Scaling in transportation networks

Subway systems span most large cities, and railway networks most countries in the world. These networks are fundamental in the development of countries and their cities, and it is therefore crucial to understand their formation and evolution. However, if the topological properties of these networks are fairly well understood, how they relate to population and socio-economical properties remains an open question. We propose here a general coarse-grained approach, based on a cost-benefit analysis that accounts for the scaling properties of the main quantities characterizing these systems (the number of stations, the total length, and the ridership) with the substrate's population, area and wealth. More precisely, we show that the length, number of stations and ridership of subways and rail networks can be estimated knowing the area, population and wealth of the underlying region. These predictions are in good agreement with data gathered for about $140$ subway systems and more than $50$ railway networks in the world. We also show that train networks and subway systems can be described within the same framework, but with a fundamental difference: while the interstation distance seems to be constant and determined by the typical walking distance for subways, the interstation distance for railways scales with the number of stations.Comment: 8 pages, 6 figures, 1 table. To appear in PLoS On

The spatial structure of networks

We study networks that connect points in geographic space, such as transportation networks and the Internet. We find that there are strong signatures in these networks of topography and use patterns, giving the networks shapes that are quite distinct from one another and from non-geographic networks. We offer an explanation of these differences in terms of the costs and benefits of transportation and communication, and give a simple model based on the Monte Carlo optimization of these costs and benefits that reproduces well the qualitative features of the networks studied.Comment: 5 pages, 3 figure

Route Planning in Transportation Networks

We survey recent advances in algorithms for route planning in transportation networks. For road networks, we show that one can compute driving directions in milliseconds or less even at continental scale. A variety of techniques provide different trade-offs between preprocessing effort, space requirements, and query time. Some algorithms can answer queries in a fraction of a microsecond, while others can deal efficiently with real-time traffic. Journey planning on public transportation systems, although conceptually similar, is a significantly harder problem due to its inherent time-dependent and multicriteria nature. Although exact algorithms are fast enough for interactive queries on metropolitan transit systems, dealing with continent-sized instances requires simplifications or heavy preprocessing. The multimodal route planning problem, which seeks journeys combining schedule-based transportation (buses, trains) with unrestricted modes (walking, driving), is even harder, relying on approximate solutions even for metropolitan inputs.Comment: This is an updated version of the technical report MSR-TR-2014-4, previously published by Microsoft Research. This work was mostly done while the authors Daniel Delling, Andrew Goldberg, and Renato F. Werneck were at Microsoft Research Silicon Valle