  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.
