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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 characterized by a non-universal 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 consider 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 non-trivial 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 were 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.
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https://hal-cea.archives-ouvertes.fr/cea-03188105
Contributor : Emmanuelle de Laborderie <>
Submitted on : Thursday, April 1, 2021 - 4:34:50 PM
Last modification on : Wednesday, April 14, 2021 - 12:13:03 PM

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Distributed under a Creative Commons Attribution 4.0 International License

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  • HAL Id : cea-03188105, version 1
  • ARXIV : 2103.10062

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Vincent Verbavatz, Marc Barthelemy. From one-way streets to percolation on random mixed graphs. Physical Review E , American Physical Society (APS), In press. ⟨cea-03188105⟩

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