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

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.

[1]  H. Robbins A Theorem on Graphs, with an Application to a Problem of Traffic Control , 1939 .

[2]  J. Hammersley,et al.  Percolation processes , 1957, Mathematical Proceedings of the Cambridge Philosophical Society.

[3]  J. W. Essam,et al.  Exact Critical Percolation Probabilities for Site and Bond Problems in Two Dimensions , 1964 .

[4]  Carsten Thomassen,et al.  Distances in orientations of graphs , 1975, J. Comb. Theory, Ser. B.

[5]  F. Boesch,et al.  ROBBINS'S THEOREM FOR MIXED MULTIGRAPHS , 1980 .

[6]  S. Obukhov The problem of directed percolation , 1980 .

[7]  H. Kesten Percolation theory for mathematicians , 1982 .

[8]  S. Redner Conductivity of random resistor-diode networks , 1982 .

[9]  S. Redner Directed and diode percolation , 1982 .

[10]  S. Redner Exact exponent relations for random resistor-diode networks , 1982 .

[11]  P. Erdos,et al.  On the evolution of random graphs , 1984 .

[12]  S. Redner,et al.  Introduction To Percolation Theory , 2018 .

[13]  F. Roberts Graph Theory and Its Applications to Problems of Society , 1987 .

[14]  Sakamoto,et al.  Percolation in two-dimensional lattices. I. A technique for the estimation of thresholds. , 1989, Physical review. B, Condensed matter.

[15]  Tomasz Luczak,et al.  The phase transition in the evolution of random digraphs , 1990, J. Graph Theory.

[16]  Mary Lay,et al.  Ways of the world : a history of the world's roads and of the vehicles that used them , 1992 .

[17]  Gary A. Talbot Applications of Percolation Theory , 1995 .

[18]  J J Stemley One-Way Streets Provide Superior Safety and Convenience , 1998 .

[19]  H. Kakuno,et al.  Critical behavior of a random diode network. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[20]  I. Jensen Low-density series expansions for directed percolation: I. A new efficient algorithm with applications to the square lattice , 1999, cond-mat/9906036.

[21]  D S Callaway,et al.  Network robustness and fragility: percolation on random graphs. , 2000, Physical review letters.

[22]  Random resistor-diode networks and the crossover from isotropic to directed percolation , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[23]  M. Newman,et al.  Random graphs with arbitrary degree distributions and their applications. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  H. Janssen,et al.  Conductivity of continuum percolating systems. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  S. N. Dorogovtsev,et al.  Giant strongly connected component of directed networks. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  A. Barabasi,et al.  Percolation in directed scale-free networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  Esther M. Arkin,et al.  A note on orientations of mixed graphs , 2002, Discret. Appl. Math..

[28]  Christophe Claramunt,et al.  Topological Analysis of Urban Street Networks , 2004 .

[29]  M. Serrano,et al.  Generalized percolation in random directed networks. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[30]  Kim Christensen,et al.  Complexity and Criticality , 2005 .

[31]  Gábor Csárdi,et al.  The igraph software package for complex network research , 2006 .

[32]  Peter Newman,et al.  The environmental impact of cities , 2006 .

[33]  V. Latora,et al.  Centrality measures in spatial networks of urban streets. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[34]  Dirk Helbing,et al.  Scaling laws in the spatial structure of urban road networks , 2006 .

[35]  Guy Theraulaz,et al.  Topological patterns in street networks of self-organized urban settlements , 2006 .

[36]  David M Levinson,et al.  Measuring the Structure of Road Networks , 2007 .

[37]  Aric Hagberg,et al.  Exploring Network Structure, Dynamics, and Function using NetworkX , 2008, Proceedings of the Python in Science Conference.

[38]  Adilson E Motter,et al.  Local structure of directed networks. , 2007, Physical review letters.

[39]  D. Dodman Blaming cities for climate change? An analysis of urban greenhouse gas emissions inventories , 2009 .

[40]  Jules Gleicher The Book of Origins , 2010 .

[41]  Vito Latora,et al.  Elementary processes governing the evolution of road networks , 2012, Scientific Reports.

[42]  Crossover from isotropic to directed percolation. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[43]  Vito Latora,et al.  Urban Street Networks, a Comparative Analysis of Ten European Cities , 2012, 1211.0259.

[44]  M. Barthelemy,et al.  A typology of street patterns , 2014, Journal of The Royal Society Interface.

[45]  Matthias Beck,et al.  On Weak Chromatic Polynomials of Mixed Graphs , 2015, Graphs Comb..

[46]  Marc Barthelemy,et al.  Morphogenesis of Spatial Networks , 2017 .

[47]  Mattia Zanella,et al.  Form and urban change – An urban morphometric study of five gentrified neighbourhoods in London , 2017 .

[48]  Geoff Boeing,et al.  OSMnx: New Methods for Acquiring, Constructing, Analyzing, and Visualizing Complex Street Networks , 2016, Comput. Environ. Urban Syst..

[49]  André P. Vieira,et al.  Percolation on an isotropically directed lattice , 2018, Physical Review E.

[50]  Dietrich Stauffer,et al.  Introduction To Percolation Theory , 2018 .

[51]  Hans Jürgen Herrmann,et al.  Perturbing the shortest path on a critical directed square lattice , 2018, Physical Review E.

[52]  Gourab Ghoshal,et al.  From the betweenness centrality in street networks to structural invariants in random planar graphs , 2017, Nature Communications.

[53]  Marc Barthelemy,et al.  Critical factors for mitigating car traffic in cities , 2019, PloS one.

[54]  Geoff Boeing,et al.  Urban spatial order: street network orientation, configuration, and entropy , 2018, Appl. Netw. Sci..

[55]  The elastic and directed percolation backbone , 2018, 1805.08201.