  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.
