arXiv subjects

Diego Rybski

Publications and source records attributed to Diego Rybski.

At least 19 recordsLinked to original sources

arXiv2026

As Cities Grow, They Spread Out Rather Than Rise

Cities and settlements are a global phenomenon, and their continued expansion is fundamentally transforming patterns of resource demand. This transformation is reflected in growing pressures on land, material use, and infrastructure. Thus, understanding how cities grow in space is essential for planning future urban development and determining all its consequences. Here, we characterize four geometrically related dimensions of urban growth: two horizontal dimensions describing the urban area, one vertical dimension represented by average building height, which together with the urban area defines building volume, and the population dimension. Analyzing a global dataset of growing cities, for the period 1975-2025 we identify three main patterns. First, urban growth is fundamentally anisotropic -- as cities gain population, they expand much more rapidly in the horizontal than in the vertical direction. Second, average building height decreases in the vast majority of cities as they grow, indicating that vertical development is not only slower than horizontal expansion but on average, can act in an effectively negative manner by giving greater weight to the horizontal dimensions. Third, despite their diverse growth trajectories, cities converge toward characteristic population densities, both with respect to urban area and building volume. We further explore alternative scenarios for future urban growth. These projections show that different assumptions about horizontal and vertical scaling can lead to substantially different demands for urban land and building volume. These findings reveal constraints on the long-term evolution of urban form and provide a quantitative assessment for understanding the spatial dynamics and development of cities.

physics.soc-ph↗

arXiv2026

Large cities lose their growth advantage as countries urbanize

The share of the world population living in cities with more than one million people rose from 11% in 1975 to 24% in 2025 (our estimates). Will this trend towards greater concentration in large cities continue or level off? We introduce two new city population datasets that use consistent city definitions across countries and over time. The first covers the world between 1975 and 2025, using satellite imagery. The second covers the U.S. between 1850 and 2020, using census microdata. We find that urban growth follows a characteristic life cycle. In the early stages of a country's urbanization process, large cities grow faster than smaller ones. At later stages, growth rates equalize across sizes. We use this life cycle to project future population concentration in large cities. Our projections suggest that 38% of the world population will be living in cities with more than one million people by 2100. This estimate is higher than the 33% implied by the well-known theory of proportional growth, but lower than the 42% obtained by extrapolating current trends.

physics.soc-ph↗

arXiv2026

Settlement percolation: global maps of Critical Distances

A substantial share of the Earth's land surface is managed by humans, with cities representing the most extreme form of anthropogenic land use. There are zillion ways in which settlements can be arranged across a given area, and their specific spatial configuration has important consequences for both urban systems and the natural environment. Here, we introduce a novel approach to characterizing settlement configuration by systematically quantifying it in terms of a transition resembling percolation -- that is, by identifying the critical distance at which isolated settlements merge into a giant, overarching settlement cluster. We estimate this critical distance across multiple spatial scales and units, including national and subnational levels, non-overlapping tiles, and moving windows, covering the entire globe. The critical distance provides an independent measure of settlement connectivity and thus adds value to spatial analyses of settlement structure and its social, economic, and ecological impacts. Accordingly, our Global Settlement Percolation (GSP) dataset is relevant to a wide range of research communities, including those studying urban morphology, land-use patterns, and landscape ecology.

physics.soc-ph↗

arXiv2025

One-Dimensional Urban Scaling

In this paper, we apply recent findings from urban scaling theory to evaluate how it could be applied to a one-dimensional archetypal city. Our focus is on how the simplicity of a one-dimensional model can provide intuitive insights that might be obscured in more complex, multidimensional models. With this instructive model, it is possible to visualize that if the population's accessibility decreases more slowly than the increase in distance, an agglomeration effect occurs. This leads to increasing returns to scale when considering interactions between people or economies of scale in the case of amenities that serve the city. These findings highlight a key characteristic of complex systems: interacting parts that together exhibit properties that are not evident when considered individually. The results presented here are derived from both numerical simulations and theoretical analysis. Although one-dimensionality is used here as a simplification, many natural (e.g., Sułoszowa in Poland) and artificial cities (e.g., The Line in Saudi Arabia) worldwide are shaped this way.

