We consider evolving networks on which players interact locally. In an evolutionary environment, players can increase their fitness either by changing their behavior or by changing their neighbors. We propose a static solution concept, what we call Stable Network Configuration (SNC). Roughlyspeaking, it requires that no player in the population distribution will change his type by imitating his best-performing neighbor, nor change his neighbor by rewiring his links. We characterize the necessary and sufficient conditions for the symmetric SNC in a network formation situation associated with the Prisoner's Dilemma game. Unlike the result by Eshel et al. (1998) that was obtained in a fixed circular network, all altruists and all egoists fare equally well and all altruists have links with some or all egoists in the symmetric SNC.