We examine networks connecting individuals, where the payoff to an individual from an economic or social activity depends on the full network of connections among individuals. Individuals can form and sever links connecting themselves to other individuals based on the improvement that the resulting network offers them relative to the current network. As individuals do this, we obtain sequences of networks called `improving paths.’ We study conditions under which such sequences cycle, and conditions under which such sequences lead to a stable network. Specifically, we give conditions necessary and sufficient to rule out cycles, which are in turn sufficient conditions for existence of pairwise stable networks.