physics.soc-ph↗

arXiv2024

How buildings change the fundamental allometry

We demonstrate that the original fundamental allometry alone cannot accurately describe the relationship between urban area and population size. Instead, building height is a third factor that interplays with area and population. To illustrate this, we propose a straightforward model based on the idea that city area is the result of people's desire to live close to one another while also having sufficient living space. This leads to a more general form of fundamental allometry (relating area, population, and building height). Our argument is supported by empirical data from different countries.

physics.soc-ph↗

arXiv2023

Analytical solution for the long- and short-range every-pair-interactions model

Many physical, biological, and social systems exhibit emergent properties that arise from the interactions between their components (cells). In this study, we systematically treat every-pair interactions (a) that exhibit power-law dependence on the Euclidean distance and (b) act in structures that can be characterized using fractal geometry. We analytically derive the mean interaction field of the cells and find that (i) in a long-range interaction regime, the mean interaction field increases following a power law with the size of the system, (ii) in a short-range interaction regime, the field saturates, and (iii) in the intermediate range it follows a logarithmic behaviour. To validate our analytical solution, we perform numerical simulations. In the case of short-range interactions, we observe that discreteness significantly impacts the continuum approximation used in the derivation, leading to incorrect asymptotic behaviour in this regime. To address this issue, we propose an expansion that substantially improves the accuracy of the analytical expression. Furthermore, our results motivate us to explore a framework for estimating the fractal dimension of unknown structures. This approach offers an alternative to established methods such as box-counting or sandbox methods. Overall, we believe that our analytical work will have broad applicability in systems where every-pair interactions play a crucial role. The insights gained from this study can contribute to a better understanding of various complex systems and facilitate more accurate modelling and analysis in a wide range of disciplines.

nlin.AO↗

arXiv2022

Employment and Land Use in the United States: cross-sectional and short-term Trends reveal the Importance of the 96th Meridian

A pervasive trend in economic development is the shift from agricultural to manufacturing and finally to service economies. Because the type of employment associated with each sector influences natural resource use this global trend may manifest itself in predictable land use transitions. We relate these to changes in urban, agricultural, and natural lands. We find that the economic transition is common in most counties and across counties. Agricultural land expands to natural areas but is eventually replaced by urban. Relating sectors to land cover, we see a strong relationship between increases in service employment and the growth of urban areas. However, we find that both large agricultural areas and large natural areas also occur with high service employment. Finally, we test if the 100th Meridian is associated with predictable relationships between land cover and economic sectors and find strong results that suggest different development patterns in the East and West.

physics.soc-ph↗

arXiv2021

Mathematical models to explain the origin of urban scaling laws: a synthetic review

The quest for a theory of cities that could offer a quantitative and systematic approach to manage cities is at the top priority, given the challenges humanity faces due to the increasing urbanization and densification of cities. If such a theory is feasible, then its formulation must be in a mathematical way. As a contribution to organizing the mathematical ideas that deal with such a systematic way of understanding urban phenomena, we present this material, concentrating on one important aspect of what recently has been called the new science of cities. In this paper, we review the main mathematical models present in the literature that aim at explaining the origin and emergence of urban scaling. We intend to present the models, identify similarities and connections between them, and find situations in which different models lead to the same output. In addition, we report situations in which some ideas initially introduced in a particular model can also be introduced in another model, generating more diversification and increasing the scope of the models. The models treated in this paper explain urban scaling from different premises: from gravity ideas, passing through densification ideas and cites' geometry, to a hierarchical organization and socio-network properties. We also investigate scenarios in which these different fundamental ideas could be interpreted as similar -- where the similarity is likely but not obvious. Furthermore, in what concerns the gravity idea, we propose a general framework that includes all gravity models analyzed as a particular case.

