The growth equation of cities.

The science of cities seeks to understand and explain regularities observed in the world's major urban systems. Modelling the population evolution of cities is at the core of this science and of all urban studies. Quantitatively, the most fundamental problem is to understand the hierarchical organization of city population and the statistical occurrence of megacities. This was first thought to be described by a universal principle known as Zipf's law1,2; however, the validity of this model has been challenged by recent empirical studies3,4. A theoretical model must also be able to explain the relatively frequent rises and falls of cities and civilizations5, but despite many attempts6-10 these fundamental questions have not yet been satisfactorily answered. Here we introduce a stochastic equation for modelling population growth in cities, constructed from an empirical analysis of recent datasets (for Canada, France, the UK and the USA). This model reveals how rare, but large, interurban migratory shocks dominate city growth. This equation predicts a complex shape for the distribution of city populations and shows that, owing to finite-time effects, Zipf's law does not hold in general, implying a more complex organization of cities. It also predicts the existence of multiple temporal variations in the city hierarchy, in agreement with observations5. Our result underlines the importance of rare events in the evolution of complex systems11 and, at a more practical level, in urban planning.

Stochastic equations and cities.

Stochastic equations constitute a major ingredient in many branches of science, from physics to biology and engineering. Not surprisingly, they appear in many quantitative studies of complex systems. In particular, this type of equation is useful for understanding the dynamics of urban population. Empirically, the population of cities follows a seemingly universal law-called Zipf's law-which was discovered about a century ago and states that when sorted in decreasing order, the population of a city varies as the inverse of its rank. Recent data however showed that this law is only approximate and in some cases not even verified. In addition, the ranks of cities follow a turbulent dynamics: some cities rise while other fall and disappear. Both these aspects-Zipf's law (and deviations around it), and the turbulent dynamics of ranks-need to be explained by the same theoretical framework and it is natural to look for the equation that governs the evolution of urban populations. We will review here the main theoretical attempts based on stochastic equations to describe these empirical facts. We start with the simple Gibrat model that introduces random growth rates, and we will then discuss the Gabaix model that adds friction for allowing the existence of a stationary distribution. Concerning the dynamics of ranks, we will discuss a phenomenological stochastic equation that describes rank variations in many systems-including cities-and displays a noise-induced transition. We then illustrate the importance of exchanges between the constituents of the system with the diffusion with noise equation. We will explicit this in the case of cities where a stochastic equation for populations can be derived from first principles and confirms the crucial importance of inter-urban migrations shocks for explaining the statistics and the dynamics of the population of cities.

Scenarios for a post-COVID-19 world airline network.

The airline industry was severely hit by the COVID-19 crisis with an average demand decrease of about 64 % (IATA, April 2020), which triggered already several bankruptcies of airline companies all over the world. While the robustness of the world airline network (WAN) was mostly studied as a homogeneous network, we introduce a new tool for analyzing the impact of a company failure: the "airline company network" where two airlines are connected if they share at least one route segment. Using this tool, we observe that the failure of companies well connected with others has the largest impact on the connectivity of the WAN. We then explore how the global demand reduction affects airlines differently and provide an analysis of different scenarios if it stays low and does not come back to its pre-crisis level. Using traffic data from the Official Aviation Guide and simple assumptions about customer's airline choice strategies, we find that the local effective demand can be much lower than the average one, especially for companies that are not monopolistic and share their segments with larger companies. Even if the average demand comes back to 60 % of the total capacity, we find that between 46 % and 59 % of the companies could experience a reduction of more than 50 % of their traffic, depending on the type of competitive advantage that drives customer's airline choice. These results highlight how the complex competitive structure of the WAN weakens its robustness when facing such a large crisis.

From global scaling to the dynamics of individual cities.

Scaling has been proposed as a powerful tool to analyze the properties of complex systems and in particular for cities where it describes how various properties change with population. The empirical study of scaling on a wide range of urban datasets displays apparent nonlinear behaviors whose statistical validity and meaning were recently the focus of many debates. We discuss here another aspect, which is the implication of such scaling forms on individual cities and how they can be used for predicting the behavior of a city when its population changes. We illustrate this discussion in the case of delay due to traffic congestion with a dataset of 101 US cities in the years 1982-2014. We show that the scaling form obtained by agglomerating all of the available data for different cities and for different years does display a nonlinear behavior, but which appears to be unrelated to the dynamics of individual cities when their population grows. In other words, the congestion-induced delay in a given city does not depend on its population only, but also on its previous history. This strong path dependency prohibits the existence of a simple scaling form valid for all cities and shows that we cannot always agglomerate the data for many different systems. More generally, these results also challenge the use of transversal data for understanding longitudinal series for cities.

