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

TL;DR

This paper compares sampling methods for analyzing attractor statistics in random Boolean networks, revealing significant differences in attractor length distributions and basin sizes depending on the sampling approach and network connectivity.

Contribution

It provides an exact enumeration approach to measure attractor properties and clarifies how sampling biases affect statistical results in RBNs.

Findings

Distribution of attractor lengths follows a power-law with exponent 1 for all K>1.

Exact enumeration reveals differences from random initial state sampling.

Basin size distributions are peaked at multiples of powers of two for K<3.

Abstract

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 $K<3$.