Cooperation in the snowdrift game on directed small-world networks under self-questioning and noisy conditions

Cooperation in the evolutionary snowdrift game with a self-questioning updating mechanism is studied on annealed and quenched small-world networks with directed couplings. Around the payoff parameter value $r=0.5$, we find a size-invariant symmetrical cooperation effect. While generally suppressing cooperation for $r>0.5$ payoffs, rewired networks facilitated cooperative behavior for $r<0.5$. Fair amounts of noise were found to break the observed symmetry and further weaken cooperation at relatively large values of $r$. However, in the absence of noise, the self-questioning mechanism recovers symmetrical behavior and elevates altruism even under large-reward conditions. Our results suggest that an updating mechanism of this type is necessary to stabilize cooperation in a spatially structured environment which is otherwise detrimental to cooperative behavior, especially at high cost-to-benefit ratios. Additionally, we employ component and local stability analyses to better understand the nature of the manifested dynamics.