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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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