Most of the world's people are now living in cities and urbanization is expected to keep increasing in the near future. The resulting challenges are complex, difficult to handle, and range from increasing dependence on energy, to the emergence of socio-spatial inequalities, to serious environmental and sustainability issues. Understanding and modeling the structure and evolution of cities is then more important than ever as policy makers are actively looking for new paradigms in urban and transport planning.

The recent advances obtained in the understanding of cities have generated increased attention to the potential implication of new theoretical models in agreement with data. Questions such as urban sprawl, effects of congestion, dominant mechanisms governing the spatial distribution of activities and residences, and the effect of new transportation infrastructures are fundamental questions that we need to understand if we want a harmonious development of cities in the future, from both social and economic points of view.

Cities were for a long time the subject of numerous studies in a large number of fields. Discussion of the ideal city can be traced back at least to the Renaissance, and more recently scientists have tried to describe quantitatively the formation and evolution of cities. Regional science and then quantitative geography addressed various problems such as the spatial organization of cities, the impact of infrastructures, and transport. It is remarkable to note that as early as the 1970s quantitative geographers realized the crucial importance of networks in these systems, and produced visionary studies about networks, their evolution, and the complexity of cities (Haggett et al. 1977).

These studies were further developed mathematically by economists who discussed the interplay between space and economic aspects in cities. Many important models find their origin in the seminal paper of Von Thunen and describe isolated, monocentric cities in terms of utility maximization subject to budget constraints. These models allowed spatial economics to get a grasp of the relations between space, income, and transportation; for example. Japanese economists Fujita and Ogawa discussed the impact of agglomeration effects between firms in a general model that deals with the location choice for individuals and companies.

In the previous chapter, we discussed the analysis and modeling of mobility patterns in cities. However, as cities expand, their transportation networks are also growing, with increasing interconnections between different transportation modes. In large cities, we can now choose the transportation mode to travel from one point to another, and this trip can even involve several different modes. This multimodality is a new aspect of large cities and brings new questions and problems. From the users, point of view, it becomes difficult to deal with the huge amount of information needed for describing the different transportation networks and their interconnections. From the transport agencies point of view, the managing task becomes harder because the different modes are usually run by separate agencies; this renders optimization difficult owing to the large number of aspects that have to be taken into account (Guo and Wilson 2011).

In particular, an important problem concerning multimodality is the synchronization between different modes. For example, on average in the UK, 23% of travel time is lost in connections for trips with more than one mode (Gallotti and Barthelemy, 2014). This lack of synchronization between modes induces differences between the theoretical quickest trip and the “time-respecting” path, which takes into account waiting times at interconnection nodes.

In order to address these problems and more generally to understand the impact of the coupling between modes, we need new tools in order to identify the main factors that govern their efficiency. The multilayer network approach seems to be the most convenient framework for studying these systems (Kivel ä et al. 2014; Boccaletti et al. 2014). In this framework, each layer represents a mode and intermodal connections are represented by inter-layer links. In this chapter we discuss some aspects of multimodality and present tools for measuring and characterizing these coupled networks and their efficiency as a whole.

A multilayer network view of urban navigation

Empirical observations of multimodality

We first describe empirical results obtained by Gallotti and Barthelemy (2014) from timetables for the whole UK and for all transport modes. We note that these results were not obtained for traffic data (that are usually difficult to get). Instead we used these timetable data and the assumption of uniform demand.

What is our “understanding”?

As we discussed in Chapter 1, “understanding” has many definitions, with a variable amount of quantitative input. For a physicist, understanding does not mean having a story consistent with reality only, but also having mathematical tools and models able to describe real phenomena and to predict the outcome of experiments. Even if a qualitative description of processes is somewhat satisfying, it is not enough for constructing a science of cities. Indeed, we would like to identify the most important parameters, not only to understand the past, but also to be able to construct a model that gives, with a reasonable confidence, the future evolution of a city and to test the impact of various policies.

