Figure 3

Stability and repair frequency in random Boolean networks. Synchronously updated random Boolean networks with k = 1, 2, and 3 regulators per node (black, dark gray, and light gray, respectively) are allowed to reach a steady state (circles) or a state that belongs to a limit cycle (squares), then are damaged by randomly selecting a node and fixing it to its opposite state. Symbol size corresponds to the fraction of observed steady states (panel (a)) vs. limit cycles (panels (b)-(d)) in 10,000 attractors selected from unique networks of a given size and value of k. (a) While steady states are less commonly observed for larger networks when k > 1, those that are observed are resilient to node knockout at a frequency that depends on k but not network size. All instances of steady states that are not stable may be repaired without ambiguity. (b) Observed limit cycles are stable with a frequency that decreases with k. For k < 3 the frequency of stable limit cycles does not depend on the network size, but it decreases as network size increases for k = 3. (c) Unstable limit cycles are repaired with decreasing frequency for increasing k, and with decreasing frequency as network size increases for k = 3. (d) Limit cycles that may not be repaired (see Methods) grow more frequent with increasing k. The values corresponding to black, dark gray, and light gray squares from panels (b)-(d) sum to 1.