Asymptotic Traffic Flow in a Hyperbolic Network: Non-uniform Traffic

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.