This paper describes the data they analyzed as following a "truncated" power law distribution:
To me, this just looks like they multiplied a power-law distribution ($\Delta r + \Delta r_0)^{-\beta}$ by an exponential distribution $e^{\frac{-\Delta r}{\kappa}}$. What exactly is being truncated? Is this related to the so-called "piecewise power law" distributions?
M.C. González et al., Understanding individual human mobility patterns, Nature 453, 779 (2008).
As explained in this wiki article (albeit briefly), a "truncated" or "cutoff" power law distribution is simply a power law multiplied by an exponential (by definition).
The term "cutoff" or "truncated" is a misnomer (when compared to distributions like truncated normal) since the range is not affected. However, as you may be aware, the moments of a power law distribution are not well behaved or are very sensitive to the power law exponent.
This behavior is caused by its tail behavior, where you are unlikely to have much data. So, if you are a researcher and you find a power law fits your data, you have to either accept a power law distribution and its heavy tails, or you recognize that you really don't know what's going on in the tails, and you assume the process is well behaved out there. In that case, you make the tails thinner by having them shrink faster than they would have otherwise. This makes the moments exist (no infinite variance) and leads to a model that does not predict wild swings in values.
Note: the choice of truncation is somewhat of a judgement call...for example, in financial applications, it may be an awful choice, but in biological or physics applications, it may make sense. Also, if you are lucky enough to have sufficient data, you can actually check the tail behavior with some degree of confidence.
