Before revision:  Within this framework , we study the evolution of cooperation in structured populations and find a simple rule: natural selection favors cooperation over defection if the ratio of the benefit provided by an altruistic behavior, b, to the corresponding cost, c, exceeds k-k', which means b/c>k-k', where k is the average number of neighbors  and k' captures the effects from game transitions.
After revision:  Within this framework , we study the evolution of cooperation in structured populations and find a simple rule: weak selection favors cooperation over defection if the ratio of the benefit provided by an altruistic behavior, b, to the corresponding cost, c, exceeds k-k', which means b/c>k-k', where k is the average number of neighbors and k' captures the effects from game transitions.