physics.soc-ph↗

arXiv2021

Commuting network effect on urban wealth scaling

Urban scaling theory explains the increasing returns to scale of urban wealth indicators by the per capita increase of human interactions within cities. This explanation implicitly assumes urban areas as isolated entities and ignores their interactions. Here we investigate the effects of commuting networks on the gross domestic product (GDP) of urban areas in the US and Brazil. We describe the urban GDP as the output of a production process where population, incoming commuters, and interactions between these quantities are the input variables. This approach significantly refines the description of urban GDP and shows that incoming commuters contribute to wealth creation in urban areas. Our research indicates that changes in urban GDP related to proportionate changes in population and incoming commuters depend on the initial values of these quantities, such that increasing returns to scale are only possible when the product between population and incoming commuters exceeds a well-defined threshold.

physics.soc-ph↗

arXiv2021

Zipf's law and urban scaling: Hypotheses towards a Unified Urban Theory

We propose hypotheses describing the empirical finding of an association between the exponents of urban GDP scaling and Zipf's law for cities. These hypotheses represent various combinations of directional or reciprocal causal links between the two phenomena and include inter- and intra-city processes. Future theories and models can be motivated with and categorized according to these hypotheses. This paper intends to stimulate the discussion around the processes behind these phenomena and pave the way to a Unified Urban Theory.

physics.soc-ph↗

arXiv2021

Association between population distribution and urban GDP scaling

Urban scaling and Zipf's law are two fundamental paradigms for the science of cities. These laws have mostly been investigated independently and are often perceived as disassociated matters. Here we present a large scale investigation about the connection between these two laws using population and GDP data from almost five thousand consistently-defined cities in 96 countries. We empirically demonstrate that both laws are tied to each other and derive an expression relating the urban scaling and Zipf exponents. This expression captures the average tendency of the empirical relation between both exponents, and simulations yield very similar results to the real data after accounting for random variations. We find that while the vast majority of countries exhibit increasing returns to scale of urban GDP, this effect is less pronounced in countries with fewer small cities and more metropolises (small Zipf exponent) than in countries with a more uneven number of small and large cities (large Zipf exponent). Our research puts forward the idea that urban scaling does not solely emerge from intra-city processes, as population distribution and scaling of urban GDP are correlated to each other.

physics.soc-ph↗

arXiv2020

Sea-level rise and continuous adaptation: residual damage rises faster than the protection costs

Damage cost curves -- relating the typical damage of a natural hazard to its physical magnitude -- represent an indispensable ingredient necessary for climate change impact assessments. Combining such curves with the occurrence probability of the considered natural hazard, expected damage and related risk can be estimated. Here we study recently published city scale damage cost curves for coastal flooding and demonstrate which insights can be gained from the functions only. Therefore, we include protection cost curves -- relating the typical investment costs necessary to protect a city against a natural hazard of certain magnitude -- which are analogous to and consistent with the above mentioned damage cost curves. Specifically, we motivate log-logistic functions, which exhibit a power-law increase at the lower end, and fit them to the cost curves. As expected, cities with large maximum potential loss (typically large cities) are also more costly to protect. Moreover, we study the idealized case of continuous adaptation, i.e.\ increasing protection levels in the same pace as sea-level rise, and compare the associated costs with residual damage from extreme events exceeding the protection. Based on the fitted exponents we find that in almost all cities the residual damage rises faster than the protection costs. Raising coastal protection can lead to lull oneself in a deceptive safety.

physics.soc-ph↗

arXiv2019

Effects of changing population or density on urban carbon dioxide emissions

The question of whether urbanization contributes to increasing carbon dioxide emissions has been mainly investigated via scaling relationships with population or population density. However, these approaches overlook the correlations between population and area, and ignore possible interactions between these quantities. Here, we propose a generalized framework that simultaneously considers the effects of population and area along with possible interactions between these urban metrics. Our results significantly improve the description of emissions and reveal the coupled role between population and density on emissions. These models show that variations in emissions associated with proportionate changes in population or density may not only depend on the magnitude of these changes but also on the initial values of these quantities. For US areas, the larger the city, the higher is the impact of changing its population or density on its emissions; but population changes always have a greater effect on emissions than population density.

