Scaling in Ordered and Critical Random Boolean Networks

Random Boolean networks, originally invented as models of genetic regulatory networks, are simple models for a broad class of complex systems that show rich dynamical structures. From a biological perspective, the most interesting networks lie at or near a critical point in parameter space that divides “ordered” from “chaotic” attractor dynamics. We study the scaling of the average number of dynamically relevant nodes and the median number of distinct attractors in such networks. Our calculations indicate that the correct asymptotic scalings emerge only for very large systems.

Figure 1

A $K=2$, $N=10$ network. Each node has two inputs, but the number of outputs (drawn from its center) can vary. See text for details.

Figure 2

(a) Average numbers of relevant nodes in $K=2$ networks with truth functions specified by $p$. Vertical dashed lines indicate theoretical predictions for the crossover from critical to ordered behavior at each $p$. Horizontal segments at the right indicate theoretical calculations of the asymptotic values of $⟨r⟩$ for each $p$. Diagonal lines are guides to the eye, showing slopes of $1/2$ and $1/3$. (b) Average numbers of unclamped nodes in $K=2$ critical ($p=1/2$) networks. The dashed line has slope $2/3$.

Figure 3

Median numbers of attractors for $K=2$ networks with truth functions specified by $p$. Error bars indicate 1 standard deviation. Note the upward curvature for the critical value $p=1/2$.