Most Networks in Wagner’s Model Are Cycling Figure 5. Binary and real matrices seem to have the same stability distribution. Equation (1) is solved for 2 sets of 376,992 random matrices each, and full enumeration of state space ð2N ~32Þ. N~5, c~1, si*f{1,1g, f (x)~sgn(x), sgn(0)~1. wij*f{1,1g for binary matrices and wij*N (0,1) for real ones. doi:10.1371/journal.pone.0034285.g005 2. Some networks never cycle while others always cycle independently of the initial conditions. 3. Sparse networks are stable more often than dense networks. 4. Using 0 instead of {1 to represent the off state induces more stable solutions. However, for well connected networks, stable states are discovered more rapidly for {1 networks. 5. Binary and real valued weight matrices have similar stability properties. 6. Discrete or continuous gene expression states may give similar results, depending on the steepness of the sigmoid function. 7. Network topology seems to have a small effect on stability 8. The distribution of attractor lengths appears to decay more slowly than the typical power law. All of these results may have implications for both the use of the RTN as a model of gene regulation and the properties of real biological networks. For brevity, we focus solely on the former. Figure 6. Steep sigmoid functions result in the same stability profiles as the sign function. Average stability (2) for different network sizes N, c~1 (a) and densities c, N~10 (b) with different normalization functions f (x). Equation (1) is solved n~106 times with wij*N (0,1), si[½{1,1 , f (x)~sgn(x) with sgn(0)~1 for the first curve, and the sigmoid f (x)~z(x; a) for all curves identified by steepness a (Text S1). For z(x; a), other parameters are t~50, e~10{5, T ~100 and ESEw0:01. Dashed lines are guides to the eye. doi:10.1371/journal.pone.0034285.g006 PLoS ONE | www.plosone.org 5 April 2012 | Volume 7 | Issue 4 | e34285