Superlinear and sublinear urban scaling in geographical networks modeling cities

Superlinear urban scaling by functional organization: A metabolic interpretation of sectoral water consumption

Prevailing view asserts that the disproportionately greater productivities of larger cities, or superlinear urban scaling, are the result of human interactions channeled by urban networks. But this view was established by considering the spatial organization of urban infrastructure and social networks-the urban "arteries" effects-but neglecting the functional organization of urban production and consumption entities-the urban "organs" effects. Here, adopting a metabolic view and using water consumption as a proxy for metabolism, we empirically quantify the scaling of entity number, size, and metabolic rate for the functionally specific urban residential, commercial, public or institutional, and industrial sectors. Sectoral urban metabolic scaling is highlighted by a disproportionate coordination between residential and enterprise metabolic rates, attributable to the functional mechanisms of mutualism, specialization, and entity size effect. The resultant whole-city metabolic scaling exhibits a constant superlinear exponent for water-abundant regions in numerical agreement with superlinear urban productivity, with varying exponent deviations for water-deficient regions explainable as adaptations to climate-driven resource constraints. These results present a functional organizational, non-social-network explanation of superlinear urban scaling.

Scaling of Energy, Water, and Waste Flows in China's Prefecture-Level and Provincial Cities

Cities have been envisioned as biological organisms as the integral part of nature's energy and material flows. Recent advances in urban scaling research have uncovered systematic changes in socioeconomic rates and infrastructural networks as urban population increases, providing predictive contents for the comparison between cities and organisms. However, it is still unclear how and why larger and smaller cities may differ in their per capita environmental impacts. Here, we study scaling patterns of urban energy, water, and waste flows as well as other relevant measures in Chinese cities. We divide cities into different groups using an algorithm that automatically assigns cities to clusters with distinct scaling patterns. Despite superlinear scaling of urban GDP, as predicted by urban scaling theories, resource consumption, such as the supply of electricity and water, and waste generation, such as wastewater and domestic waste, do not show significant deviations from linear scaling. The lengths of resource pipelines scale linearly in most cases, as opposed to sub-linearity predicted by theory. Furthermore, we show two competing forces underlying the overall observed effects of scale: a higher population density tends to decrease per capita resource consumption and infrastructure provisions, while intensified socioeconomic activities have the opposite effect.

Data-driven stochastic simulation leading to the allometric scaling laws in complex systems

We propose a data-driven stochastic method that allows the simulation of a complex system's long-term evolution. Given a large amount of historical data on trajectories in a multi-dimensional phase space, our method simulates the time evolution of a system based on a random selection of partial trajectories in the data without detailed knowledge of the system dynamics. We apply this method to a large data set of time evolution of approximately one million business firms for a quarter century. Accordingly, from simulations starting from arbitrary initial conditions, we obtain a stationary distribution in three-dimensional log-size phase space, which satisfies the allometric scaling laws of three variables. Furthermore, universal distributions of fluctuation around the scaling relations are consistent with the empirical data.

Rediscovering the Scaling Law of Urban Land from a Multi-Scale Perspective—A Case Study of Wuhan

The law of urban scaling implies that there is a universally applicable nonlinear scaling relationship between population size and urban indicators, which is a method of quantitative analysis that can reflect the growth law and internal logic of the urban system. However, most present research is conducted at the municipal scale, and studies of scaling law in the inner-city system are scarce, especially from the perspective of compact urban form development. The goal of this paper is to discover the scaling law within urban systems from a multi-scale perspective. Through the empirical analysis of Wuhan, this paper examines the internal scale law of the urban system from the municipal and district scales. Moreover, we use the landscape expansion index to perform spatial autocorrelation analysis. In this way, we assess the relationship between the compactness of urban morphological development and the urban scaling law. The results indicate that the temporal scaling law on the city scale has a more significant linear law than the single-year scaling law. The analysis also shows the scaling law relationship within the inner-city system. Nevertheless, there is a deviation between the temporal scaling law and the cross-section scaling law. Namely, the time series development of a district does not follow the section scaling law of the urban system. Furthermore, the urban scaling law shows a negative correlation with the compactness of the urban form development. It is crucial to understand the current economic development and resource endowment of an urban system in the urbanization process, as it significantly contributes to urban development and regional coordinated planning.

Comment on Bettignies et al. The Scale-Dependent Behaviour of Cities: A Cross-Cities Multiscale Driver Analysis of Urban Energy Use. Sustainability 2019, 11, 3246

Bettignies et al. examine power-law relationships between drivers of energy use and urban features at city and infra-city levels for ten different cities in six countries across four continents, featuring a wide distribution of urban indicators from various data sources. The authors employ univariate linear regression models using selected log-transformed indicators to investigate whether the intensity of energy use scales with urban indicators such as population size, density, and income. Bettignies et al. suggest that based on their findings, the urban energy-use drivers are in fact scale-dependent, and that their results reveal a substantial heterogeneity across and within cities. They reference this as why more consideration needs to be paid to local factors when devising urban policy. On this note, we argue that Bettignies et al. appear to have not only misunderstood the urban scaling literature they have cited, but have also employed flawed methodological design in their analysis that ultimately leaves their conclusions unsubstantiated.

