A Condition for Cooperation in a Game on Complex Networks

We study a condition of favoring cooperation in a Prisoner's Dilemma game on complex networks. There are two kinds of players: cooperators and defectors. Cooperators pay a benefit b to their neighbors at a cost c, whereas defectors only receive a benefit. The game is a death-birth process with weak selection. Although it has been widely thought that b/c> <k> is a condition of favoring cooperation, we find that b/c> <knn> is the condition. We also show that among three representative networks, namely, regular, random, and scale-free, a regular network favors cooperation the most, whereas a scale-free network favors cooperation the least. In an ideal scale-free network where network size is infinite, cooperation is never realized. Whether or not the scale-free network and network heterogeneity favor cooperation depends on the details of a game, although it is occasionally believed that scale-free networks favor cooperation irrespective of game structures. If the number of players are small, then the cooperation is favored in scale-free networks.