Attractor and basin entropies of random Boolean networks under asynchronous stochastic update

We introduce a method to study random Boolean networks with asynchronous stochastic update. Each node in the state space network starts with equal occupation probability, which then evolves to a steady state. Attractors and the sizes of their basins are determined by the nodes left occupied at late times. As for synchronous update, the basin entropy grows with system size only for critical networks. We determine analytically the distribution for the number of attractors and basin sizes for networks with connectivity $K=1$. These differ from the case of synchronous update for all $K$.

Figure 1

(Color online) The distribution of the number of even loops $P\left(n_{even}\right)$, obtained using $Q\left(\mathbf{m}\right)$—compared to the distribution of the number of attractors, $P\left({\text{log}}_{2}A\right)$, and the distribution of inverse basin sizes, $P\left({\text{log}}_2\frac{N}{b}\right)$, obtained using our SSN method for $K=1$ ARBNs with $N=16$. The number of realizations is $2\times {10}^4$ and the error bars correspond to two standard deviations.

Figure 2

(Color online) (a) $P\left(\lambda \right)$ for $K=1,2,6$ and $N=16$. Both $K=1$ and $K=2$ ARBNs show a broad distribution while all attractor lengths for $K=6$ are either of order unity or of order $2^N$. (b) Rescaled $P\left(\lambda \right)$ vs. rescaled $\lambda$ for $K=2$ and $N=12–22$. The approximate collapse is consistent with a power law decay up to a cutoff which is a stretched exponential in $N$.

Figure 3

(Color online) The basin entropy $⟨h⟩$ for ARBNs with various $K$, and $N=8–20$ (curves with error bars). $⟨h⟩$ grows with $N$ only for $K=2$. The solid (black) line shows $⟨h⟩$ for $K=1$ derived from $Q\left(\mathbf{m}\right)$. The dashed (black) line has a slope of 0.15 and is a guide for the eyes.