Los desafíos

Power-law relationships can be used to describe two kinds of distributions of values: continuous and discrete distributions. A continuous power-law distribution describes continuous real values, while a discrete power-law distribution is for discrete values, ordinarily positive integers.

Let _X be an independent variable and a function _p(_x) be a corresponding func- tion that measures the quantity of the observed value _xof _X. A continuous power- law distribution uses _p(_x) to indicate the sum of the corresponding values of _p(_x) when _x is within a particular interval (_{a, b}]. When a continuous power-law distri- bution is used to describe a probability distribution, _p(_x) is the probability density function (pdf) of the probability distribution, which can be expressed as:

P r(_{a < X b}) = b a p (_x)_dx= b a Cx −α_{dx , (3.2)} where

• x is the observed value of_X;

• C is the normalizing constant;

• α is the exponent, or scaling, or scaling exponent parameter.

For example, the Pareto distribution is a Power-law continuous probability distri- bution whose probability density function is

p(_x) = αx α m

x(α+1) , (3.3)

where _xm is the minimum possible positive real value of _X and _α is a positive real value.

Supposing that _{α >}0, _α = 1 and_xmin is the lower bound where the power-law

distribution holds, the normalizing constant can be easily computed:

C = _+∞1 xminx−αdx

= α−1

xmin(1−α).

(3.4)

Thus, the continuous power-law distribution becomes:

p(_x) = α−1 xmin x xmin −α . (3.5)

A discrete power-law distribution indicates the corresponding value of_p(_x) when

x represents the value of a discrete random variable. When a discrete power-law distribution is used to represent a probability distribution, _p(_x) is the probability mass function (pmf) of the probability distribution, which has the form:

p(_x) = _{P r}(_X=_x) = _Cx−α_. (3.6) For instance, the Zipf distribution is one of the family of discrete power-law proba- bility distributions. The probability mass function of the Zipf distribution is

p(_x) = x −s

HN,s ,

(3.7) where _N is a natural number, _s is a real value and _HN,s is the _Nth generalized harmonic number (i.e., N_n=1n−s). The Zipf distribution may be considered as a discrete counterpart of the Pareto distribution.

Given a lower bound_xmin on the discrete power-law distribution, after computing

the normalizing constant

C = _+∞1 i=xmin(i−α)

= 1

ζ(_{α, x}min) ,

(3.8)

with _{α >}1 and _xmin>0, the discrete integer power-law distribution becomes:

p(_x) = x −α

ζ(_{α, x}min), (3.9)

where _ζ(_{α, x}min) is a Hurwitz zeta function [55] or generalized Riemann zeta func-

tion [101], i.e., ζ(_{α, xmin}) = +∞ i=0 (_i+_xmin)−α_. (3.10) In many cases, it is worth studying the complementary distributions of power- law distributions, which indicate the sum of the corresponding values of _p(_x) when

x is above a particular value. In probability theory and statistics, the cumulative distribution function (cdf) _P(_x) is used to describe the kind of distribution and can be written as:

P(_x) =_{P r}(_{X x})_. (3.11) For the continuous random variable, a cumulative distribution function is

P(_x) =_{P r}(_{X x}) = _C +∞ x p (_t)_dt = α−1 xmin−α+1 +∞ x t −α_dt = x xmin (−α+1) . (3.12)

Note that Equation 3.12 indicates that the cumulative distribution of a continu- ous power-law distribution is also a power-law distribution with a smaller scaling exponent value (_α−1).

For discrete random variables, the cumulative distribution function is:

P(_x) = _{P r}(_{X x}) = +∞ i=x p(_i) = +∞ i=x i−α ζ(_{α, x}min)

= ζ(α, x)

ζ(_{α, x}min).

(3.13) The form of Equation 3.13 is too complicated to directly show that the cumulative distribution of a discrete power-law distribution roughly displays a power-law re- lationship. However, since a discrete power-law distribution can be approximately computed by a continuous power-law distribution, the cumulative distribution of the discrete power-law distribution can be approximately computed by the cumu- lative distribution of the continuous power-law distribution as well. That indicates that the cumulative distribution of a discrete power-law distribution with a scaling exponent value _α approximately follows a power-law distribution with the scaling exponent value (_α−1). This inference can also be revealed by using the qualitative appraisal based on visualizations without loss of generality.

For instance, let _α = 2_.5. Figure 3.1(a) shows pairs (_{x, ζ}(2_.5_{, x})) in the natural log-log plot, where 25_,000 _x250_,000. Those pairs (_{x, ζ}(2_.5_{, x})) in Figure 3.1(a) follow a straight line. Also, compare _ζ(2_.5_{, x}) with the power-law distribution _x−1.5 in Figure 3.1(b) where 25_,000_x250_,000. Note that lim x→+∞ ζ(2_.5_{, x})−_x−1.5 = lim x→+∞ _+∞ n=0 (_x+_n)−2.5−_x−1.5 = lim x→+∞ −x−1.5+_x−2.5 + (_x+ 1)−2.5+ (_x+ 2)−2.5 +· · · = 0 (3.14) and lim x→1 ζ(2_.5_{, x})−_x−1.5=_ζ(2_.5_,1)−1 =_ζ(2_.5)−1 = 0_.3415_. (3.15) Since 0 and 0_.3415 are very small values, _ζ(2_.5_{, x}) is very close to the power-law distribution _x−1.5, when _x ∈ [1_,+∞). Moreover, _ζ(2_.5_{, x}min) is a constant with a

certain value _xmin. Thus, P(x) = _ζ(2ζ(2_.5._,x5,x_min)₎, which is the cumulative distribution

of the discrete integer power-law distribution _p(_x) = _ζ(2x_.5−_,x2.5

min), is very close to the power-law distribution _ζ(2x_.5−_,x1.5

min). This example demonstrates that the cumulative distribution of a discrete power-law distribution roughly follows a power-law dis- tribution with a smaller scaling exponent value, which is approximately equal to

(a) Pairs (x, ζ(2.5, x)) in the natural log-log plot

(b) Functionζ(2.5, x)−x−1.5 where 25,000x250,000

Figure 3.1: The behaviour of Hurwitz zeta function _ζ(_{α, x}), where _α = 2_.5 and 25_,000 _x250_,000

(_α−1). That is consistent with the inference based on that a discrete power-law distribution can be approximately computed by a continuous power-law distribution. In practice, a discrete power-law distribution is often approximately computed by a corresponding continuous power-law distribution. From the formulas above, it is clear that the computation of the continuous power-law distribution is much easier than that of the discrete power-law distribution, since the computation of the discrete power-law distribution involves the calculation of a Hurwitz zeta function. In general, calculating the value of a Hurwitz zeta function requires complicated calculation. Consequently, for the sake of computational simplicity, in many ap- plications, a discrete power-law distribution is approximated by a corresponding continuous power-law distribution [30].

There are many approximation methods and not all of them can generate ac- curate results in this case. For example, the approximations using rounding up or down normally generate poor results. A better approximation is to round the real values of_x of a corresponding continuous power-law distribution to the nearest integer [30].