= Power law
{title2=$f(x)=Cx^p$}
{wiki}
A <power law> is a relation $f(x)=Cx^p$, with fixed constants $C$ and $p$. For positive $x$, any <real number> $p$ is allowed. Its useful feature is a simple scaling rule:
$$
f(\lambda x)=\lambda^p f(x)\qquad(\lambda>0).
$$
The same proportional change of input produces the same proportional change of output, independently of the starting value. The exponent controls that response; the coefficient fixes the magnitude.
For example, the <area> of a <circle> is $A(r)=\pi r^2$. Doubling the <radius> multiplies its <area> by four, and tripling it multiplies the <area> by nine. The <volume> of a <sphere> is $V(r)=4\pi r^3/3$: the same changes multiply its <volume> by eight and twenty-seven. The distinction explains why similarly shaped objects do not preserve their ratio of <surface area> to <volume> when their size changes.
When the output is positive, a <logarithm> turns the <power law> into a straight-line relation. Using positive reference values $x_0$ and $f_0=f(x_0)$,
$$
\frac{f(x)}{f_0}=\left(\frac{x}{x_0}\right)^p,
\qquad
\log\frac{f(x)}{f_0}=p\log\frac{x}{x_0}.
$$
These ratios are <dimensionless quantities>, so the formula also applies to physical measurements with units. Plotting both ratios on a <logarithmic scale> gives a line of slope $p$. Changing a unit changes the numerical coefficient, while the exponent stays the same. Positive $p$ describes increasing values; negative $p$ describes decreasing values; $p=0$ gives a constant. The zero coefficient is a degenerate case with no useful logarithmic slope.
Exact scaling is stronger than <regular variation>. The <function> $x^2\log x$ has a doubling ratio tending to four, but its logarithmic correction keeps growing. The diagram contrasts it with the exact <power law> $x^2$.
\Image[/regular-variation-rescaling-ratios.png]
{title=Exact quadratic scaling and a logarithmic correction have different finite doubling ratios, even though the correction leaves the limiting exponent unchanged}
A <power law> for a <probability density> and a <power law> for a <tail probability> have different exponents. For a <Pareto distribution> with $\alpha>0$ and lower endpoint one,
$$
\mathbb P(X>x)=x^{-\alpha},
\qquad
p_X(x)=\alpha x^{-\alpha-1}
\qquad(x>1).
$$
The <tail probability> has exponent $-\alpha$; differentiating it and changing sign gives the <probability density> exponent $-\alpha-1$. Confusing them changes the interpretation of a fitted exponent. The normalizing coefficient and lower endpoint also matter: a decreasing <power law> is not automatically a valid <probability density> over the entire positive <real line>.
The appearance of a line over a finite logarithmic range is evidence to investigate, rather than a proof of an exact <power law> or its mechanism. In their study of <power-law distributions>, Clauset, Shalizi and Newman combine <maximum likelihood estimation>, <goodness-of-fit tests> and comparisons with alternative models; their abstract explains why ordinary least-squares fitting can give misleading results for tail data. See https://arxiv.org/abs/0706.1062[Aaron Clauset, Cosma Rohilla Shalizi and M. E. J. Newman, Power-law distributions in empirical data, SIAM Review 51 (2009), 661–703]. The scaling rule is mathematically exact when it defines a model; whether a dataset supports that model is a question of <statistical inference>.
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