MOVING THE WORLD. Traffic Science Faculty "Friedrich List" Institute for Economics and Traffic Chair for Traffic Modelling and Econometrics Dresden University of Technology, Germany Stefan Lämmer Scaling Laws in the Spatial Structure of Urban Road Networks Physica A 363(1) 89-95, 2006

2 Institute for Economics and Traffic Chair for Traffic Modeling and Econometrics Stefan Lämmer 2 Scope of Study ► Urban road network analysis of 20 largest cities of Germany (ranked by population) ► Geographical database: Tele Atlas MultiNet TM City of Dresden with 9643 nodes and links S. Lämmer, B. Gehlsen, and D. Helbing (2006) Scaling laws in the spatial structure of urban road networks, Physica A 363(1)

3 Institute for Economics and Traffic Chair for Traffic Modeling and Econometrics Stefan Lämmer 3 Distinct Classes of Networks ► Random networks (Erdős and Rényi 1959) Exponential node-degree distribution High vulnerability to random failures ► Scale-free networks (Barabasi and Albert 1999) Short distances (small world phenomenon) High clustering coefficients Power-law node-degree distribution ► Urban road networks (Gastner and Newman 2004) Mainly planar cellular structure Limited node-degrees (average strictly less than 6) Very high network diameter, high redundancy

4 Institute for Economics and Traffic Chair for Traffic Modeling and Econometrics Stefan Lämmer 4 Generating a Road Network ► Combinatorial Optimization Problem Given: node positions and total link length. Minimize the average length of all shortest paths. Distances are the most fundamental property of road networks! (a) Link cost proportional to network distance (b)Link cost proportional to Euclidian distance Scale-free network Road network

5 Institute for Economics and Traffic Chair for Traffic Modeling and Econometrics Stefan Lämmer 5 Scaling of Neighborhood Sizes With a travel-distance-budget of r kilometers, a car driver can reach a neighborhood of size Dresden: d = 2.2 With a travel-time-budget of τ minutes, a car driver can reach a neighborhood of size

6 Institute for Economics and Traffic Chair for Traffic Modeling and Econometrics Stefan Lämmer 6 Effective Dimension ► Computation of effective dimension d: ► High values of d imply: Neighborhoods grow fast by small increases of travel-time budget. High accessibility of distant places. Heterogeneous distribution of road speeds, e.g. the city might have an underlying highway system. Typical values for d Berlin2,330Bremen2,220 Hamburg2,350Duisburg2,050 Munich2,463Leipzig2,304 Cologne2,372Nuremberg2,399 Frankfurt2,388Dresden2,205 Dortmund2,091Bochum2,279 Stuttgart2,008Wuppertal2,040 Essen2,243Bielefeld2,337 Düsseldorf2,700Bonn2,134

7 Institute for Economics and Traffic Chair for Traffic Modeling and Econometrics Stefan Lämmer 7 Spatial concentration of traffic flow ► Traffic flows can be estimated by betweenness centrality b, which is the number of shortest paths visiting a link or a node. (We assume homogeneous OD-flows and ignore congestion effects)

8 Institute for Economics and Traffic Chair for Traffic Modeling and Econometrics Stefan Lämmer 8 Scaling of traffic flow on nodes ► High values of β imply: Traffic flow concentrates on few highly important intersections. Low redundancy (lack of alternative routes) High vulnerability to failures of traffic control Typical values for β Berlin1.481 Hamburg1.469 Munich1.486 Cologne1.384 Frankfurt1.406 Dortmund1.340 Stuttgart1.377 Essen1.368 Düsseldorf1.380 Bremen1.351 Duisburg1.480 Leipzig1.320 Nuremberg1.420 Dresden1.355 Bochum1.337 Wuppertal1.279 Bielefeld1.337 Bonn1.374

9 Institute for Economics and Traffic Chair for Traffic Modeling and Econometrics Stefan Lämmer 9 Concentration of Traffic on Road Meters ► Lorenz-Curve ► High values of Gini-Coefficient g imply Bundling of traffic on arterial roads Existence of bottlenecks (bridges) Reduced traffic (in residential areas) Distinct hierarchy of roads Typical values for g Berlin0,871Stuttgart0,894Nuremberg0,854 Hamburg0,869Essen0,892Dresden0,870 Munich0,869Düsseldorf0,849Bochum0,847 Cologne0,875Bremen0,909Wuppertal0,881 Frankfurt0,873Duisburg0,900Bielefeld0,872 Dortmund0,875Leipzig0,880Bonn0,889 Half of the total traffic volume is handled by only 3.2% of the road meters. 50% of all road meters have only 0.2% of the total traffic volume. 80% of total traffic volume is concentrated on no more than 10% of all road meters.

10 Institute for Economics and Traffic Chair for Traffic Modeling and Econometrics Stefan Lämmer 10 Cellular Structures ► Distribution of cell-degrees (number of neighboring cells) Crack patterns Dragon fly wings Road networks Honey combs

11 Institute for Economics and Traffic Chair for Traffic Modeling and Econometrics Stefan Lämmer 11 Scaling of Cell Areas Typical values for α Berlin Bremen Hamburg Duisburg Munich Leipzig Cologne Nuremberg Frankfurt Dresden Dortmund Bochum Stuttgart Wuppertal Essen Bielefeld Düsseldorf Bonn Cells with half the cell area are two times more frequent

12 Institute for Economics and Traffic Chair for Traffic Modeling and Econometrics Stefan Lämmer 12 Distribution of Form Factors ► Form Factor φ: Fraction of the circumscribed circle that is covered by the cell φ=0 … long and narrow φ=1 … compact and round ► High values of Var(φ) imply Irregular network structure, e.g. city has grown over many epochs Difficult to navigate from car driver’s point of view Var(φ) Berlin0,159 Hamburg0,164 Munich0,159 Cologne0,165 Frankfurt0,169 Dortmund0,166 Stuttgart0,170 Essen0,169 Düsseldorf0,175 Bremen0,166 Duisburg0,169 Leipzig0,153 Nuremberg0,172 Dresden0,156 Bochum0,171 Wuppertal0,162 Bielefeld0,161 Bonn0,173

13 Institute for Economics and Traffic Chair for Traffic Modeling and Econometrics Stefan Lämmer 13 Conclusions ► Scaling of neighborhood sizes Fast roads let neighborhoods grow fast Effective dimension of urban space is significantly higher than two (although mainly planar) ► Scaling of traffic flow Estimation of traffic based on betweenness centrality Power-law distribution of traffic flow on nodes We found quantitative measures to characterize concentration of traffic ► Scaling of cell areas Power-law distribution of cell areas We found quantitative measures to characterize irregularity of cellular structure

MOVING THE WORLD. Traffic Science Faculty "Friedrich List" Institute for Economics and Traffic Chair for Traffic Modelling and Econometrics Dresden University of Technology, Germany Stefan Lämmer Scaling Laws in the Spatial Structure of Urban Road Networks Physica A 363(1) 89-95