From one-way streets to percolation on random mixed graphs.

Abstract

In most studies, street networks are considered as undirected graphs while one-way streets and their effect on shortest paths are usually ignored. Here, we first study the empirical effect of one-way streets in about 140 cities in the world. Their presence induces a detour that persists over a wide range of distances and is characterized by a nonuniversal exponent. The effect of one-ways on the pattern of shortest paths is then twofold: they mitigate local traffic in certain areas but create bottlenecks elsewhere. This empirical study leads naturally to considering a mixed graph model of 2d regular lattices with both undirected links and a diluted variable fraction p of randomly directed links which mimics the presence of one-ways in a street network. We study the size of the strongly connected component (SCC) versus p and demonstrate the existence of a threshold p_{c} above which the SCC size is zero. We show numerically that this transition is nontrivial for lattices with degree less than 4 and provide some analytical argument. We compute numerically the critical exponents for this transition and confirm previous results showing that they define a new universality class different from both the directed and standard percolation. Finally, we show that the transition on real-world graphs can be understood with random perturbations of regular lattices. The impact of one-ways on the graph properties was already the subject of a few mathematical studies, and our results show that this problem has also interesting connections with percolation, a classical model in statistical physics.