At this point, we certainly have a number of pieces of the puzzle, and we have discussed some of them in this book. It doesn't however mean that we have solved the full puzzle. New data sources and large datasets allow us to get a precise idea of what is happening in cities. We are currently experiencing an exciting time during which we can challenge the purely theoretical developments made these last decades. In many empirical studies, the identification of relevant factors was essentially done statistically, and we can now hope to go beyond and to have a more mechanistic approach, where a model based on simple processes is able to reproduce empirical observations.

Concerning the spatial structure of cities, new data sources give us a real-time, high-resolution picture of mobility. The structure of mobility flows that come out from these datasets departs from the usual image of a monocentric city where flows converge towards the central business district. Instead, for large cities, the main flows appear to be far from the localization between centers of residence and activities, that we could have na ïvely expected. This massive amount of data also allows us to quantitatively assess the degree of polycentricity of an urban system. A simple model showed that congestion is a crucial factor in understanding the evolution of polycentricity and mobility patterns with the population size.

Statistical physics of complex systems

The beginning of statistical physics can be traced back to thermodynamics in the nineteenth century. The field is still very active today, with modern problems occurring in out-of-equilibrium systems. The first problems (up to c. 1850) were to describe the exchange of heat through work and to define concepts such as temperature and entropy. A little later many studies were devoted to understanding the link between a microscopic description of a system (in terms of atoms and molecules) and a macroscopic observation (e.g., the pressure or the volume of a system). The concepts of energy and entropy could then be made more precise, leading to an important formalization of the dynamics of systems and their equilibrium properties.

More recently, during the twentieth century, statistical physicists invested much time in understanding phase transitions. The typical example is a liquid that undergoes a liquid-to-solid transition when the temperature is lowered. This very common phenomenon turned out, however, to be quite complex to understand and to describe theoretically. Indeed, this type of “emergent behavior” is not easily predictable from the properties of the elementary constituents and as Anderson (1972) put it: ”… the whole becomes not only more than but very different from the sum of its parts.” In these studies, physicists understood that interactions play a critical role: without interactions there is usually no emergent behavior, since the new properties that appear at large scales result from the interactions between constituents. Even if the interaction is “simple,” the emergent behavior might be hard to predict or describe. In addition, the emergent behavior depends, not on all the details describing the system, but rather on a small number of parameters that are actually relevant at large scales (see for example Goldenfeld 1992).

Statistical physics thus primarily deals with the link between microscopic rules and macroscopic emergent behavior and many techniques and concepts have been developed in order to understand this translation – among them the notion of relevant parameters, but also the idea that at each level of description of a system there is a specifically adapted set of tools and concepts.

Mobility is obviously a crucial phenomenon in cities. In fact, it is probably one of the most important mechanisms that govern the structure and dynamics of cities. Indeed, individuals go to cities to buy, sell or exchange goods, to work, or to meet with other individuals and for this they need different transportation means. This is where technology enters the problem through the (average) velocity of transportation modes. This average velocity increased during the evolution of technology and modified the structure and organization of cities. For example, we see in Fig. 5.1 that the “horizon” of an individual depends strongly on her transportation mode. For a walker, the horizon is essentially isotropic and small, while the car allows for a wider exploration but one which is anisotropic and follows transportation infrastructures. This correlation between the spatial structure of the city and the available technology at the moment of its creation is clearly illustrated by Anas et al. (1998) for US cities. Many major cities, such as Denver or Oklahoma City, developed around rail terminals that triggered the formation of central business districts. In contrast, automobile-era cities that developed later, such as Dallas or Houston, have a spatial organization that is essentially determined by the highway system.

In terms of mobility, the city center is also the location that mimimizes the average distance to all other locations in the city. Very naturally, it is then the main attraction for businesses and residences, which leads to competition for space between individuals or firms, giving rise to the real-estate market. There is also a well-known relation between land-use and accessibility, as was discussed some time ago by Hansen (1959), and new, extensive datasets will certainly enable us in the future to characterize precisely the relation between these important factors.

It is of course very difficult to make an exhaustive review about all studies on mobility and we will focus in this chapter on several specific points. We will mostly describe the general features of mobility and will leave the discussion about multimodal aspects for Chapter 6.