Estimating the drivers of urban economic complexity and their connection to economic performance

Estimating the capabilities, or inputs of production, that drive and constrain the economic development of urban areas has remained a challenging goal. We posit that capabilities are instantiated in the complexity and sophistication of urban activities, the know-how of individual workers, and the city-wide collective know-how. We derive a model that indicates how the value of these three quantities can be inferred from the probability that an individual in a city is employed in a given urban activity. We illustrate how to estimate empirically these variables using data on employment across industries and metropolitan statistical areas in the USA. We then show how the functional form of the probability function derived from our theory is statistically superior when compared with competing alternative models, and that it explains well-known results in the urban scaling and economic complexity literature. Finally, we show how the quantities are associated with metrics of economic performance, suggesting our theory can provide testable implications for why some cities are more prosperous than others.

How the geometry of cities determines urban scaling laws

Urban scaling laws relate socio-economic, behavioural and physical variables to the population size of cities. They allow for a new paradigm of city planning and for an understanding of urban resilience and economics. The emergence of these power-law relations is still unclear. Improving our understanding of their origin will help us to better apply them in practical applications and further research their properties. In this work, we derive the basic exponents for spatially distributed variables from fundamental fractal geometric relations in cities. Sub-linear scaling arises as the ratio of the fractal dimension of the road network and of the distribution of the population embedded in three dimensions. Super-linear scaling emerges from human interactions that are constrained by the geometry of a city. We demonstrate the validity of the framework with data from 4750 European cities. We make several testable predictions, including the relation of average height of cities and population size, and the existence of a critical density above which growth changes from horizontal densification to three-dimensional growth.

Understanding the mesoscopic scaling patterns within cities

Understanding quantitative relationships between urban elements is crucial for a wide range of applications. The observation at the macroscopic level demonstrates that the aggregated urban quantities (e.g., gross domestic product) scale systematically with population sizes across cities, also known as urban scaling laws. However, at the mesoscopic level, we lack an understanding of whether the simple scaling relationship holds within cities, which is a fundamental question regarding the spatial origin of scaling in urban systems. Here, by analyzing four extensive datasets covering millions of mobile phone users and urban facilities, we investigate the scaling phenomena within cities. We find that the mesoscopic infrastructure volume and socioeconomic activity scale sub- and super-linearly with the active population, respectively. For a same scaling phenomenon, however, the exponents vary in cities of similar population sizes. To explain these empirical observations, we propose a conceptual framework by considering the heterogeneous distributions of population and facilities, and the spatial interactions between them. Analytical and numerical results suggest that, despite the large number of complexities that influence urban activities, the simple interaction rules can effectively explain the observed regularity and heterogeneity in scaling behaviors within cities.

Scaling in words on Twitter

Scaling properties of language are a useful tool for understanding generative processes in texts. We investigate the scaling relations in citywise Twitter corpora coming from the metropolitan and micropolitan statistical areas of the United States. We observe a slightly superlinear urban scaling with the city population for the total volume of the tweets and words created in a city. We then find that a certain core vocabulary follows the scaling relationship of that of the bulk text, but most words are sensitive to city size, exhibiting a super- or a sublinear urban scaling. For both regimes, we can offer a plausible explanation based on the meaning of the words. We also show that the parameters for Zipf's Law and Heaps' Law differ on Twitter from that of other texts, and that the exponent of Zipf's Law changes with city size.

The Statistical Analysis of Top Hubs in Growing Geographical Networks with Optimal Policy

Many practical networks, such as city networks, road networks and neural networks, usually grow up on basis of topological structures and geographical measures. Big hubs, importance of which have been well known in complex networks, still play crucial roles in growing networks with geographical measures. Therefore, it is very necessary to investigate the underlying mechanisms of statistical features of different top hubs in such networks. Here, we propose a growing network model based on optimal policy in geographical ground. Through the statistics of a great number of geographical networks, we find that the degree and position distributions of top four hubs are diverse between them and closely interrelated with each other, and further gain the relationships between the upper limits of top hubs and the size of networks. Then, the underlying mechanisms are explored. Meanwhile, we are diligent to obtain the corresponding relationships of different spatial distribution areas for different top hubs, and compute their abnormal average degrees at different spatial positions, which show significant differences and imply the advantage of spatial positions and intense competition between top hubs. We hope our results could offer useful inspirations for related practical network studies.

