Economic agents care about their relative well-being, and the comparisons are usually local. We capture this using a network model, in which an agent's payoff depends on the ranking of their allocation among their network neighbors’. Given a network, an allocation is called α-stable if no blocking coalition whose size is an α fraction of the population can strictly improve their payoffs. We find a sufficient and necessary condition for a network to permit an α-stable allocation: the network has an independent set whose size is at least 1−α of the network population. The characterization of permissive networks holds not only for our baseline ranking preference but also for a range of preferences under which the sets of stable allocations are expanded. We also provide a sufficient condition for an allocation to be stable. Extensions of the model concern directed networks and the case where agents have limited enforcement power.