Relation between Power Laws and Fractals 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!
 A: $$y=c \cdot x^{d}$$
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.
