What is the difference between a power law exponent and the Pareto exponent?
A random variable $X$ has a Pareto distribution with Pareto exponent
  $\theta$ if  $$\text{P}(X&gt;x)=\begin{cases} 
 \left(\frac{x}{x_m}\right)^\theta \quad \text{ if } x\geq x_{min}\\  \
 \ 1 \quad \quad \quad  \text{ if } x&lt;x_{min} \end{cases}$$


In this case, the Pareto exponent is $\theta = \alpha - 1$. Remember that $P(X&gt;x)=\int_x^\infty p(x')\mathrm{d}x'$.