Emergence of hierarchy in cost-driven growth of spatial networks.

One of the most important features of spatial networks--such as transportation networks, power grids, the Internet, and neural networks--is the existence of a cost associated with the length of links. Such a cost has a profound influence on the global structure of these networks, which usually display a hierarchical spatial organization. The link between local constraints and large-scale structure is not elucidated, however, and we introduce here a generic model for the growth of spatial networks based on the general concept of cost-benefit analysis. This model depends essentially on a single scale and produces a family of networks that range from the star graph to the minimum spanning tree and are characterized by a continuously varying exponent. We show that spatial hierarchy emerges naturally, with structures composed of various hubs controlling geographically separated service areas, and appears as a large-scale consequence of local cost-benefit considerations. Our model thus provides the basic building blocks for a better understanding of the evolution of spatial networks and their properties. We also find that, surprisingly, the average detour is minimal in the intermediate regime as a result of a large diversity in link lengths. Finally, we estimate the important parameters for various world railway networks and find that, remarkably, they all fall in this intermediate regime, suggesting that spatial hierarchy is a crucial feature for these systems and probably possesses an important evolutionary advantage.

Structure of road networks and the shape of the macroscopic fundamental diagram.

The macroscopic fundamental diagram (MFD) is a large-scale description of the traffic in an urban area and relates the average car flow to the average car density. This MFD has been observed empirically in several cities but how its properties are related to the structure of the road network has remained unclear so far. The MFD displays in general a maximum flow q^{*} for an optimal car density k^{*} which are crucial quantities for practical applications. Here, using numerical modeling and dimensional arguments, we propose scaling laws for these quantities q^{*} and k^{*} in terms of the road density, the intersection density, the average car size and the maximum velocity. This framework is able to explain the scaling observed empirically for several cities in the world, such as the scaling of k^{*} with the road density, the relation between q^{*} and k^{*} and the impact of buses on the overall capacity q^{*}. This work opens the way to a better understanding of the traffic on a road network at a large urban scale.

Symmetry breaking in optimal transport networks.

Engineering multilayer networks that efficiently connect sets of points in space is a crucial task in all practical applications that concern the transport of people or the delivery of goods. Unfortunately, our current theoretical understanding of the shape of such optimal transport networks is quite limited. Not much is known about how the topology of the optimal network changes as a function of its size, the relative efficiency of its layers, and the cost of switching between layers. Here, we show that optimal networks undergo sharp transitions from symmetric to asymmetric shapes, indicating that it is sometimes better to avoid serving a whole area to save on switching costs. Also, we analyze the real transportation networks of the cities of Atlanta, Boston, and Toronto using our theoretical framework and find that they are farther away from their optimal shapes as traffic congestion increases.

Modeling the polycentric transition of cities.

Empirical evidence suggests that most urban systems experience a transition from a monocentric to a polycentric organization as they grow and expand. We propose here a stochastic, out-of-equilibrium model of the city, which explains the appearance of subcenters as an effect of traffic congestion. We show that congestion triggers the instability of the monocentric regime and that the number of subcenters and the total commuting distance within a city scale sublinearly with its population, predictions that are in agreement with data gathered for around 9000 U.S. cities between 1994 and 2010.

Disentangling collective trends from local dynamics.

A single social phenomenon (such as crime, unemployment, or birthrate) can be observed through temporal series corresponding to units at different levels (i.e., cities, regions, and countries). Units at a given local level may follow a collective trend imposed by external conditions, but also may display fluctuations of purely local origin. The local behavior is usually computed as the difference between the local data and a global average (e.g, a national average), a viewpoint that can be very misleading. We propose here a method for separating the local dynamics from the global trend in a collection of correlated time series. We take an independent component analysis approach in which we do not assume a small average local contribution in contrast with previously proposed methods. We first test our method on synthetic series generated by correlated random walkers. We then consider crime rate series (in the United States and France) and the evolution of obesity rate in the United States, which are two important examples of societal measures. For the crime rates in the United States, we observe large fluctuations in the transition period of mid-70s during which crime rates increased significantly, whereas since the 80s, the state crime rates are governed by external factors and the importance of local specificities being decreasing. In the case of obesity, our method shows that external factors dominate the evolution of obesity since 2000, and that different states can have different dynamical behavior even if their obesity prevalence is similar.

