2013/06/13 by Arratia, Richard, Liggett, Thomas M., Williamson, Malcolm J.
#05C80 #60B10 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1306.3017
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.