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We propose a simple model for a hypergraph generalization of the stochastic block model, using the clustering connection probability $P_{ij}$ between communities $i$ and $j$, and integrating explicitly the hyperedge formation process. Indeed, hyperedges are groups of nodes and we can expect that different real-world networks correspond to different formation processes of these groups and to different levels of homophily between nodes. We describe a simple model where we can explicitly introduce the hyperedge formation process and study its impact on the composition of hyperedges. We focus on the standard case where $P_{ij}=p\delta_{ij}+q(1-\delta_{ij})$, and when $0\leq q\leq p$, we show that the degree and hyperedge size distributions can be approximated by binomials with effective parameters that depend on the number of communities and on $q/p$. Also, the composition of hyperedges goes for $q=0$ from `pure' hyperedges (comprising nodes belonging to the same community) to `mixed' hyperedges that comprise nodes from different communities for $q=p$. We tested various formation processes and our results suggest that when they depend on the composition of the hyperedge, they tend to favor the dominant community and lead to hyperedges with a smaller diversity. In contrast, for formation processes that are independent from the hyperedge structure, we obtain hyperedges comprising a larger diversity of communities. The advantages of the model proposed here are its simplicity and flexibility that make it a good candidate for testing community-related problems, from their detection to their impact on various dynamics., Comment: 10 pages, 10 figures

Despite its importance for practical applications, not much is known about the optimal shape of a network that connects in an efficient way a set of points. This problem can be formulated in terms of a multiplex network with a fast layer embedded in a slow one. To connect a pair of points, one can then use either the fast or slow layer, or both, with a switching cost when going from one layer to the other. We consider here distributions of points in spaces of arbitrary dimension d and search for the fast-layer network of given size that minimizes the average time to reach a central node. We discuss the d = 1 case analytically and the d > 1 case numerically, and show the existence of transitions when we vary the network size, the switching cost and/or the relative speed of the two layers. Surprisingly, there is a transition characterized by a symmetry breaking indicating that it is sometimes better to avoid serving a whole area in order to save on switching costs, at the expense of using more the slow layer. Our findings underscore the importance of considering switching costs while studying optimal network structures, as small variations of the cost can lead to strikingly dissimilar results. Finally, we discuss real-world subways and their efficiency for the cities of Atlanta, Boston, and Toronto. We find that real subways are farther away from the optimal shapes as traffic congestion increases., Comment: 11 pages, 5 figures + Supplementary Material

The macroscopic fundamental diagram (MFD) is a large scale description of the traffic in a 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., Comment: 9 pages, 12 figures, 1 table

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., Comment: Review paper for Reports on Progress in Physics

Urban inequality is a major challenge for cities in the 21st century. This inequality is reflected in the spatial income structure of cities which evolves in time through various processes. Gentrification is a well-known illustration of these dynamics in which the population of a low income area changes as wealthier residents arrive and old-settled residents are expelled. Less understood but very important is the reverse process of gentrification through which areas of cities get impoverished. Gentrification has been widely studied among social sciences, especially in case studies, but there have been fewer quantitative analyses of this phenomenon, and more generally about the spatial dynamics of income in cities. Here, we first propose a quantitative analysis of these income dynamics in cities based on household incomes in 45 American and 9 French Functional Urban Areas (FUA). We found that an important ingredient that determines the evolution of the income level of an area is the income level of its immediate neighboring areas. This empirical finding leads to the idea that these dynamics can be modeled by the voter model of statistical physics. We show that such a model constitutes an interesting tool for both describing and predicting evolution scenarios of urban areas with a very limited number of parameters (two for the US and one for France). We illustrate our results by computing the probability that areas will change their income status in the case of Boston and Paris at the horizon of 2030., Comment: 12 pages, 5 figures

