In this work we study the asymptotic traffic behavior for Gromov's hyperbolic
networks as the size of the network increases. We prove that under certain mild
hypothesis the traffic in a large hyperbolic network tends to pass through a
finite set of highly congested nodes. These nodes will be called the ``core" of
the network. We provide a formal definition of the core in a very general
context and we study the properties of this set for hyperbolic graphs.