Relationship between exponent of power-law distributions and exponent of cumulative distributions

TL;DR

This paper investigates the relationship between the exponents of power-law and cumulative distributions, proving their equality for geometrically growing domains and validating with numerical simulations.

Contribution

It provides a theoretical proof linking power-law exponents to cumulative distribution exponents for specific distributions, correcting previous approaches.

Findings

Exponent of power-law equals exponent of cumulative distribution for geometrically growing domains

Numerical simulations confirm the theoretical relationship

Clarifies previous misconceptions in the estimation of degree distributions

Abstract

We commented on Ref.[Andrade J S, Herrmann H J, Andrade R F S, et al. Phys. Rev. Lett. 94, 018702 (2005)] and corrected the approach to estimate the degree distribution of the Apollonian network. However, after reading our manuscript, Herrmann indicated that it was due to a small typographic error and Herrmann et al. published Ref. [Andrade J S, Herrmann H J, Andrade R F S, et al. Phys. Rev. Lett. 102, 079901 (2009)]. In this paper, the relationship between an exponent of power-law distributions and the exponent of cumulative distributions is studied. For power-law distribution with geometrically growing domain, we prove that its exponent is equal to the exponent of its cumulative distribution. We carried out numerical simulations and obtain results that are in good agreement with the theoretical analysis.