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

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.

10 pages, 19 references, Electronic Communications in Probability, volume 19, 2014