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

TL;DR

This paper investigates how traffic flow behaves in hyperbolic networks with exponentially decaying traffic, revealing a phase transition between localized and global traffic regimes depending on decay rate.

Contribution

It establishes the existence of a phase transition in traffic flow in hyperbolic graphs based on decay parameters, linking network geometry to congestion patterns.

Findings

Existence of a critical decay parameter $eta_c$ for phase transition.

Global traffic with a congested core occurs when $eta<eta_c$.

Traffic remains local with minimal congestion when $eta>eta_c$.

Abstract

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.