From one-way streets to percolation on random mixed graphs

• Published in: arXiv.org
• Main Authors: Verbavatz, Vincent, Barthelemy, Marc
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 18.03.2021
• Subjects: Empirical analysis; Graphs; Mathematical models; Percolation; Perturbation; Physics - Disordered Systems and Neural Networks; Physics - Physics and Society; Physics - Statistical Mechanics; Shortest-path problems; Streets; Two dimensional models
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2103.10062

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.