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Scale-free and power law distributions via fixed points and convergence of (thinning and conditioning) transformations

   
@article{ECP2923,
	author = {Richard Arratia and Thomas Liggett and Malcolm Williamson},
	title = {Scale-free and power law distributions via  fixed points and convergence of  (thinning and conditioning) transformations},
	journal = {Electron. Commun. Probab.},
	fjournal = {Electronic Communications in Probability},
	volume = {19},
	year = {2014},
	keywords = {thinning, power-law, scale-free, degree distribution, Pareto distribution},
	abstract = {
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.
},
	pages = {no. 39, 1-10},
	issn = {1083-589X},
	doi = {10.1214/ECP.v19-2923},    
        url = {http://ecp.ejpecp.org/article/view/2923}}