Asymptotic Traffic Flow in an Hyperbolic Network II: Non-uniform Traffic

In this work we study the asymptotic traffic behaviour in Gromov’s hyperbolic spaces when the traffic decays exponentially with the distance. We prove that under general conditions, there exist 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 D that depends on the geometry of the network such that if 1 < β < D the traffic is global and there is a small set of highly congested nodes called the core. However, if β > D then the traffic is essentially local and the core is empty.