  In this work we study the asymptotic traffic flow in Gromov's hyperbolic
graphs when the traffic decays exponentially with the distance. We prove that
under general conditions, there exists a phase transition between local and
global traffic. More specifically, assume that the traffic rate between two
nodes $u$ and $v$ is given by $R(u,v)=\beta^{-d(u,v)}$ where $d(u,v)$ is the
distance between the nodes. Then there exists a constant $\beta_c$ that depends
on the geometry of the network such that if $1<\beta<\beta_c$ the traffic is
global and there is a small set of highly congested nodes called the core.
However, if $\beta>\beta_c$ then the traffic is essentially local and the core
is empty which implies very small congestion.
