Competitive interactions represent one of the driving forces behind evolution and natural selection in biological and sociological systems. For example, animals in an ecosystem may vie for food or mates; in a market economy, firms may compete over the same group of customers; sensory stimuli may compete for limited neural resources in order to enter the focus of attention. Here, we derive rules based on the spectral properties of the network governing the competitive interactions between groups of agents organized in networks. In the scenario studied here the winner of the competition, and the time needed to prevail, essentially depend on the way a given network connects to its competitors and on its internal structure. Our results allow assessing the extent to which real networks optimize the outcome of their interaction, but also provide strategies through which competing networks can improve on their situation. The proposed approach is applicable to a wide range of systems that can be modeled as networks.