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Recently, De Martino et al have presented a general framework for the studyof transportation phenomena on complex networks. One of their most significantachievements was a deeper understanding of the phase transition from theuncongested to the congested phase at a critical traffic load. In this paper,we also study phase transition in transportation networks using a discrete timerandom walk model. Our aim is to establish a direct connection between thestructure of the graph and the value of the critical traffic load. Applyingspectral graph theory, we show that the original results of De Martino et alshowing that the critical loading depends only on the degree sequence of thegraph -- suggesting that different graphs with the same degree sequence havethe same critical loading if all other circumstances are fixed -- is valid onlyif the graph is dense enough. For sparse graphs, higher order corrections,related to the local structure of the network, appear.

The effect of network structure on phase transitions in queuing networks