Several factors that promote cooperation on scale-free networks have been studied, and the mechanisms of the evolution of cooperation are revealed by numerical simulations and theoretical works. However, influence of the power-law exponent on the evolution of cooperation has not been manifested. In our previous study, it is found by numerical simulations that there exists an optimal value of the power law exponent which promotes cooperation. In this paper, we consider the reason why there exists the optimal value, using a theoretical analysis.