Moves in chess games are usually analyzed on a case-by-case basis by professional players, but thanks to the availability of large game databases, we can envision another approach of the game. Here, we indeed adopt a very different point of view, and analyze moves in chess games from a statistical point of view. We first focus on spatial properties and the location of pieces and show that the number of possible moves during a game is positively correlated with its outcome. We then study heatmaps of pieces and show that the spatial distribution of pieces varies less between human players than with engines (such as Stockfish): engines seem to use pieces in a very different way as human did for centuries. These heatmaps also allow us to construct a distance between players that characterizes how they use their pieces. In a second part, we focus on the best move and the second best move found by Stockfish and study the difference $\Delta$ of their evaluation. We found different regimes during a chess game. In a `quiet' regime, $\Delta$ is small, indicating that many paths are possible for both players. In contrast, there are also `volatile' regimes characterized by a `tipping point', for which $\Delta$ becomes large. At these tipping points, the outcome could then switch completely depending on the move chosen. We also found that for a large number of games, the distribution of $\Delta$ can be fitted by a power law $P(\Delta)\sim \Delta^{-\beta}$ with an exponent that seems to be universal (for human players and engines) and around $\beta\approx 1.8$. The probability to encounter a tipping point in a game is therefore far from being negligible. Finally, we conclude by mentioning possible directions of research for a quantitative understanding of chess games such as the structure of the pawn chain, the interaction graph between pieces, or a quantitative definition of critical points., Comment: 10 pages, 12 figures. Comments welcome

Using global wheat trade data and a network model for shock propagation, we study the impact of the Ukrainian crisis on food security. Depending on the level of reduction in Ukrainian wheat exports, the number of additional individuals falling under the minimum dietary energy requirement varies from 1 to 9 millions, and reaches about 4.8 millions for a $50\%$ reduction in exports. In the most affected countries, supply reductions are mainly related to indirect trade restrictions., Comment: Main paper and supplementary material

The description of complex human mobility patterns is at the core of many important applications ranging from urbanism and transportation to epidemics containment. Data about collective human movements, once scarce, has become widely available thanks to new sources such as Phone CDR, GPS devices, or Smartphone apps. Nevertheless, it is still common to rely on a single dataset by implicitly assuming that it is a valid instance of universal dynamics, regardless of factors such as data gathering and processing techniques. Here, we test such an overarching assumption on an unprecedented scale by comparing human mobility datasets obtained from 7 different data-sources, tracing over 500 millions individuals in 145 countries. We report wide quantifiable differences in the resulting mobility networks and, in particular, in the displacement distribution previously thought to be universal. These variations -- that do not necessarily imply that the human mobility is not universal -- also impact processes taking place on these networks, as we show for the specific case of epidemic spreading. Our results point to the crucial need for disclosing the data processing and, overall, to follow good practices to ensure the robustness and the reproducibility of the results.

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 $r_c^*\sim 1/\sqrt{E}$. 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., Comment: 11 pages, 11 figures (final version with additional results)

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.

Most cities in the US and in 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, working conditions, the future of these structures and the organization of large urban areas is uncertain. Here, we analyze empirical data for US cities and show that they display two transitions at different thresholds. For the first threshold of order $T_c^{FW}\sim 10^4$ commuters, we observe the emergence of a urban freeway. The second threshold is larger and of the order $T_c^{RR}\sim 10^5$ 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 predicts indeed such transitions and allows us to compute explicitly the commuter's thresholds in terms of critical parameters such as the average value of time, average capacity of roads, typical construction cost, etc. Furthermore, this analysis allows us to discuss possible scenarios for the future evolution of these structures. In particular, we show that in many cases it is beneficial to remove urban freeways due to their large social cost (that includes pollution, health cost, etc). This type of information is particularly useful at a time when many cities must confront with the dilemma of renovating these aging structures or converting them into another use., Comment: 11 pages, 6 figures, 1 Table (Supp. Info is available upon request)

