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In this work we study the asymptotic traffic flow in Gromov 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) = β-d(u, v), where d(u, v) is the distance between the nodes. Then there exists a constant βc, that depends on the geometry of the network, such that if 1 <; β <; βc the traffic is global and there is a small set of highly congested nodes called the core. However, if β >; βc then the traffic is essentially local and the core is empty which implies very small congestion.