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

Random sampling vs. exact enumeration of attractors in random Boolean networks