Zhou and Lipowsky, 10.1073/pnas.0409296102. |
Fig. 5.
Three networks obtained via the configuration model. (Upper Left) Random scale-free network with n = 64 vertices and M = 140 edges generated from a scale-free vertex distribution with g = 2.5 and k0 = 2. (Upper Right) Random scale-free network with n = 64 vertices and M = 96 edges generated from a scale-free vertex degree distribution with g = 3 and k0 = 2. (Lower) Random Poissonian network with n = 64 vertices and M = 82 edges.
Supporting Figure 6
Fig. 6.
Distribution of relaxation times for the same random networks as in Figs. 1 and 2: (i) scale-free with g = 2.25 and k0 = 5 (circles), (ii) scale-free with g = 3 and k0 = 10 (squares), and (iii) Poissonian (diamonds). All three networks have the same mean vertex degree á kñ = 20. For each network, 2,000 individual trajectories have been generated, each starting from a strongly disordered pattern as defined by Eq. 9. For each of these trajectories, the relaxation time is equal to the number of time steps until the trajectory has reached one of the two completely ordered patterns. The probability to observe a relaxation time that exceeds 100 steps is smaller than 10–2 in all three cases.
Supporting Figure 7
Fig. 7.
Probability P> of not reaching one of the two completely ordered patterns within 100 time steps as a function of vertex number (or network size) N. Each data point corresponds to an average of 100 different networks with g = 2.25 and k0 = 5. For each network, 2,000 individual pattern trajectories have been generated starting from a strongly disordered patterns as defined by Eq. 9.
Supporting Figure 8
Fig. 8.
Distribution of decay times for three different random networks with n = 218 vertices and mean vertex degree ákñ = 10. For each network, 2,000 individual pattern trajectories have been generated starting from a strongly disordered pattern as defined by Eq. 9. The decay time is equal to the number of time steps until a pattern with |Q – 1/2| ³ 1/4 is reached. The three networks are (i) scale-free with g = 2.148 and k0 = 2 (circles), (ii) scale-free with g = 2.828 and k0 = 5 (squares), and (iii) Poissonian (diamonds). Inset shows the same data but with the horizontal axis extended to 200 decay steps.
Supporting Figure 9
Fig. 9.
Distribution of relaxation times for the same networks and for the same ensembles of trajectories as in Fig. 8. The relaxation time is equal to the number of time steps until one of the two ordered patterns has been reached. In contrast to the distribution of the decay times as shown in Fig. 8, the distribution of the relaxation times is similar for all three networks. This implies that, for this realization of the scale-free network with g = 2.148 (circles), the approach toward the two completely ordered states is slowed down for larger times (see Fig. 9).
Supporting Figure 10
Fig. 10.
Absolute value of the order parameter, |y| = |Q – 1/2| as a function of time (in units of iteration steps) for the same three networks as in Figs. 8 and 9. Each data set represents an average over 2,000 individual trajectories, which all start initially from strongly disordered spin patterns. For the scale-free network with g = 2.148 (circles), the absolute value of the order parameter, |y|, quickly exceeds the value |y| = 1/4, but the subsequent approach toward |y| = 1 is slower than in the case of the Poissonian network (diamonds).