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A simple stochastic mechanism that produces exact and approximate power-law distributions is presented. The model considers radially symmetric Gaussian, exponential and power-law functions inn= 1, 2, 3 dimensions. Randomly sampling these functions with a radially uniform sampling scheme produces heavy-tailed distributions. For two-dimensional Gaussians and one-dimensional exponential functions, exact power-laws with exponent –1 are obtained. In other cases, densities with an approximate power-law behaviour close to the origin arise. These densities are analyzed using Padé approximants in order to show the approximate power-law behaviour. If the sampled function itself follows a power-law with exponent –α, random sampling leads to densities that also follow an exact power-law, with exponent -n/a – 1. The presented mechanism shows that power-laws can arise in generic situations different from previously considered specialized systems such as multi-particle systems close to phase transitions, dynamical systems at bifurcation points or systems displaying self-organized criticality. Thus, the presented mechanism may serve as an alternative hypothesis in system identification problems.

Across scientific disciplines, heavy-tailed and in particular, power-law distributed quantities have received spe- cial attention due to their association with phenomena such as phase transitions, self-organized criticality and fractal patterns in space and time [

Let

where

functions (“signal shapes”) to be randomly sampled, i.e. a Gaussian, an exponential or a power-law function in one, two or three dimensions. In higher dimensions, these functions are assumed to follow the given law in any direction, i.e. to have radial symmetry. In the context of this article,

We choose the following representations, valid in any dimension.

Gaussian:

Exponential:

Power-law:

For the shape parameters it is assumed that

Assuming a radially uniform sampling on

Padé approximants of the transformed densities

We assume radially symmetric Gaussian functions in one, two and three dimensions. The radial distribution in arbitrary dimensions is given by (1.2). Let us now assume the Gaussian function is randomly sampled with the radially uniform sampling scheme (1.5), where the sampling volume is given by

and derivative

In

We observe an exact power-law distribution

In this section, radially symmetric exponential shapes as given by (1.3) in

Radially uniform random sampling of Gaussian functions in n = 1, 2, 3 dimensions yield exact and approximate power-law distributions (black curves). In the case n = 1, an exact power-law with exponent −1 is obtained. The blue curves are the Padé approximants to the exact distrubutions P(y). For visualization purposes, the blue curves are offset by a fixed amount

In the exponential case, an exact power-law distribution

Finally, we ask which amplitude distribution

Random sampling of exponential functions in n = 1, 2, 3 dimensions yield an exact power-law distribution with exponent −1 for n = 1. For n = 2, 3, an approximate power-law be- ha- viour is observed for. Blue curves are the Padé approximants to the exact distrubutions P(y). For visualization purposes, the blue curves are offset by a fixed amount

In this case, exact power-laws with exponents

shows the randomly sampled densities

In

the form

dimension, exact power-laws with exponent

A small numerical example is presented to illustrate the connection between the theoretically derived results and possible implications for experimental data. Consider an experiment where Gaussian shaped signals occur at a random distance

Random sampling of power-law functions in n dimensions produces exact power-law dis- tributions P(y) with exponent

Numerical example. In the left panel, a generic experimental setting is illu- strated. A sensor (S) is placed at a fixed location and Gaussian shaped events occur at random distances x from the sensor S, within a disc shaped 2D region of radius R. The amplitude of the Gaussian y measured at the sensor site decreases with increasing dis- tance x. The right panel shows the empirical distribution of event amplitudes P(y) (blue circles, n = 10^{4} samples) in double logarithmic coordinate axes to emphasize the exact power-law character of the empirical distribution. A power-law fit to the data (black solid line) yields an exponent of, a close fit to the theoretically de- rived exponent α = −1

in the right panel of

In the present work, a simple mechanism for the generation of power-law distributions is derived. The idea is

Table 1. First-order Padé approximants of the densities P(y) are given by. The table shows the coefficients for Gaussian and exponential functions in n dimensions, denoted Gaussian-n and Exponential-n.

Gaussian-1 | |||

Gaussian-2 | |||

Gaussian-3 | |||

Exponential-1 | |||

Exponential-2 | |||

Exponential-3 |

based upon a realistic scenario in experimental sciences. A signal of a given shape, e.g. a Gaussian or an ex- ponential, is measured by a sensor at a random distance x to the signal maximum. Random sampling arises when the Gaussian or exponentially shaped signal occurs randomly distributed across space (with density