Difference between power law distribution and exponential decay This is probably a silly one, I've read in Wikipedia about power law and exponential decay. I really don't see any difference between them. For example, if I have a histogram or a plot that looks like the one in the Power law article, which is the same as the one for $e^{-x}$, how should I refer to it?

How is a power law different from an exponential? (I'm putting this answer here mainly for my own future reference. Hopefully someone else may find it useful.)
Power Law function
(notice the exponent, $k,$ is a constant)
$$
y = x^k
$$
Exponential function
(notice the exponent is a variable)
$$
y = a^x
$$
Technical definition of Power Law:
A power law is any polynomial relationship that exhibits the property of scale invariance. 
Scale invariance (from Wikipedia)
One attribute of power laws is their scale invariance. Given a relation $f(x) = ax^k,$ scaling the argument $x$ by a constant factor $c$ causes only a proportionate scaling of the function itself. That is,
$$
f(c x) = a(c x)^k = c^k f(x) \propto f(x)
$$
That is, scaling by a constant $c$ simply multiplies the original power-law relation by the constant $c^k.$
Thus, it follows that all power laws with a particular scaling exponent are equivalent up to constant factors, since each is simply a scaled version of the others. This behavior is what produces the linear relationship when logarithms are taken of both $f(x)$ and $x,$ and the straight-line on the log-log plot is often called the signature of a power law. 