Evolution of road infrastructure in large urban areas.

Most cities in the United States and around the world were organized around car traffic. In particular, large structures such as urban freeways or ring roads were built for reducing car traffic congestion. With the evolution of public transportation and working conditions, the future of these structures and the organization of large urban areas is uncertain. Here we analyze empirical data for U.S. urban areas and show that they display two transitions at different thresholds. For the first threshold of order T_{c}^{FW}∼10^{4} commuters, we observe the emergence of a urban freeway. The second threshold is larger and on the order T_{c}^{RR}∼10^{5} commuters above which a ring road emerges. In order to understand these empirical results, we propose a simple model based on a cost-benefit analysis which relies on the balance between construction and maintenance costs of infrastructures and the trip duration decrease (including the effect of congestion). This model indeed predicts such transitions and allows us to compute explicitly the commuter thresholds in terms of critical parameters such as the average value of time, average capacity of roads, and typical construction cost. Furthermore, this analysis allows us to discuss possible scenarios for the future evolution of these structures. In particular, we show that because of the externalities associated with freeways (pollution, health costs, etc.), it might become economically justified to remove urban freeways. This type of information is particularly useful at a time when many cities are confronted with the dilemma of renovating these aging structures or converting them into other uses.

Microdynamics in stationary complex networks.

Many complex systems, including networks, are not static but can display strong fluctuations at various time scales. Characterizing the dynamics in complex networks is thus of the utmost importance in the understanding of these networks and of the dynamical processes taking place on them. In this article, we study the example of the US airport network in the time period 1990-2000. We show that even if the statistical distributions of most indicators are stationary, an intense activity takes place at the local ("microscopic") level, with many disappearing/appearing connections (links) between airports. We find that connections have a very broad distribution of lifetimes, and we introduce a set of metrics to characterize the links' dynamics. We observe in particular that the links that disappear have essentially the same properties as the ones that appear, and that links that connect airports with very different traffic are very volatile. Motivated by this empirical study, we propose a model of dynamical networks, inspired from previous studies on firm growth, which reproduces most of the empirical observations both for the stationary statistical distributions and for the dynamical properties.

Class of models for random hypergraphs.

Despite the recently exhibited importance of higher-order interactions for various processes, few flexible (null) models are available. In particular, most studies on hypergraphs focus on a small set of theoretical models. Here, we introduce a class of models for random hypergraphs which displays a similar level of flexibility of complex network models and where the main ingredient is the probability that a node belongs to a hyperedge. When this probability is a constant, we obtain a random hypergraph in the same spirit as the Erdos-Renyi graph. This framework also allows us to introduce different ingredients such as the preferential attachment for hypergraphs, or spatial random hypergraphs. In particular, we show that for the Erdos-Renyi case there is a transition threshold scaling as 1/sqrt[EN] where N is the number of nodes and E the number of hyperedges. We also discuss a random geometric hypergraph which displays a percolation transition for a threshold distance scaling as r_{c}^{*}∼1/sqrt[E]. For these various models, we provide results for the most interesting measures, and also introduce new ones in the spatial case for characterizing the geometrical properties of hyperedges. These different models might serve as benchmarks useful for analyzing empirical data.

Betweenness centrality in dense spatial networks.

The betweenness centrality (BC) is an important quantity for understanding the structure of complex large networks. However, its calculation is in general difficult and known in simple cases only. In particular, the BC has been exactly computed for graphs constructed over a set of N points in the infinite density limit, displaying a universal behavior. We reconsider this calculation and propose an expansion for large and finite densities. We compute the lowest nontrivial order and show that it encodes how straight are shortest paths and is therefore nonuniversal and depends on the graph considered. We compare our analytical result to numerical simulations obtained for various graphs such as the minimum spanning tree, the nearest neighbor graph, the relative neighborhood graph, the random geometric graph, the Gabriel graph, or the Delaunay triangulation. We show that in most cases the agreement with our analytical result is excellent even for densities of points that are relatively low. This method and our results provide a framework for understanding and computing this important quantity in large spatial networks.

Local impacts on road networks and access to critical locations during extreme floods.

