  The solution-space structure of the 3-Satisfiability Problem (3-SAT) is
studied as a function of the control parameter alpha (ratio of number of
clauses to the number of variables) using numerical simulations. For this
purpose, one has to sample the solution space with uniform weight. It is shown
here that standard stochastic local-search (SLS) algorithms like "ASAT" and
"MCMCMC" (also known as "parallel tempering") exhibit a sampling bias.
Nevertheless, unbiased samples of solutions can be obtained using the
"ballistic-networking approach", which is introduced here. It is a
generalization of "ballistic search" methods and yields also a cluster
structure of the solution space. As application, solutions of 3-SAT instances
are generated using ASAT plus ballistic networking. The numerical results are
compatible with a previous analytic prediction of a simple solution-space
structure for small values of alpha and a transition to a clustered phase at
alpha_c ~ 3.86, where the solution space breaks up into several non-negligible
clusters. Furthermore, in the thermodynamic limit there are, for values of
alpha close to the SATUNSAT transition alpha_s ~ 4.267, always clusters without
any frozen variables. This may explain why some SLS algorithms are able to
solve very large 3-SAT instances close to the SAT-UNSAT transition.
