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Are all power laws (i.e., of the general form $y=cx^{\alpha}$) fractal (exhibiting some form of self-similarity)?

Does the scalability of power laws also mean by definition that they are also fractal? If someone could provide a little intuition on connecting the dots between these two concepts I would be very grateful.

Thanks!

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  • $\begingroup$ What do you mean by "fractal" or "some form of self-similarity" in this context? E.g., the law that the area of a circle is $A = \pi r^2$ is certainly a power law, but has nothing to do with fractals as commonly understood. $\endgroup$ Commented Aug 9, 2015 at 2:50
  • $\begingroup$ As an example, let's say that the power law describes wealth in the United States. So the distribution for some alpha, as we get further out into the right tail, is "similar" (where fractal comes from) for the number of people worth over 1 billion dollars although not exactly the same as the number of those who are worth 100 million dollars. I guess what I am asking is "Is scalable and fractal the same thing?" $\endgroup$ Commented Aug 9, 2015 at 3:05
  • $\begingroup$ If you don't provide a definition of fractal (and there is not really a universally accepted one), this question is ill-posed. Most people would not call a general power-law in the mentioned examples fractal, though. $\endgroup$ Commented Aug 9, 2015 at 4:03
  • $\begingroup$ @LukasGeyer Fractals have no general definition, self-similarity does. You should answer, if that was the intention of your questioning, along the lines of self-similarity. The concept of "Fractal" is just about as ambiguous as "Closed-Form"... $\endgroup$
    – Zach466920
    Commented Aug 10, 2015 at 15:27
  • $\begingroup$ The word "fractal" is related to the word "fractured" i.e. broken. This would seem to suggest to me that the word was intended for non-smooth structures. As @Zach466920 points out in his answer, the case for smooth structures is rather boring. So while the power law is self-similar, I would hesitate to call it a "fractal", which I would reserve for strange objects like Sierpinski's gasket or Cantor's dust. $\endgroup$ Commented Aug 10, 2015 at 15:56

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$$y=c \cdot x^{d}$$

Is indeed self-similar. This is because scaling the independent variable by a factor $\lambda$ scales the dependent variable by a factor $\lambda^d$. This property is referred to as being homogeneous. What people get tripped up over is the difference between discrete and continuous self-similarity. The former means that applying an operator, for instance multiplying by $\lambda$, will yield some kind of relation with the original object, but only if applied in a certain way, i.e. at specific points. Continuous self-similarity is less exciting, it just means you can apply the transformation at any point and always be within some nontrivial factor of the original object.

What does this practically mean? It means that power laws give an indication of the behavior of a system. If you're looking for "fractional" scaling, sometimes considered to be fractal, having a fractional exponent will indeed indicate a fractal. However the dimensionality of the relation need not be geometric, and fractals are usually considered to geometric.

For example, if you run an experiment plotting the change in a stock price according to some change in time, you'll have a lot of numbers to analyze. If you remember the power law we discussed, you'd most likely find that the amount of change in the stock price, $\Delta P$ scales with $\Delta t$. To determine the scaling exponent you take logarithms, or use a plotting method.

$$d={{\ln(\Delta P)} \over {\ln(\Delta t)}}$$

If you got a fractional value for $d$ then you'll probably wonder whether or not a fractal process underlies the values of stocks. What you wouldn't say is that the relation between the variables is fractal.

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  • $\begingroup$ Great answer Zach, thanks for the insight! Am I right thinking that the fractal dimension could be considered a shape parameter of a power law? $\endgroup$ Commented Aug 13, 2015 at 20:58
  • $\begingroup$ @user2597291 Depends on what "shape parameter" means. The fractal dimension is just an exponent that relates some kind of "mass" with respect to "scale". The actual shape of the object is not defined by the fractal dimension. For instance, a parabola and a sine wave are both "1d". $\endgroup$
    – Zach466920
    Commented Aug 13, 2015 at 21:48

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