Most Networks in Wagner’s Model Are Cycling Figure 3. Sparser networks are more stable. Average stability (Equation (2)) for different network densities c~fK=N : K~1,2,:::, Ng and sizes N~5,10,20. Equation (1) is solved up to n~108 times for each c and N. Other parameters as in Figure 1. Dashed lines are guides to the eye. Shaded region represents the density of biological networks [32,45]. doi:10.1371/journal.pone.0034285.g003 compare these different normalization functions in Figure S13. In lim fact, z(x; a)~sgn(x). We also note that stability starts to a?? behave differently for low a~1 (Text S1, Text S2 and Figure S9). More results are presented in Text S2. Discussion In this study, we have conducted extensive simulation analysis of a subclass of random Boolean networks, known as Random Threshold Networks (RTNs). We defined stability as the probability of reaching a steady state and investigated the dependence of stability on network size and density, types of regulatory interactions and gene expressions and parameter values. The main findings are: 1. There are vastly more cyclic solutions than steady state solutions; only the latter have been assumed to be ‘‘viable’’ networks in previous studies. Figure 4. Stability is 2*4 fold higher with the f0,1g (squares) than the f{1,1g (circles) maps. Equation (1) is solved n~106 times for each N, c~1 (a) and c, N~10 (b) with wij*N (0,1). For the f{1,1g map: si*f{1,1g, f (x)~sgn(x), sgn(0)~1. For the f0,1g map si*f0,1g, f (x)~H(x), H(0)~1 (Text S1). Dashed lines are guides to the eye. doi:10.1371/journal.pone.0034285.g004 PLoS ONE | www.plosone.org 4 April 2012 | Volume 7 | Issue 4 | e34285