The Scale-Dependent Behaviour of Cities: A Cross-Cities Multiscale Driver Analysis of Urban Energy Use

Hosting more than half of the world population, cities are currently responsible for two thirds of the global energy use and three quarters of the global CO2 emissions related to energy use. As humanity becomes more urbanized, urban systems are becoming a major nexus of global sustainability. Various studies have tried to pinpoint urban energy use drivers in order to find actionable levers to mitigate consumption and its associated environmental effects. Some of the approaches, mainly coming from complexity science and industrial ecology disciplines, use city-scale data to find power-laws relating to different types of energy use metrics with urban features at a city-scale. By doing so, cities’ internal complexity and heterogeneity are not explicitly addressed. Moreover, to our knowledge, no studies have yet explicitly addressed the potential scale dependency of such drivers. Drivers might not be transferable to other scales and yield undesired effects. In the present study, power-law relations are examined for 10 cities worldwide at city scale and infra-city scale, and the results are compared across scales. Relations are made across three urban features for three energy use intensity metrics. The results show that energy use drivers are in fact scale-dependent and are city-dependent for intra-urban territories.

Urban scaling and the regional divide

Superlinear growth in cities has been explained as an emergent consequence of increased social interactions in dense urban environments. Using geocoded microdata from Swedish population registers, we remove population composition effects from the scaling relation of wage income to test how much of the previously reported superlinear scaling is truly attributable to increased social interconnectivity in cities. The Swedish data confirm the previously reported scaling relations on the aggregate level, but they provide better information on the micromechanisms responsible for them. We find that the standard interpretation of urban scaling is incomplete as social interactions only explain about half of the scaling parameter of wage income and that scaling relations substantively reflect differences in cities' sociodemographic composition. Those differences are generated by selective migration of highly productive individuals into larger cities. Big cities grow through their attraction of talent from their hinterlands and the already-privileged benefit disproportionally from urban agglomeration.

Scaling tunable network model to reproduce the density-driven superlinear relation

Previous works have shown the universality of allometric scaling under total and density values at the city level, but our understanding of the size effects of regions on the universality of allometric scaling remains inadequate. Here, we revisit the scaling relations between the gross domestic production (GDP) and the population based on the total and density values and first reveal that the allometric scaling under density values for different regions is universal. The scaling exponent β under the density value is in the range of (1.0, 2.0], which unexpectedly exceeds the range observed by Pan et al. [Nat. Commun. 4, 1961 (2013)]. For the wider range, we propose a network model based on a 2D lattice space with the spatial correlation factor α as a parameter. Numerical experiments prove that the generated scaling exponent β in our model is fully tunable by the spatial correlation factor α. Our model will furnish a general platform for extensive urban and regional studies.

The scaling of crime concentration in cities

Crime is a major threat to society’s well-being but lacks a statistical characterization that could lead to uncovering some of its underlying mechanisms. Evidence of nonlinear scaling of urban indicators in cities, such as wages and serious crime, has motivated the understanding of cities as complex systems—a perspective that offers insights into resources limits and sustainability, but that usually neglects details of the indicators themselves. Notably, since the nineteenth century, criminal activities have been known to occur unevenly within a city; crime concentrates in such way that most of the offenses take place in few regions of the city. Though confirmed by different studies, this concentration lacks broad analyses on its characteristics, which hinders not only the comprehension of crime dynamics but also the proposal of sounding counter-measures. Here, we developed a framework to characterize crime concentration which divides cities into regions with the same population size. We used disaggregated criminal data from 25 locations in the U.S. and the U.K., spanning from 2 to 15 years of longitudinal data. Our results confirmed that crime concentrates regardless of city and revealed that the level of concentration does not scale with city size. We found that the distribution of crime in a city can be approximated by a power-law distribution with exponent α that depends on the type of crime. In particular, our results showed that thefts tend to concentrate more than robberies, and robberies more than burglaries. Though criminal activities present regularities of concentration, we found that criminal ranks have the tendency to change continuously over time—features that support the perspective of crime as a complex system and demand analyses and evolving urban policies covering the city as a whole.

Is this scaling nonlinear?

One of the most celebrated findings in complex systems in the last decade is that different indexes y (e.g. patents) scale nonlinearly with the population x of the cities in which they appear, i.e. y∼x (β) ,β≠1. More recently, the generality of this finding has been questioned in studies that used new databases and different definitions of city boundaries. In this paper, we investigate the existence of nonlinear scaling, using a probabilistic framework in which fluctuations are accounted for explicitly. In particular, we show that this allows not only to (i) estimate β and confidence intervals, but also to (ii) quantify the evidence in favour of β≠1 and (iii) test the hypothesis that the observations are compatible with the nonlinear scaling. We employ this framework to compare five different models to 15 different datasets and we find that the answers to points (i)-(iii) crucially depend on the fluctuations contained in the data, on how they are modelled, and on the fact that the city sizes are heavy-tailed distributed.