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.
Heavy-Tailed Distributions Random Sampling Gaussian Exponential Power-Law1. Introduction
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 [1] -[5] . Power-laws are often contrasted with exponential and Gaussian dis- tributions that typically occur in spatiotemporal correlation functions and as distributions of characteristic quan- tities in standard equilibrium kinetics [1] [6] . However, there exists no unique mechanism for the generation of power-law behaviour [2] [6] -[9] . Therefore, in the context of system identification, the occurrence of a power- law cannot be used to infer the mechanisms governing the generating process. We here present a simple mecha- nism producing exact and approximate power-law distributions. In the presented model, Gaussian, exponential and power-law functions in one, two and three dimensions are uniformly random-sampled. The resulting am- plitude distributions of the random samples show exact and approximate power-law functional forms. Exact po- wer-law distributions with exponent −1 are obtained for one-dimensional exponential and two-dimensional Gaus- sian distributions. Generalized power-laws with arbitrary scaling exponents are obtained from randomly sam- pled power-law functions. The presented mechanism can easily be imagined to occur in diverse experimental settings where a sensor at a fixed location samples a signal, of Gaussian shape for instance, which occurs at a random distance of the sensor site. Given this generic mechanism for the generation of power-law distributions, our model may serve as an alternative mechanism to be accounted for whenever a power-law distribution is found in an experimental setting.
2. Background
Let be a random variable over, where represents the Borel sets over and let be the probability density of. By conservation of probability, for any monotonous, differentiable transformation, the density is obtained from the random variable transformation theorem [10] :
where is the continuous derivative of the inverse of. In the following, will be one of the
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, represents the radial variable, commonly denoted as, in polar or spherical coordinates.
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, we obtain the follow- ing expressions for in dimensions:
Padé approximants of the transformed densities were calculated with the CAS maxima (http://maxima.sourceforge.net/).
3. Randomly Sampled Gaussian, Exponential and Power-Law Functions3.1. Gaussians
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. The function
is a continuous, bijective mapping with inverse
and derivative
In dimensions, random variable transformation (1.1) yields the densities:
We observe an exact power-law distribution with power-law exponent in dimen- sions. Figure 1 shows the randomly sampled densities in dimensions.
3.2. Exponentials
In this section, radially symmetric exponential shapes as given by (1.3) in dimensions are analyzed. The exponential defines a continuous, bijective mapping. The inverse is given by
Figure 1
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 with exponent is obtained in one dimen- sion (in n = 1). In dimensions, the distributions approximately follow a power-law for. This behaviour is visualized in Figure 2 and analyzed quantitatively using Padé approximants further below. Figure 2 shows the randomly sampled densities in dimensions.
3.3. Power-Laws
Finally, we ask which amplitude distribution is obtained by randomly sampling functions that already follow a power-law. The function in arbitrary dimensions is given by (1.4). In order to obtain a con- tinuous, bijective mapping, domain and co-domain are set accordingly, , where. The inverse is given by
Figure 2
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 are obtained in any dimension. Figure 3
shows the randomly sampled densities.
4. Padé Approximants
In Figure 1 and Figure 4, an approximate power-law behaviour of is observed for small values of. In order to quantify this behaviour, we computed Padé approximants of order of the densities [11] . The approximants were calculated from the Taylor expansions of at the left border of the codomain of. For first order approximations, the densities can be approximated by functions of
the form. As derived above, sampling a Gaussian in two dimensions or an exponential in one
dimension, exact power-laws with exponent are obtained. In these cases, the Padé approximants yield the exact result. In the other cases, the Padé approximants yield functions that follow the density close to the origin. The Padé coefficients are given in Table 1.
5. A Numerical Example
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 to a fixed sensor. This situation is illustrated in the left panel of Figure 3, with the sensor at the center. In analogy to the analytical derivations, all events are assumed to occur within a two-di- mensional disc of radius. The amplitude y of the event measured at site depends on the random distance between the sensor and the center of the event (dashed line). We simulated events at random dis- tances from S and recorded the amplitude y as measured at S. The right panel of Figure 4 shows the resulting empirical distribution (blue circles) in double logarithmic coordinates. The linear shape suggests a power-law behaviour of the distribution. Estimating the exponent yields (fitted distribution as black solid line
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 = 104 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 Figure 4), a result close to the theoretically derived distribution with ex- ponent.
6. Discussion
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.
