During the study of the LLS agent based model of financial markets we realized that in certain conditions, when the time variations of the individuals wealths w (t) are random fractions of the wealth
w (t+1) = λ w(t) (1)
(with λ > 0 a random variable extracted from a probability distribution P(λ))
the wealth distribution in the system is scaling:
P(w> W) ~ W -µ (2)
Basically we rediscovered 100 later the Pareto law and the mechanisms leading to skew distributions as described by Yule, Gibrat, Simon , Kesten, etc.
The first, not very pedagogical papers were:
Dynamical Explanation for the Emergence of Power Law in a Stock Market Model, M. Levy and S. Solomon, Int J of Mod Phys C, Vol. 7, No. 1 (1996) 65-72
Spontaneous Scaling Emergence in Generic Stochastic Systems, S. Solomon and M. Levy Int. J. Mod. Phys. C , Vol. 7, No. 5 (1996) 745 ;
POWER LAWS ARE LOGARITHMIC BOLTZMANN LAWS, M. LEVY and S. SOLOMON, Int. J. Mod. Phys. C V7, 595 (1996)
New evidence for the power-law distribution of wealth, M. Levy and S. Solomon Physica A 242 (1997) 90.
However, the particular context in which we rediscovered it, solved a few problems outstanding in the previous treatments:
By introducing an interaction (a kind of social security), between the wealth wi (t) of each individual i and the current average wealth wave = Σi wi (t) /N (t) one makes the power law hold (with fixed exponent μ) even in arbitrary non-stationary situations (time dependent Π(λ,t)) and even if the number of elements N(t) is variable too:
Power laws in cities population, financial markets and internet sites (scaling in systems with a variable number of components), Aharon Blank and Sorin Solomon, Physica A 287 (1-2) (2000) pp.279-288
If one supplements the dynamics
wi (t+1) = λi (t) wi (t) i=1,…, N(t) (3)
[where λi (t)> 0 are extracted from the same, possibly time dependent but i-independent arbitrary probability distribution Π(λ,t)]
with the condition
wi (t+1) > c wave(t+1) (4)
( where c an arbitrary constant 0<c<1 )
[i.e. each time that a wi (t+1) falls because of (3) below (4) it will be restored to wi (t+1) = c wave (t+1)]
Then the wi will converge quickly to the power law
P(wi > W) = W-1/(1-c) for W > c wave(t+1) (5)
This is in spite of wave (t) having an arbitrarily irregular time variations induced by the arbitrary variations in the probability distribution Π(λ,t).
Another version of the above mechanism made the contact to the celebrated logistic equation (we called Generalized Lotka Volterra equation GLV):
Generic emergence of power law distributions and Lévy-stable intermittent fluctuations in discrete logistic systems. O. Biham, O. Malcai, M. Levy and S. Solomon, Phys. Rev. E 58, 1352 (1998).
More precisely the dynamics
wi (t+1) = λi (t) wi (t) + a wave + d(w1,…,wN,t) wi (6)
leads for large wi to a power law
P(wi > W) = W-1- 2 a/D (7)
independent on the arbitrary function d(w1,…,wN,t)
For a more general dynamics and the functional dependence in (7) for arbitrary W see
Stable power laws in variable economies, S. Solomon and P. Richmond, Eur. Phys. J. B 27, 257-261 (2002)
We continued the study of these systems in many papers listed below but a crucial moment was the confirmation in
Pioneers on a new continent: on physics and economics, S Solomon and M Levy Quantitative Finance (IoP) V 3, N 1, C12 Feb 2003
of the theoretical prediction that the time fluctuations of wave (t) [identified with the stock market index] are characterized by a fractal exponent
equal to the Pareto exponent in (7) [representing the wealth distribution in the same economic system].
Power-law distributions and L\’evy-stable intermittent fluctuations in stochastic systems of many autocatalytic elements. O. Malcai O. Biham and S. Solomon, Phys. Rev. E, 60, 1299 (1999).
Finite market size as a source of extreme wealth inequality and market instability Zhi-Feng Huang and Solomon Physica A , 294 (3-4) (2001) pp. 503
Power Laws of Wealth, Market Order Volumes and Market Returns, Sorin Solomon, Peter Richmond Physica AVol. 299 (1-2) (2001) pp. 188-197
Power Laws are Boltzmann Laws in Disguise Powed Laws Are Disguised Boltzmann Laws, Peter Richmond, Sorin Solomon, International Journal of Modern Physics C, Vol. 12, No. 3 (2001) 333-343
Stochastic Multiplicative Processes for Financial Markets, Zhi-Feng Huang, Sorin Solomon Physica A 306 (2002) 412-422 [cond-mat/0110273] Download PDF
Co-Evolutionist Stochastic Dynamics: Emergence of Power-LawsPower-laws in stochastic Lotka-Volterra-Eigen-Schuster Systems S. Solomon, P. Richmond, O. Biham and O. Malcai, in: “Towards Cognitive Economics”, Eds: P. Bourgine and J.-P. Nadal, Springer (2003). Download PDF
Non-equilibrium and irreversible simulation of competition among languages, D. Stauffer, C. Schulze, F.W.S. Lima, S. Wichmann, S. Solomon; Physica A 371 (2006) 719–724 Download PDF
Copying nodes versus editing links: the source of the difference between genetic regulatory networks and the WWW, Yoram. Louzoun, LevMuchnickand Sorin Solomon Bioinformatics (Oxford University Press 2006) 22(5):581-588; Download PDF
The Forbes 400 and the Pareto wealth distribution, Economics Letters Volume 90, Issue 2, February 2006, Pages 290-295, Download PDF
The Forbes 400, the Pareto power-law and efficient markets, O.S. Klass, O. Biham, M. Levy, O. Malcai, and S. Solomon; Eur. Phys. J. B 55(2),143–147 (2007) Download PDF
Slicing and Dicing the Genome: A Statistical Physics Approach to Population Genetics YE Maruvka, NM Shnerb, S Solomon Journal of Statistical Physics, 142(6), 1302-1316, Springer, 2011
Runaway events dominate the heavy tail of citation distributions, M Golosovsky, S Solomon The Eur Phys Jou-Special Topics 205 (1), 303, 3 2012
Stochastic Dynamical Model of a Growing Citation Network Based on a Self-Exciting Point Process , M Golosovsky, S Solomon Physical Review Letters 109 (9), 98701 3 2012
The transition towards immortality: non-linear autocatalytic growth of citations to scientific papers. Golosovsky, M, and S Solomon. Journal of Statistical Physics,Feb 2013.