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Encyclopedia of Complexity and Systems Science pp 7009–7024Cite as

Probability Distributions in Complex Systems

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This core article for the Encyclopedia of Complexity and System Science (Springer Science) reviews briefly the concepts underlying complex systemsand probability distributions. The latter are often taken as the first quantitative characteristics of complex systems, allowing one to detect thepossible occurrence of regularities providing a step toward defining a classification of the different levels of organization (the“universality classes”). A rapid survey covers the Gaussian law, the power law and the stretched exponential distributions. Thefascination for power laws is then explained, starting from the statistical physics approach to critical phenomena, out-of‐equilibrium phasetransitions, self‐organized criticality, and ending with a large, but not exhaustive, list of mechanisms leading to power lawdistributions. A checklist for testing and qualifying a power law distribution from data is described in seven steps. This essay enlarges thedescription of...

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Abbreviations

Complex system:

A system with a large number of mutually interacting parts, often open to its environment, which self‐organizes its internal structure and its dynamics with novel and sometimes surprising macroscopic “emergent” properties.

Criticality (in physics):

A state in which spontaneous fluctuations of the order parameter occur at all scales, leading to diverging correlation length and susceptibility of the system to external influences.

Power law distribution:

A specific family of statistical distribution appearing as a straight line in a log-log plot; exhibits the property of scale invariance and therefore does not possess characteristic scales.

Self‐organized criticality:

Occurs when the system dynamics are attracted spontaneously, without any obvious need for parameter tuning to a critical state with infinite correlation length and power law statistics.

Stretched‐exponential distribution:

A specific family of sub‐exponential distribution interpolating smoothly between the exponential distribution and the power law family.

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Authors and Affiliations

  1. Department of Management, Technology and Economics, ETH Zurich, Zurich, Switzerland

    Didier Sornette

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  1. Didier Sornette
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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Sornette, D. (2009). Probability Distributions in Complex Systems. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_418

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