Scale-free and power law distributions via fixed points and convergence of (thinning and conditioning) transformations

TL;DR

This paper introduces a new interpretation of scale-free distributions as fixed points of a thinning and conditioning transformation, identifying unique power-law fixed points for each exponent between 1 and 2 and proving convergence of the transformation.

Contribution

It provides a novel interpretation of scale-free distributions as fixed points of a specific transformation and characterizes these fixed points for exponents in (1,2), distinct from traditional models.

Findings

Unique power-law fixed points for each β in (1,2)

Convergence of iterative transformations to these fixed points

Different from Yule–Simon power law distributions

Abstract

In discrete contexts such as the degree distribution for a graph, \emph{scale-free} has traditionally been \emph{defined} to be \emph{power-law}. We propose a reasonable interpretation of \emph{scale-free}, namely, invariance under the transformation of $p$-thinning, followed by conditioning on being positive. For each $\beta \in (1,2)$, we show that there is a unique distribution which is a fixed point of this transformation; the distribution is power-law-$\beta$, and different from the usual Yule--Simon power law-$\beta$ that arises in preferential attachment models. In addition to characterizing these fixed points, we prove convergence results for iterates of the transformation.