We study distributed editing of network topologies from a game theoretic perspective. The nodes are the agents/players in the game and the editing decisions of each node are to add a new link to another node, remove an existing link, or maintain it. The aim of each node is to achieve a balance between the network property reward and the editing costs for changing and maintaining one's neighborhood. We study several variants of the potential game that result from repeated interactions between agents over the network and describe algorithms that ensure convergence to equilibrium topologies. Simulation results demonstrate the relevant properties of the limiting networks and its dependence on the cost structure.