Floods affected more than 2 billion people worldwide from 1998 to 2017 and their occurrence is expected to increase due to climate warming, population growth and rapid urbanization. Recent approaches for understanding the resilience of transportation networks when facing floods mostly use the framework of percolation but we show here on a realistic high-resolution flood simulation that it is inadequate. Indeed, the giant connected component is not relevant and instead, we propose to partition the road network in terms of accessibility of local towns and define new measures that characterize the impact of the flooding event. Our analysis allows to identify cities that will be pivotal during the flooding by providing to a large number of individuals critical services such as hospitalization services, food supply, etc. This approach is particularly relevant for practical risk management and will help decision makers for allocating resources in space and time.

Spatial structure of city population growth.

We show here that population growth, resolved at the county level, is spatially heterogeneous both among and within the U.S. metropolitan statistical areas. Our analysis of data for over 3,100 U.S. counties reveals that annual population flows, resulting from domestic migration during the 2015-2019 period, are much larger than natural demographic growth, and are primarily responsible for this heterogeneous growth. More precisely, we show that intra-city flows are generally along a negative population density gradient, while inter-city flows are concentrated in high-density core areas. Intra-city flows are anisotropic and generally directed towards external counties of cities, driving asymmetrical urban sprawl. Such domestic migration dynamics are also responsible for tempering local population shocks by redistributing inflows within a given city. This spill-over effect leads to a smoother population dynamics at the county level, in contrast to that observed at the city level. Understanding the spatial structure of domestic migration flows is a key ingredient for analyzing their drivers and consequences, thus representing a crucial knowledge for urban policy makers and planners.

Empirical evidence for a jamming transition in urban traffic.

Understanding the mechanisms leading to the formation and the propagation of traffic jams in large cities is of crucial importance for urban planning and traffic management. Many studies have already considered the emergence of traffic jams from the point of view of phase transitions, but mostly in simple geometries such as highways for example or in the framework of percolation where an external parameter is driving the transition. More generally, empirical evidence and characterization for a congestion transition in complex road networks are scarce, and here, we use traffic measures for Paris (France) during the period 2014-2018 for testing the existence of a jamming transition at the urban level. In particular, we show that the correlation function of delays due to congestion is a power law (with exponent η ≈ 0.4) combined with an exponential cut-off ξ. This correlation length is shown to diverge during rush hours, pointing to a jamming transition in urban traffic. We also discuss the spatial structure of congestion and identify a core of congested links that participate in most traffic jams and whose structure is specific during rush hours. Finally, we show that the spatial structure of congestion is consistent with a reaction-diffusion picture proposed previously.

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.

Emerging dynamics from high-resolution spatial numerical epidemics.

Simulating nationwide realistic individual movements with a detailed geographical structure can help optimise public health policies. However, existing tools have limited resolution or can only account for a limited number of agents. We introduce Epidemap, a new framework that can capture the daily movement of more than 60 million people in a country at a building-level resolution in a realistic and computationally efficient way. By applying it to the case of an infectious disease spreading in France, we uncover hitherto neglected effects, such as the emergence of two distinct peaks in the daily number of cases or the importance of local density in the timing of arrival of the epidemic. Finally, we show that the importance of super-spreading events strongly varies over time.

Fluctuation effects in metapopulation models: percolation and pandemic threshold.

Metapopulation models provide the theoretical framework for describing disease spread between different populations connected by a network. In particular, these models are at the basis of most simulations of pandemic spread. They are usually studied at the mean-field level by neglecting fluctuations. Here we include fluctuations in the models by adopting fully stochastic descriptions of the corresponding processes. This level of description allows to address analytically, in the SIS and SIR cases, problems such as the existence and the calculation of an effective threshold for the spread of a disease at a global level. We show that the possibility of the spread at the global level is described in terms of (bond) percolation on the network. This mapping enables us to give an estimate (lower bound) for the pandemic threshold in the SIR case for all values of the model parameters and for all possible networks.

Access to mass rapid transit in OECD urban areas.

As mitigating car traffic in cities has become paramount to abate climate change effects, fostering public transport in cities appears ever-more appealing. A key ingredient in that purpose is easy access to mass rapid transit (MRT) systems. So far, we have however few empirical estimates of the coverage of MRT in urban areas, computed as the share of people living in MRT catchment areas, say for instance within walking distance. In this work, we clarify a universal definition of such a metrics - People Near Transit (PNT) - and present measures of this quantity for 85 urban areas in OECD countries - the largest dataset of such a quantity so far. By suggesting a standardized protocol, we make our dataset sound and expandable to other countries and cities in the world, which grounds our work into solid basis for multiple reuses in transport, environmental or economic studies.