This paper analyzes the stability and efficiency of social and economic networks when self-interested individuals choose to form or sever links. It presents two stylized models to explore the relationship between stable and efficient networks. In the first model, the "Connections Model," individuals benefit from communication through direct and indirect links, but must weigh the benefits against the costs of maintaining links. The paper shows that the most efficient network (complete graph) is not always stable, and that stable networks may not be efficient. In the second model, the "Co-Author Model," researchers' productivity depends on their collaborations, and indirect connections can negatively affect productivity. The paper finds that the most efficient network (a set of pairs) is not always stable, and that stable networks may not be efficient. The paper also discusses the general model of network stability and efficiency, showing that the sets of stable and efficient networks do not always intersect. It demonstrates that to ensure at least one efficient network is stable, resources must be allocated to nodes not responsible for production. The paper characterizes one such allocation rule: the equal split rule. It also characterizes another rule that fails to ensure efficient graphs are stable but arises naturally from bargaining among players. The paper concludes that the tension between stability and efficiency is a general feature of network models, and that the conflict depends on the nature of the value function and the conditions imposed on the allocation rule. It shows that under certain conditions, such as critical link monotonicity, efficient networks can be stable. The paper also discusses the role of anonymity and component balance in the results, and shows that relaxing these conditions can help avoid the conflict between efficiency and stability.This paper analyzes the stability and efficiency of social and economic networks when self-interested individuals choose to form or sever links. It presents two stylized models to explore the relationship between stable and efficient networks. In the first model, the "Connections Model," individuals benefit from communication through direct and indirect links, but must weigh the benefits against the costs of maintaining links. The paper shows that the most efficient network (complete graph) is not always stable, and that stable networks may not be efficient. In the second model, the "Co-Author Model," researchers' productivity depends on their collaborations, and indirect connections can negatively affect productivity. The paper finds that the most efficient network (a set of pairs) is not always stable, and that stable networks may not be efficient. The paper also discusses the general model of network stability and efficiency, showing that the sets of stable and efficient networks do not always intersect. It demonstrates that to ensure at least one efficient network is stable, resources must be allocated to nodes not responsible for production. The paper characterizes one such allocation rule: the equal split rule. It also characterizes another rule that fails to ensure efficient graphs are stable but arises naturally from bargaining among players. The paper concludes that the tension between stability and efficiency is a general feature of network models, and that the conflict depends on the nature of the value function and the conditions imposed on the allocation rule. It shows that under certain conditions, such as critical link monotonicity, efficient networks can be stable. The paper also discusses the role of anonymity and component balance in the results, and shows that relaxing these conditions can help avoid the conflict between efficiency and stability.