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 $, we show that there is a unique distribution which is a fixed point of this transformation; the distribution is power-law-$\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.
"Scale-free and power law distributions via fixed points and convergence of (thinning and conditioning) transformations." Electron. Commun. Probab. 19 1 - 10, 2014. https://doi.org/10.1214/ECP.v19-2923