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Scale-free and power law distributions via fixed points and convergence of (thinning and conditioning) transformations. (English) Zbl 1320.60010
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.

MSC:
 60B10 Convergence of probability measures 05C82 Small world graphs, complex networks (graph-theoretic aspects)
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