an:06349197
Zbl 1320.60010
Arratia, Richard; Liggett, Thomas M.; Williamson, Malcolm J.
Scale-free and power law distributions via fixed points and convergence of (thinning and conditioning) transformations
EN
Electron. Commun. Probab. 19, Paper No. 39, 10 p. (2014).
00336550
2014
j
60B10 05C82
thinning; power-law; scale-free; degree distribution; Pareto distribution
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.