  A solution to a 3-satisfiability (3-SAT) formula can be expanded into a
cluster, all other solutions of which are reachable from this one through a
sequence of single-spin flips. Some variables in the solution cluster are
frozen to the same spin values by one of two different mechanisms: frozen-core
formation and long-range frustrations. While frozen cores are identified by a
local whitening algorithm, long-range frustrations are very difficult to trace,
and they make an entropic belief-propagation (BP) algorithm fail to converge.
For BP to reach a fixed point the spin values of a tiny fraction of variables
(chosen according to the whitening algorithm) are externally fixed during the
iteration. From the calculated entropy values, we infer that, for a large
random 3-SAT formula with constraint density close to the satisfiability
threshold, the solutions obtained by the survey-propagation or the walksat
algorithm belong neither to the most dominating clusters of the formula nor to
the most abundant clusters. This work indicates that a single solution cluster
of a random 3-SAT formula may have further community structures.
