Figure 2.

(a) Probability of reaching the successful state (blue color) as a function of the initial cooperation rate C ₀ and the payoff b. Here, we generated networks using the model described in Appendix with 10000 nodes, a power law degree distribution with exponent γ = 2.5, temperature T = 0.4, and mean degree <k> ≈ 6. The results are averaged over 50 simulations. A realization is said to have approached the functional state if the performance of the system in the stationary state is larger than 0.5. To compute the quantities in the stationary state, we first let 250 rounds pass, which avoids the initial transient phase, and then we average over another 250 rounds (we observe that this initial phase usually takes approx. 25 steps, and therefore allow 10 × 25 steps to pass, which indeed ensures that the system is in one of the two described states). Note that we only observe states that either perform very well with success rates $\gg 0.5$ or very poorly with success rates $\ll 0.5$ (see Supplementary Materials). Initial conditions are randomly assigned cooperators. (b) Same as before but starting from clustered cooperators according to Eq. (3). (c) Threshold value of C ₀ for the system to approach the functional state with at least 50% probability (black lines in a, b) for different values of the power-law exponent γ. Solid lines correspond to the random distribution of initial cooperators, whereas dashed lines represent the clustered initial configuration. Arrows show the reduction of the initial “critical mass”. The network parameters are the same as in (a). (d,e) Same as a–c but for the IPv6 Internet network (averaged over 200 realizations). (f) The same as in (c) but for the IPv6 Internet topology. The solid line corresponds to the black line in (d) and the dashed line is the same as the black line in (e). Arrows show the reduction of the initial “critical mass” due to localization of the initial cooperators.