  We focus on the random generation of SAT instances that have properties
similar to real-world instances. It is known that many industrial instances,
even with a great number of variables, can be solved by a clever solver in a
reasonable amount of time. This is not possible, in general, with classical
randomly generated instances. We provide a different generation model of SAT
instances, called \emph{scale-free random SAT instances}. It is based on the
use of a non-uniform probability distribution $P(i)\sim i^{-\beta}$ to select
variable $i$, where $\beta$ is a parameter of the model. This results into
formulas where the number of occurrences $k$ of variables follows a power-law
distribution $P(k)\sim k^{-\delta}$ where $\delta = 1 + 1/\beta$. This property
has been observed in most real-world SAT instances. For $\beta=0$, our model
extends classical random SAT instances.
  We prove the existence of a SAT-UNSAT phase transition phenomenon for
scale-free random 2-SAT instances with $\beta<1/2$ when the clause/variable
ratio is $m/n=\frac{1-2\beta}{(1-\beta)^2}$. We also prove that scale-free
random k-SAT instances are unsatisfiable with high probability when the number
of clauses exceeds $\omega(n^{(1-\beta)k})$. %This implies that the SAT/UNSAT
phase transition phenomena vanishes when $\beta>1-1/k$, and formulas are
unsatisfiable due to a small core of clauses. The proof of this result suggests
that, when $\beta>1-1/k$, the unsatisfiability of most formulas may be due to
small cores of clauses. Finally, we show how this model will allow us to
generate random instances similar to industrial instances, of interest for
testing purposes.
