order, ancient or
modern, which has not followed the Pareto pattern
at least approximately. (p. 395) Snyder
1939 Paretos curve is destined to take its
place as one of the great generalizations of
human knowledge
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LOGISTIC EQUATIONS History, Applications
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Malthus autocatalitic proliferation

dX/dt a X with a birth rate -
death rate
exponential solution X(t) X(0)ea t
contemporary estimations doubling of the
population every 30yrs
Verhulst way out of it dX/dt a X c X2
Solution exponential ?saturation at X
a / c
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c X2 competition for resources and
other the adverse feedback effects
?saturation of the population to the value X a
/ c
For humans data at the time could not
discriminate between exponential growth of
Malthus and logistic growth of Verhulst But
data fit on animal population sheep in
Tasmania exponential in the first 20 years
after their introduction and saturated
completely after about half a century.
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Confirmations of Logistic Dynamics pheasants turtl
e dove humans world population for the last
2000 yrs and US population for the last 200 yrs,
bees colony growth escheria coli
cultures, drossofilla in bottles, water flea at
various temperatures, lemmings etc.
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Montroll Social dynamics and quantifying of
social forces almost all the social phenomena,
except in their relatively brief abnormal times
obey the logistic growth''. - default universal
logistic behavior generic to all social systems
- concept of sociological force which induces
deviations from it Social Applications of the
Logistic curve technological change
innovations diffusion (Rogers) new product
diffusion / market penetration (Bass) social
change diffusion
dX/dt X(N
X )
X number of people that have already adopted
the change and N the total population
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Sir Ronald Ross ? Lotka generalized the
logistic equation to a system of coupled
differential equations for malaria in
humans and mosquitoes
a11 spread of the disease from humans to humans
minus the percentages of deceased and healed
humans a12 rate of humans infected by
mosquitoes a112 saturation (number of humans
already infected becomes large one cannot count
them among the new infected). The second
equation same effects for the mosquitoes
infection
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Volterra
d Xi Xi (ai - ci F ( X1 , X n ) )
Xi the population of species i ai growth
rate of population i in the absence of
competition and other species F interaction
with other species predation
competition symbiosis Volterra assumed F a1 X1
an Xn
more rigorous ? Kolmogorov.
MPeshel and W Mende The Predator-Prey Model Do
we live in a Volterra World? Springer Verlag,
Wien , NY 1986
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Eigen Darwinian selection and evolution in
prebiotic environments. Autocatalyticity The
fundamental property of life capital
ideas institutions each type i of DNA and RNA
molecules replicate in the presence of proteins
at rate ai Typos instead of generating an
identical molecule of type i probability rate aij
for a mutated molecule of type j d Xi Xi (ai -
cii Xi - ?j cij Xj) ?j aij Xj -?j aij Xi cij
competition of the replicators for various
resources ? saturation
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Mikhailov Eigen equations relevant to market
economics i agents that produce a certain kind
of commodity Xi amount of commodity the agent
i produces per unit time The net cost to an
individual agent of the produced commodity is
Vi ai Xi
ai specific cost which includes expenditures
for raw materials machine depreciation labor
payments research etc Price of the commodity on
the market is c c (X.,t) ?i ai Xi / ?i Xi
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The profits of the various agents will then be
ri c (X.,t) Xi - ai Xi
Fraction k of it is invested to expand production
at rate d Xi k (c (X.,t) Xi - ai Xi )
These equations describe the competition between
agents in the free market This ecology market
analogy was postulated already in Schumpeter and
Alchian See also Nelson and Winter Jimenez and
Ebeling Silverberg Ebeling and Feistel Jerne Aoki
etc account for cooperation exchange between the
agents d Xi k (c (X.,t) Xi - ai Xi ) ?j aij Xj
-?j aij Xi
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Eigen aij ?variability of the population
adaptability and survival
economic system social security or
some form of mutual help conglomerates
with aij ? more stable in a
stochastically changing environment
agents that are not fit now might become later
the fittest
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GLV and interpretations
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wi (tt) wi (t) ri (t) wi (t) a w(t)
c(w.,t) wi (t) w(t) is the average of wi (t)
over all i s at time t a and c(w.,t) are of
order t c(w.,t) means c(w1,. . ., wN,t) ri (t)
random numbers distributed with the same
probability distribution independent of i with a
square standard deviation lt ri (t) 2gt D of
order t One can absorb the average ri (t) in
c(w.,t) so lt ri (t) gt 0
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wi (tt) wi (t) ri (t) wi (t) a w(t)
c(w.,t) wi (t) admits a few practical
interpretations wi (t) the individual wealth of
the agent i then ri (t) the random part of the
returns that its capital wi (t) produces during
the time between t and tt a the autocatalytic
property of wealth at the social level the
wealth that individuals receive as members of the
society in subsidies, services and social
benefits. This is the reason it is proportional
to the average wealth This term prevents the
individual wealth falling below a certain minimum
