Stability (Equation (2) in Methods) has a maximum of 0:26 for N~4, K ~2, and decreases monotonically in both parameters. For K~2, stability decreases almost as a power-law in N, and this decrease is faster for larger K . A minimal stability value of 2:5|10{6 is found for N~K ~60. Small networks of N~K ~4 genes are about 3 times more likely to reach a cycle than a fixed point steady state. For N~K~10, cycles are *12 times more likely. Also represented is a non-regular biological topology, with exponential in-degree distribution, and scale-free out-degrees (Text S1, Text S2, Figure S1 and Figure S2; see Table S2 and Figure S3 for transient times and Figure S4 for sample sizes). Figure 1 depicts the average stability for networks of different sizes. We next ask if this average is a good proxy for typical network behavior. In other words, what is the stability distribution for networks of a given size? Figure 2 shows that this distribution is bimodal, where some matrices are never stable, independently of the initial state, while others always reach a fixed point from every initial state. This suggests there could be two type of matrices in this model: unstable and stable ones, with the former being much more common than the latter. Interestingly, if we increase network size to N~10 and sample random binary matrices, we still find both types of matrices, but with a more uneven distribution. We find 1928 matrices with stability~0 in our random sample of N~10, against 87 with stability~1: a 22-fold difference. We next ask how network density, c, affects stability. As we can see in Figure 3, stability goes down with increasing c, and has a maximum value of 0:23 for N~5 and minimal c. Again, cycles dominate the dynamics, independent of network density, but sparser networks seem more stable. Although sparse networks of size N~10 with one or two regulatory inputs per gene (c~0:1 and Most Networks in Wagner’s Model Are Cycling 0:2) only have stability *0:14, they are almost twice as stable as dense networks with c~1. So far we have represented the off state of a gene by {1, and although this seems to be the most common choice in the literature, some authors use 0 to represent the off state [18,19]. As shown in Figure 4, stability is higher in the f0,1g than the f{1,1g map, and it goes down linearly with both N and c. The slope of this decay is about the same for both maps, resulting in a 2*4 fold difference in stability between the two. Interestingly, the f0,1g map produces the only instances for which reaching a fixed point is actually more likely than a cycle. This is the case for N~4, c~1 and N~10, c~0:1. In fact, using the f0,1g map, stability is always greater than 0:5 for the sparsest (K ~1) network of any size (Figure S6; see Text S2, Figure S7 and Figure S8 for more comparative studies of the two representations). In Figure 5 we show that stability distributions are very similar for binary and real matrices, with the real set having slightly more unstable and fewer stable matrices than the binary one. To see whether the same is true of the normalized means of the distributions, i.e., the networks’ average stability, we return to Figure 1 and see that for N~4 we find a stability value around 0:23, estimated by randomly sampling 106 real matrices. Random sampling of 106 binary matrices estimates stability around 0:31. A full enumeration of the total 65,536 binary matrices (N~4) yields *0:37 stability. It seems that random sampling over-represents unstable matrices, which are the most frequent ones in the full distribution. Finally, we compare stability for binary and real states. Figure 6 shows that the stability distributions are similar with either the sign (binary) or the steep sigmoid (real) functions with a~10,100 and even a~2 for all N and cw0:1. This is somewhat expected if you Figure 2. Stability distribution is bimodal. Two types of matrices: stable (stability~1) and unstable (stability~0). Full enumeration of binary network space for N~4 2N2 ~65,536 and a random sample of 3876 binary matrices for N~10. Full enumeration of the state space for both cases ð2N Þ. For N~4, each bin corresponds to a specific stability value. There are a total of 26,135 unstable (first bin) and 16,574 stable (last bin) matrices in the genotype space of binary matrices of size N~4. For N~10, different stability values are binned together in 17 bins of width *0:06 each. For that reason, not all N~10 matrices included in the first bin have stability~0, for example. There are 1928 unstable matrices versus 87 stable ones in our random sample of N~10. c~1; other parameters are as in Figure 1. doi:10.1371/journal.pone.0034285.g002 PLoS ONE | www.plosone.org 3 April 2012 | Volume 7 | Issue 4 | e34285