physics.soc-ph↗

arXiv2018

A Gini approach to spatial CO2 emissions

Combining global gridded population and fossil fuel based CO2 emission data at 1km scale, we investigate the spatial origin of CO2 emissions in relation to the population distribution within countries. We depict the correlations between these two datasets by a quasi-Lorenz curve which enables us to discern the individual contributions of densely and sparsely populated regions to the national CO2 emissions. We observe pronounced country-specific characteristics and quantify them using an indicator resembling the Gini-index. Relating these indices with the degree of socio-economic development, we find that in developing countries locations with large population tend to emit relatively more CO2 and in developed countries the opposite tends to be the case. Based on the relation to urban scaling we discuss the connection with CO2 emissions from cities. Our results show that general statements with regard to the (in)efficiency of large cities should be avoided as it is subject to the socio-economic development of respective countries. Concerning the political relevance, our results suggest a differentiated spatial prioritization in deploying climate change mitigation measures in cities for developed and developing countries.

physics.soc-ph↗

arXiv2018

The efficient, the intensive, and the productive: insights from urban Kaya scaling

Urban areas play an unprecedented role in potentially mitigating climate change and supporting sustainable development. In light of the rapid urbanisation in many parts on the globe, it is crucial to understand the relationship between settlement size and CO2 emission efficiency of cities. Recent literature on urban scaling properties of emissions as a function of population size have led to contradictory results and more importantly, lacked an in-depth investigation of the essential factors and causes explaining such scaling properties. Therefore, in analogy to the well-established Kaya Identity, we develop a relation combining the involved exponents. We demonstrate that application of this Urban Kaya Relation will enable a comprehensive understanding about the intrinsic factors determining emission efficiencies in large cities by applying it to a global dataset of 61 cities. Contrary to traditional urban scaling studies which use Ordinary Least Squares (OLS) regression, we show that the Reduced Major Axis (RMA) is necessary when complex relations among scaling exponents are to be investigated. RMA is given by the geometric mean of the two OLS slopes obtained by interchanging the dependent and independent variable. We discuss the potential of the Urban Kaya Relation in main-streaming local actions for climate change mitigation.

physics.soc-ph↗

arXiv2017

On the relation between UHI intensity and city proximity

Recently, D. Zhou et al. 2015 studied empirically the land surface temperature of 32 Chinese cities and reported an exponentially decaying residual effect of the urban heat island (UHI) around the cities. Here we show analytically that such a form is equivalent to a previously proposed two-dimensional Gaussian function. The reason for this seeming contradiction is the way how the distance from the considered city is defined. While in the former, consecutive equal area belts around the city are used, in the latter it is the euclidean distance. In simplified terms, the definition of the belts implies a transformation of the independent variable $ρ\sim d^α$ with $α\approx 1/2$, where $ρ$ is the euclidean distance from the center and $d$ is the index of the belt. Since the belts have equal area, outer ones become more narrow. As a consequence, the Gaussian function $\sim \mathrm{e}^{-ρ^2}$ becomes an exponential cone $\sim \mathrm{e}^{-d}$. This paper provides an explicit derivation of the equivalence.

physics.soc-ph↗

arXiv2016

Relating urban scaling, fundamental allometry, and density scaling

We study the connection between urban scaling, fundamental allometry (between city population and city area), and per capita vs.\ population density scaling. From simple analytical derivations we obtain the relation between the 3 involved exponents. We discuss particular cases and ranges of the exponents which we illustrate in a "phase diagram". As we show, the results are consistent with previous work.

physics.soc-ph↗

arXiv2016

The Area and Population of Cities: New Insights from a Different Perspective on Cities

The distribution of the population of cities has attracted a great deal of attention, in part because it sharply constrains models of local growth. However, to this day, there is no consensus on the distribution below the very upper tail, because available data need to rely on the "legal" rather than "economic" definition of cities for medium and small cities. To remedy this difficulty, in this work we construct cities "from the bottom up" by clustering populated areas obtained from high-resolution data. This method allows us to investigate the population and area of cities for urban agglomerations of all sizes using clustering methods from percolation theory. We find that Zipf's law (a power law with exponent close to 1) for population holds for cities as small as 12,000 inhabitants in the USA and 5,000 inhabitants in Great Britain. In addition the distribution of city areas is also close to a Zipf's law. We provide a parsimonious model with endogenous city area that is consistent with those findings.

physics.soc-ph↗