The effect of network structure on phase transitions in queuing networks

Recently, De Martino et al. [J. Stat. Mech. (2009) P08023; Phys. Rev. E 79, 015101 (2009)] have presented a general framework for the study of transportation phenomena on random networks with annealed disorder. One of their most significant achievements was a deeper understanding of the phase transition from the uncongested to the congested phase at a critical traffic load on uncorrelated networks. In this paper, we also study phase transition in transportation networks using a discrete time random walk model. Our aim is to establish a direct connection between the structure of an uncorrelated random graph with quenched disorder and the value of the critical traffic load. We show that if the network is dense, the quenched and annealed formulas for the critical loading probability coincide. For sparse graphs, higher-order corrections, related to the local structure of the network, appear.