Abstract
Many complex systems are characterized by power-law distributions. In this article, we show that for various examples of power-law distributions, including the two probably most popular ones, the Pareto law for the wealth distribution and Zipf’s law for the occurrence frequency of words in a written text, the power-law tails of the probability distributions can be decomposed into a statistical mixture of canonical equilibrium probability densities of the subsystems. While the interacting units or subsystems have canonical distributions at equilibrium, as predicted by canonical statistical mechanics, the heterogeneity of the shapes of their distributions leads to the appearance of a power-law.
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Acknowledgments
M.P., E.H., and A.C. acknowledge support from the Estonian Science Foundation Grant no. 9462 and the Institutional Research Funding IUT (IUT39-1) of the Estonian Ministry of Education and Research. A.C. acknowledges financial support from grant number BT/BI/03/004/2003(C) of Government of India, Ministry of Science and Technology, Department of Biotechnology, Bioinformatics Division. L.M. acknowledges the Estonian Research Council for supporting his work with the grant PUTJD110.
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Appendix: Text analyzed
Appendix: Text analyzed
We constructed the text to be analyzed by merging in a single text file all the crime stories available on-line from the Gutenberg Project (www.gutenberg.org), mostly novels, by e.g. Christie, Collins, Davies, etc. The final file was a plain text file with a size of about 27 MB, containing about 50 millions words of which about 57,000 words were different from each other.
In order to extract the probability distribution of the occurrence frequency, we constructed a set of texts of equal size by dividing the original file into parts, each one containing a number \(N_W =\) 3,000 of words. The last part containing less than \(N_W\) words was neglected.
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Patriarca, M., Heinsalu, E., Marzola, L., Chakraborti, A., Kaski, K. (2016). Power-Laws as Statistical Mixtures. In: Battiston, S., De Pellegrini, F., Caldarelli, G., Merelli, E. (eds) Proceedings of ECCS 2014. Springer Proceedings in Complexity. Springer, Cham. https://doi.org/10.1007/978-3-319-29228-1_23
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