Figure 3

Analysis of simple and complex cycles. Filled circles represent states of the system, not to individual nodes. In asynchronous update mode, two neighboring states differ by the value of exactly one node. Thus, the shortest possible cycle (0, 0) → (0, 1) → (1, 1) → (1, 0) → (0, 0) → …, composed of two changing nodes, has length 4 in state space. A: The cycle “abcd” is fixed only in the absence of the dashed transition a → g. In the presence of this transition, the cycle leaks into state g, and the system has only a single fixed point, g. B: In the cycle “abcd”, here state c is 3-discontent and, hence, can transition to three different successor states d, e, and f. For the cycle to be fixed, as shown here, all the successor states of d, e, and f have to return to the cycle. As a consequence, complex fixed cycles are expected to be rare in biologically motivated networks.