A classification of the natural and social distributions Part 2: the explanations

TL;DR

This paper surveys various models explaining the statistical regularities in natural and social distributions, highlighting the rarity of models producing complete power laws and the uncommonness of Zipf's law with an exponent of 1.

Contribution

It provides a comprehensive classification of models explaining natural and social distributions, emphasizing the scarcity of models that produce full power laws and the rarity of Zipf's law with an exponent of 1.

Findings

Few models produce a power law for the entire distribution.

Models with Zipf exponent 1 are rare.

Power law models are mostly limited to distribution tails.

Abstract

In this second part of our survey on the social and natural distributions, we investigate some models, which intend to explain the statistical regularity of the natural and social distributions. There is a large variety of models and in their majority, they look for a power law, at least in the tail, although there are several real distributions which are not described by a power law. Among the power law models, we discuss a) the two basic models and their variants: the random multiplicative model and the preferential attachment model; b) models based on the BoseEinstein statistics; c) geographical, economical, and criticality models. We present also some models, which do not intend to explain a power law, and among them lognormal-like distributions, exponential and stretched exponential distributions. The interesting findings of this survey are that there are few models giving a power law for the complete distribution and that among them, the Zipf exponent 1 is rare.