  We clarify the effect different sampling methods and weighting schemes have
on the statistics of attractors in ensembles of random Boolean networks (RBNs).
We directly measure cycle lengths of attractors and sizes of basins of
attraction in RBNs using exact enumeration of the state space. In general, the
distribution of attractor lengths differs markedly from that obtained by
randomly choosing an initial state and following the dynamics to reach an
attractor. Our results indicate that the former distribution decays as a
power-law with exponent 1 for all connectivities $K>1$ in the infinite system
size limit. In contrast, the latter distribution decays as a power law only for
K=2. This is because the mean basin size grows linearly with the attractor
cycle length for $K>2$, and is statistically independent of the cycle length
for K=2. We also find that the histograms of basin sizes are strongly peaked at
integer multiples of powers of two for $K<3$.

Statistical Mechanics;Disordered Systems and Neural Networks