Table 1
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) and the sensor resides at a fixed site. Our derivation shows that two-dimensional Gaussians and one-dimensional exponentials lead to exact power-law densities with exponent and that approximate power-law densities arise in other dimensions. Indeed, this mechanism has been observed experimentally in dynamic fluorescence microscopy of subcellular calcium currents [12] . The result is of interest as it provides a simple and realistic mechanism that produces exact power-laws. Power-laws are often associated with specialized mechanisms such as phase transitions in complex systems, bifurcation points of dynamical systems or systems displaying self- organized criticality and relatively few authors have considered alternative mechanisms [6] . The mechnism pre- sented here is generic and may serve as an alternative hypothesis in cases where power-law distributions are ob- served in experimental settings.
ReferencesHOHENBERG P.C. , HALPERIN B.I. ,et al. (1977)THEORY OF DYNAMIC CRITICAL PHENOMENA. REVIEWS OF MODERN PHYSICS 49, 435-479.HTTP://DX.DOI.ORG/10.1103/REVMODPHYS.49.435MITZENMACHER M. ,et al. (2003)A BRIEF HISTORY OF GENERATIVE MODELS FOR POWER LAW AND LOGNORMAL DISTRIBUTIONS. INTERNET MATHEMATICS 1, 226-251.HTTP://DX.DOI.ORG/10.1080/15427951.2004.10129088MONTROLL M. , SHLESINGER M.F. ,et al. (1983)MAXIMUM ENTROPY FORMALISM, FRACTALS, SCALING PHENOMENA, AND 1/F NOISE: A TALE OF TAILS. THE JOURNAL OF CHEMICAL PHYSICS 32, 209-230.HTTP://DX.DOI.ORG/10.1007/BF01012708NEWMAN, M.E.J. (2005) POWER LAWS, PARETO DISTRIBUTIONS AND ZIPFS LAW, CONTEMPORARY PHYSICS.HTTP://DX.DOI.ORG/10.1080/00107510500052444STANLEY H.E. ,et al. (1999)SCALING, UNIVERSALITY, AND RENORMALIZATION: THREE PILLARS OF MODERN CRITICAL PHENOMENA. REVIEWS OF MODERN PHYSICS 71, S358-S366.HTTP://DX.DOI.ORG/10.1103/REVMODPHYS.71.S358SORNETTE, D. (2004) CRITICAL PHENOMENA IN NATURAL SCIENCES: CHAOS, FRACTALS, SELFORGANIZATION, AND DISORDER: CONCEPTS AND TOOLS. SPRINGER, NEW YORK.MALMGREN R.D., STOUFFER D.B., MOTTER A.E. , AMARAL L.A.N. ,et al. (2008)A POISSONIAN EXPLANATION FOR HEAVY TAILS IN E-MAIL COMMUNICATION. PROCEEDINGS OF NATIONAL ACADEMY SCIENCE OF USA 105, 18153-18158.HTTP://DX.DOI.ORG/10.1073/PNAS.0800332105SORNETTE D. ,et al. (1998)MULTIPLICATIVE PROCESSES AND POWER LAWS. PHYSICAL REVIEW E 57, 4811-4813.HTTP://DX.DOI.ORG/10.1103/PHYSREVE.57.4811TOUBOUL, J. AND DESTEXHE, A. (2010) CAN POWER-LAW SCALING AND NEURONAL AVALANCHES ARISE FROM STOCHASTIC DYNAMICS? PLOS ONE, 5, E8982.RAMACHANDRAN, K.M. AND TSOKOS, C.P. (2009) MATHEMATICAL STATISTICS WITH APPLICATIONS. ACADEMIC PRESS.JR. BAKER, G.A. AND GRAVES-MORRIS, P. (1996) PADÉ APPROXIMANTS. CAMBRIDGE UNIVERSITY PRESS, NEW YORK.RÍOS E., SHIROKOVA N., KIRSCH W.G., PIZARRO G., STERN M.D., CHENG H. , GONZÁLEZ A. ,et al. (2001)A PREFERRED AMPLITUDE OF CALCIUM SPARKS IN SKELETAL MUSCLE. BIOPHYSICAL JOURNAL 80, 169-183.HTTP://DX.DOI.ORG/10.1016/S0006-3495(01)76005-5
. Blue curves are the Padé approximants to the exact distrubutions P(y). For visualization purposes, the blue curves are offset by a fixed amount
, a close fit to the theoretically de- rived exponent α = −1
. The table shows the coefficients for Gaussian and exponential functions in n dimensions, denoted Gaussian-n and Exponential-n.
