Power-law distributions and Lévy-stable intermittent fluctuations in stochastic systems of many autocatalytic elements

Abstract

A generic model of stochastic autocatalytic dynamics with many degrees of freedom wi, i = 1, . . . , N is studied using computer simulations. The time evolution of the wi's combines a random multiplicative dynamics wi(t + 1) = λwi(t) at the individual level with a global coupling through a constraint which does not allow the wi's to fall below a lower cutoff given by c ·w, wherew is their momentary average and 0 < c < 1 is a constant. The dynamic variables wi are found to exhibit a power-law distribution of the form p(w) ∼ w −1−α . The exponent α(c, N ) is quite insensitive to the distribution Π(λ) of the random factor λ, but it is non-universal, and increases monotonically as a function of c. The "thermodynamic" limit N → ∞ and the limit of decoupled free multiplicative random walks c → 0 do not commute: α(0, N ) = 0 for any finite N while α(c, ∞) ≥ 1 (which is the common range in empirical systems) for any positive c. The time evolution ofw(t) exhibits intermittent fluctuations parametrized by a (truncated) Lévy-stable distribution Lα(r) with the same index α. This non-trivial relation between the distribution of the wi's at a given time and the temporal fluctuations of their average is examined and its relevance to empirical systems is discussed.

Theoretical analysis and simulations of the generalized Lotka-Volterra model

“…(29), and the ex- This provides an effective lower cutoff for the range of x in which a power-law behavior is observed. This result can be compared to a somewhat simpler model studied earlier, in which the value of the lower cutoff x min is imposed as a constraint [15]. In this model, using the sum rules for the probability and the total wealth, it was found that x min = 1 − 1/α.…”

“…. , w n , t) ≡ 0, wherew(t) does not reach a steady state and diverges to infinity (for λ > 0) or collapses to 0 (for λ < 0) [15]. This can lead to changes by orders of magnitude in the total wealth or the population size without affecting the exponent α.…”

“…The generalized Lotka-Volterra system [13,14,15,18,19,20] describes the evolution in discrete time of N dynamic variables w i , i = 1, . .…”

“…These results stimulated theoretical studies in attempt to construct models that reproduce the power-law behavior and predict the value of α [13,14,15,16,17,18,19].…”

The virtues and vices of equilibrium and the future of financial economics

“…For low to medium income it has a functional form that has been variously described as exponential or log-normal [102,103], but for very high incomes it is better approximated by a power law. Since the early efforts of Champernowne, Simon, and others, the most successful theories for explaining this have been random process models for the acquisition and transfer of wealth [102,[104][105][106][107][108]. If these theories are right, then the distribution of wealth, which is one of the most remarkable and persistent properties of the economy, has little to do with the principles of equilibrium theory, and indeed little to do with human cognition.…”