How skew distributions emerge in evolving systems

Abstract

Despite the ubiquitous emergence of skew distributions such as power law, log-normal, and Weibull distributions, there still lacks proper understanding of the mechanism as well as relations between them. It is studied how such distributions emerge in general evolving systems and what makes the difference between them. Beginning with a master equation for general evolving systems, we obtain the time evolution equation for the size distribution function. Obtained in the case of size changes proportional to the current size are the power law stationary distribution with an arbitrary exponent and the evolving distribution, which is of either log-normal or Weibull type asymptotically, depending on production and growth in the system. This master equation approach thus gives a unified description of those three types of skew distribution observed in a variety of systems, providing physical derivation of them and disclosing how they are related.

Comment on “How skew distributions emerge in evolving systems” by Choi M. Y. et al.

“…Namely, it is straightforward to verify that eqs. (1) to (5) in [1] can be consistent with each other only at a constant N . Performing the summation over all x 1 , x 2 , .…”

“…In [1] some interesting ideas are developed how skew distributions such as power law, log-normal, and Weibull distributions emerge in general evolving systems and what makes the difference between them. However, we have found several problematic points in this consideration.…”

“…First, one has to note that the starting equation (1) in [1] is the usual master equation for a set of stochastic variables (sizes) x 1 , x 2 , . .…”

“…Nevertheless, it is used in [1] to derive equations for a system with exponentially growing total number of elements N according to dN/dt = rN . Therefore, the derivation of the basic equation (7) in [1] is not selfconsistent at r = 0. Namely, it is straightforward to verify that eqs.…”

Time evolution of entropy in a growth model: Dependence on the description

“…The general master equation governing the time evolution of the probability allows one to probe the time evolution of the entropy as well. To clarify the difference between the two entropies, we analyze the simple growth model, with no production or with uniform size production [18,19]. The state of each element is specified by its 'size', which can in general take continuous values.…”