fraction of the average. c(w.,t) parametrizes the
general state of the economy large and positive
correspond boom periods negative recessions
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c(w.,t) limits the growth of w(t) to values
sustainable for the current
conditions and resources external limiting
factors finite amount of resources and money
in the economy technological inventions wars
, disasters etc internal market effects
competition between investors adverse
influence of self bids on prices
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A different interpretation a set of companies i
1, , N wi (t) shares prices
capitalization of the company i total
wealth of all the market shares of the company
ri (t) fluctuations in the market worth of the
company relative changes in individual
share prices (typically fractions of
the nominal share price) aw correlation
between wi and the market index w c(w.,t) usually
of the form c w ? represents competition Time
variations in global resources may lead to lower
or higher values of c ?increases or decreases in
the total w
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Yet another interpretation investors herding
behavior wi (t) number of traders adopting a
similar investment policy or position. they
comprise herd i one assumes that the sizes of
these sets vary autocatalytically according to
the random factor ri (t) This can be justied by
the fact that the visibility and social
connections of a herd are proportional to its
size aw represents the diffusion of traders
between the herds c(w.,t) popularity of the
stock market as a whole competition
between various herds in attracting individuals
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POWER LAWS IN GLV
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Crucial surprising fact as long as the term
c(w.,t) and the distribution of the ri (t) s
are common for all the i s the Pareto power
law P(wi) wi 1-a holds and its exponent
a is independent on c(w.,t) This an important
finding since the i-independence corresponds to
the famous market efficiency property in
financial markets
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take the average in both members of wi (tt)
wi (t) ri (t) wi (t) a w(t) c(w.,t) wi (t)
assuming that in the N ? limit the random
fluctuations cancel w(tt) w(t)
a w(t) c(w.,t) w (t) It is of a
generalized Lotka-Volterra type with quite
chaotic behavior
x i (t) w i (t) / w(t)
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and applying the chain rule for differentials d
xi (t) dxi (t) dwi (t) / w(t) - w i (t) d
(1/w) dwi (t) / w(t) - w i (t) d
w(t)/w2 ri (t) wi (t) a w(t) c(w.,t) wi
(t)/ w(t) -w i (t)/w a w(t) c(w.,t) w
(t)/w ri (t) xi (t) a c(w.,t)
xi (t) -x i (t) a c(w.,t)
crucial cancellation the system splits into a
set of independent linear stochastic differential
equations with constant coefficients
(ri (t) a ) xi (t) a
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dxi (t) (ri (t) a ) xi (t) a Rescaling in t
means rescaling by the same factor in lt ri (t) 2gt
D and a therefore the stationary asymptotic time
distribution P(xi ) depends only on the ratio
a/D Moreover, for large enough xi the additive
term a is negligible and the equation reduces
formally to the Langevin equation for ln xi (t)
d ln xi (t) (ri (t) a ) Where temperature
D/2 and force -a gt Boltzmann distribution
P(ln xi ) d ln xi exp(-2 a/D ln xi ) d ln xi
xi -1-2 a/D d xi In fact, the exact
solution is P(xi ) exp-2 a/(D xi ) xi -1-2
a/D
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P(w)
10-1
t0
10-2
10-3
10-4
10-5
10-6
10-7
10-8
10-9
100
10-3
10-2
10-1
10-4
w
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P(w)
10-1
10-2
10-3
10-4
t10 000
10-5
10-6
10-7
10-8
10-9
100
10-3
10-2
10-1
10-4
w
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P(w)
10-1
10-2
10-3
10-4
10-5
10-6
10-7
t100 000
10-8
10-9
100
10-3
10-2
10-1
10-4
w
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P(w)
10-1
10-2
10-3
10-4
10-5
10-6
10-7
10-8
t1 000 000
10-9
100
10-3
10-2
10-1
10-4
w
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P(w)
10-1
10-2
10-3
10-4
10-5
10-6
t30 000 000
10-7
10-8
10-9
100
10-3
10-2
10-1
10-4
w
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P(w)
10-1
t0
10-2
10-3
10-4
10-5
10-6
t10 000
t30 000 000
10-7
t100 000
10-8
t1 000 000
10-9
100
10-3
10-2
10-1
10-4
w
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K amount of wealth necessary to keep 1 alive
If wmin lt K gt revolts
L average number of dependents per average
income
Their consuming drive the food, lodging,
transportation and services prices to values that
insure that at each time wmean gt KL
Yet if wmean lt KL they strike and overthrow
governments.
So cx min 1/L
Therefore a 1/(1-1/L) L/(L-1)
For L 3 - 4 , a 3/2 4/3 for internet L
average nr of links/ site
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In Statistical Mechanics, if not detailed
balance ? no Boltzmann
In Financial Markets, if no efficient market
?no Pareto
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Further Analogies
Thermal Equilibrium
Efficient Market
Pareto Law
Boltzmann law
One cannot extract energy from systems in thermal
equilibrium
One cannot gain systematically wealth from
efficent markets
Except for Maxwell Demons with microscopic
information
Except if one has access to detailed private
information
By extracting energy from non-equilibrium systems
, one brings them closer to equilibrium
By exploiting arbitrage opportunities, one
eliminates them (makes market efficient)
Irreversibility II Law of Theromdynamics Entropy
Irreversibility ? ?
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Market Fluctuations Scaling
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Feedback Volatility ?? Returns gt Long range
Volatility correlations
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