Hamilton-like rule for coevolution of strategy and structure

In this paper, we investigate coevolution of strategy and structure in Prisoner´s Dilemma. We concentrate on the noise effect in the topological evolution on cooperation. We assume individuals can either update their strategies by imitating their partners or adjust their social ties. At each time in the strategy evolution, pairwise comparison is employed. While at each time in the topological evolution, with some probability, an individual dumps off one of its partner and makes a new social relationship. We show that a Hamilton-like rule is obtained, quantitatively saying the more often dissatisfied links break off than satisfied ones, the more likely cooperation can prevail, provided the linking dynamics proceeds much faster than strategy evolution. Furthermore, by investigating the upper bound of the strategy updating probability, we also show how much faster linking dynamics proceeds than strategy evolution to make the Hamilton-like rule valid. Interestingly, we unveil that the probability is dependent on the frequency of cooperators. Our work may shed light on the ubiquitous cooperation in societies.