Empirical examples
The distributions of a wide variety of physical, biological, and man-made phenomena approximately follow a power law over a wide range of magnitudes: these include the sizes of craters on the moon and of solar flares, the foraging pattern of various species, the sizes of activity patterns of neuronal populations, the frequencies of words in most languages, frequencies ofProperties
Scale invariance
One attribute of power laws is theirLack of well-defined average value
A power-law has a well-defined mean over _only_if_Universality
The equivalence of power laws with a particular scaling exponent can have a deeper origin in the dynamical processes that generate the power-law relation. In physics, for example, phase transitions in thermodynamic systems are associated with the emergence of power-law distributions of certain quantities, whose exponents are referred to as thePower-law functions
Scientific interest in power-law relations stems partly from the ease with which certain general classes of mechanisms generate them. The demonstration of a power-law relation in some data can point to specific kinds of mechanisms that might underlie the natural phenomenon in question, and can indicate a deep connection with other, seemingly unrelated systems; see also universality above. The ubiquity of power-law relations in physics is partly due to dimensional constraints, while inExamples
More than a hundred power-law distributions have been identified in physics (e.g. sandpile avalanches), biology (e.g. species extinction and body mass), and the social sciences (e.g. city sizes and income). Among them are:Astronomy
*Physics
*ThePsychology
* Stevens's power law of psychophysics ( challenged with demonstrations that it may be logarithmic) * The power law of forgettingBiology
*Meteorology
* The size of rain-shower cells, energy dissipation in cyclones, and the diameters of dust devils on Earth and MarsGeneral science
* Exponential growth and random observation (or killing) *Progress through exponential growth and exponential diffusion of innovations * Highly optimized tolerance *Proposed form of experience curve effects * Pink noise *The law of stream numbers, and the law of stream lengths ( Horton's laws describing river systems) *Populations of cities (Mathematics
*Economics
* Population sizes of cities in a region orFinance
* The mean absolute change of the logarithmic mid-prices * Number of tick counts over time * Size of the maximum price move * Average waiting time of a directional change * Average waiting time of an overshootVariants
Broken power law
A broken power law is a piecewise function, consisting of two or more power laws, combined with a threshold. For example, with two power laws: :Power law with exponential cutoff
A power law with an exponential cutoff is simply a power law multiplied by an exponential function: :Curved power law
:Power-law probability distributions
In a looser sense, a power-lawGraphical methods for identification
Although more sophisticated and robust methods have been proposed, the most frequently used graphical methods of identifying power-law probability distributions using random samples are Pareto quantile-quantile plots (or ParetoPlotting power-law distributions
In general, power-law distributions are plotted on doubly logarithmic axes, which emphasizes the upper tail region. The most convenient way to do this is via the (complementary)Estimating the exponent from empirical data
There are many ways of estimating the value of the scaling exponent for a power-law tail, however not all of them yield unbiased and consistent answers. Some of the most reliable techniques are often based on the method ofMaximum likelihood
For real-valued,Kolmogorov–Smirnov estimation
Another method for the estimation of the power-law exponent, which does not assumeTwo-point fitting method
This criterion can be applied for the estimation of power-law exponent in the case of scale free distributions and provides a more convergent estimate than the maximum likelihood method. It has been applied to study probability distributions of fracture apertures. In some contexts the probability distribution is described, not by theValidating power laws
Although power-law relations are attractive for many theoretical reasons, demonstrating that data does indeed follow a power-law relation requires more than simply fitting a particular model to the data. This is important for understanding the mechanism that gives rise to the distribution: superficially similar distributions may arise for significantly different reasons, and different models yield different predictions, such as extrapolation. For example,See also
* Fat-tailed distribution * Heavy-tailed distributions *References
Notes Bibliography * * * * * * * * * *External links