The Rank-Size Scaling Law and Entropy-Maximizing Principle

TL;DR

This paper derives the rank-size scaling law, including Zipf's law, from entropy-maximizing principles within hierarchical city models, providing a theoretical foundation for observed scaling behaviors in social systems.

Contribution

It introduces a new derivation of Zipf's law based on local and global entropy maximization, linking hierarchical city structures to scaling laws.

Findings

Derivation of Zipf's law from entropy principles.

Identification of a scaling range with outliers.

Hierarchical laws imply rank-size regularities.

Abstract

The rank-size regularity known as Zipf's law is one of scaling laws and frequently observed within the natural living world and in social institutions. Many scientists tried to derive the rank-size scaling relation by entropy-maximizing methods, but the problem failed to be resolved thoroughly. By introducing a pivotal constraint condition, I present here a set of new derivations based on the self-similar hierarchy of cities. First, I derive a pair of exponent laws by postulating local entropy maximizing. From the two exponential laws follows a general hierarchical scaling law, which implies general Zipf's law. Second, I derive a special hierarchical scaling law with exponent equal to 1 by postulating global entropy maximizing, and this implies the strong form of Zipf's law. The rank-size scaling law proved to be one of the special cases of the hierarchical law, and the derivation suggests a certain scaling range with the first or last data point as an outlier. The entropy maximization of social systems differs from the notion of entropy increase in thermodynamics. For urban systems, entropy maximizing suggests the best equilibrium state of equity for parts/individuals and efficiency for the whole.