Power Laws are very frequent in data mining. For instance, you can find a power law in many properties of the web graph. I found a quite interesting explanation of power laws for modelling particles and cosmic radiation (Enrico Fermi 1949)
In this model, particles are produced at constant rate and the particles' distribution at time t is an exponential of the form \lamda * e ^ - (\lamda * t) (in other words there is a negative exponential contribution). In addition, particles gain energy when time is passing such as C e ^ (\alpha * t) (in other words this is giving a positive exponential contribution).
Under these two forces the particles will have a power law density. Can you explain why?