Summary: 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 \(\beta \in (1,2)\), 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.