Celebratio Mathematica — Liggett — Complete Bibliography

Complete Bibliography

Works connected to Malcolm J. Williamson.

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R. Arratia, T. M. Liggett, and M. J. Williamson: “Scale-free and power law distributions via fixed points and convergence of (thinning and conditioning) transformations,” Electron. Commun. Probab. 19 (2014). Article no. 39, 10 pages. MR 3225870 Zbl 1320.60010 ArXiv 1306.3017 article

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.

Malcolm J. Williamson.	Related
Richard Alejandro Arratia	Related