Phase transitions in the Prisoner's Dilemma game on scale-free networks

We study stochastic dynamics of the Prisoner's Dilemma game on random Erd\"{o}s-R\'{e}nyi and Barab\'{a}si-Albert networks with a cost of maintaining a link between interacting players. Stochastic simulations show that when the cost increases, the population of players located on Barab\'{a}si-Albert network undergoes a sharp transition from an ordered state, where almost all players cooperate, to a state in which both cooperators and defectors coexist. At the critical cost, the population oscillates in time between these two states. Such a situation is not present in the Erd\"{o}s-R\'{e}nyi network. We provide some heuristic analytical arguments for the phase transition and the value of the critical cost in the Barab\'{a}si-Albert network.