Characterizing the spatial organization of urban systems is a challenge which points to the more general problem of describing marked point processes in spatial statistics. We propose a non-parametric method that goes beyond standard tools of point pattern analysis and which is based on a mapping between the points and a "dominance tree", constructed from a recursive analysis of their Voronoi tessellation. Using toy models, we show that the height of a node in this tree encodes both its mark and the structure of its neighborhood, reflecting its importance in the system. We use historical population data in France (1876-2018) and the US (1880-2010) and show that the method highlights multiscale urban dynamics experienced by these countries. These include non-monotonous city trajectories in the US, as revealed by the evolution of their height in the tree. We show that the height of a city in the tree is less sensitive to different statistical definitions of cities than its rank in the urban hierarchy. The method also captures the attraction basins of cities at successive scales, and while in both countries these basin sizes become more homogeneous at larger scales, they are also more heterogeneous in France than in the US. Finally, we introduce a simple graphical representation - the height clock - that monitors the evolution of the role of each city in its country., Comment: Final published version (11 pages, 5 figures)

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.

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 non-trivial order and show that it encodes how straight are shortest paths and is therefore non-universal 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., Comment: 10 pages, 10 figures

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 is 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 $\eta\approx 0.4$) combined with an exponential cut-off $\xi$. This correlation length $\xi$ 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., Comment: Main text + appendices, 14 pages, 11 figures

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 characterized by a non-universal 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 consider 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 non-trivial 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 were 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., Comment: 11 pages, 3 tables, 11 figures

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 cities and the statistical occurrence of megacities, first thought to be described by a universal law due to Zipf, but whose validity has been challenged by recent empirical studies. A theoretical model must also be able to explain the relatively frequent rises and falls of cities and civilizations, and despite many attempts these fundamental questions have not been satisfactorily answered yet. Here we fill this gap by introducing a new kind of stochastic equation for modelling population growth in cities, which we construct from an empirical analysis of recent datasets (for Canada, France, UK and USA) that reveals how rare but large interurban migratory shocks dominate city growth. This equation predicts a complex shape for the city distribution and shows that Zipf's law does not hold in general due to finite-time effects, implying a more complex organization of cities. It also predicts the existence of multiple temporal variations in the city hierarchy, in agreement with observations. Our result underlines the importance of rare events in the evolution of complex systems and at a more practical level in urban planning., Comment: 12 pages, 2 figures, 3 tables. Minor corrections added for the US case (Full paper and supplementary material can be found at: https://www.nature.com/articles/s41586-020-2900-x )

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, the "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., Comment: 17 pages, 2 figures, 4 tables

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 an 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 its stays low and does not come back to its pre-crisis level. Using traffic data from the Official Aviation Guide (OAG) 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., Comment: 12 pages, 7 main figures (+5 figures in appendix)

Understanding how interurban movements can modify the spatial distribution of the population is important for transport planning but is also a fundamental ingredient for epidemic modeling. We focus here on vacation trips (for all transportation modes) during the Chinese Lunar New Year and compare the results for 2019 with the ones for 2020 where travel bans were applied for mitigating the spread of a novel coronavirus (COVID-19). We first show that these travel flows are broadly distributed and display both large temporal and spatial fluctuations, making their modeling very difficult. When flows are larger, they appear to be more dispersed over a larger number of origins and destinations, creating de facto hubs that can spread an epidemic at a large scale. These movements quickly induce (in about a week) a very strong population concentration in a small set of cities. We characterize quantitatively the return to the initial distribution by defining a pendular ratio which allows us to show that this dynamics is very slow and even stopped for the 2020 Lunar New Year due to travel restrictions. Travel restrictions obviously limit the spread of the diseases between different cities, but have thus the counter-effect of keeping high concentration in a small set of cities, a priori favoring intra-city spread, unless individual contacts are strongly limited. These results shed some light on how interurban movements modify the national distribution of populations, a crucial ingredient for devising effective control strategies at a national level., Comment: 15 pages, 6 figures (+supp. material with 14 figures)