Scale-free and Power Law Distributions via Fixed Points and Convergence of (thinning and Conditioning) Transformations

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