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Register a new userOn the relation between Transversal and Longitudinal Scaling in Cities
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Given that a group of cities follows a scaling law connecting urban population with socio-economic or infrastructural metrics (transversal scaling), should we expect that each city would follow the same behavior over time (longitudinal scaling)? This assumption has important policy implications, although rigorous empirical tests have been so far hindered by the lack of suitable data. Here, we advance the debate by looking into the temporal evolution of the scaling laws for 5507 municipalities in Brazil. We focus on the relationship between population size and two urban variables, GDP and water network length, analyzing the time evolution of the system of cities as well as their individual trajectory. We find that longitudinal (individual) scaling exponents are city-specific, but they are distributed around an average value that approaches to the transversal scaling exponent when the data are decomposed to eliminate external factors, and when we only consider cities with a sufficiently large growth rate. Such results give support to the idea that the longitudinal dynamics is a micro-scaling version of the transversal dynamics of the entire urban system. Finally, we propose a mathematical framework that connects the microscopic level to global behavior, and, in all analyzed cases, we find good agreement between theoretical prediction and empirical evidence.
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Cities can be characterised and modelled through different urban measures. Consistency within these observables is crucial in order to advance towards a science of cities. Bettencourt et al have proposed that many of these urban measures can be predicted through universal scaling laws. We develop a framework to consistently define cities, using commuting to work and population density thresholds, and construct thousands of realisations of systems of cities with different boundaries for England and Wales. These serve as a laboratory for the scaling analysis of a large set of urban indicators. The analysis shows that population size alone does not provide enough information to describe or predict the state of a city as previously proposed, indicating that the expected scaling laws are not corroborated. We found that most urban indicators scale linearly with city size regardless of the definition of the urban boundaries. However, when non-linear correlations are present, the exponent fluctuates considerably.
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.
Cities are centers for the integration of capital and incubators of invention, and attracting venture capital (VC) is of great importance for cities to advance in innovative technology and business models towards a sustainable and prosperous future. Yet we still lack a quantitative understanding of the relationship between urban characteristics and VC activities. In this paper, we find a clear nonlinear scaling relationship between VC activities and the urban population of Chinese cities. In such nonlinear systems, the widely applied linear per capita indicators would be either biased to larger cities or smaller cities depends on whether it is superlinear or sublinear, while the residual of cities relative to the prediction of scaling law is a more objective and scale-invariant metric. %(i.e., independent of the city size). Such a metric can distinguish the effects of local dynamics and scaled growth induced by the change of population size. The spatiotemporal evolution of such metrics on VC activities reveals three distinct groups of cities, two of which stand out with increasing and decreasing trends, respectively. And the taxonomy results together with spatial analysis also signify different development modes between large urban agglomeration regions. Besides, we notice the evolution of scaling exponents on VC activities are of much larger fluctuations than on socioeconomic output of cities, and a conceptual model that focuses on the growth dynamics of different sized cities can well explain it, which we assume would be general to other scenarios.
Numerous urban indicators scale with population in a power law across cities, but whether the cross-sectional scaling law is applicable to the temporal growth of individual cities is unclear. Here we first find two paradoxical scaling relationships that urban built-up area sub-linearly scales with population across cities, but super-linearly scales with population over time in most individual cities because urban land expands faster than population grows. Different cities have diverse temporal scaling exponents and one city even has opposite temporal scaling regimes during two periods, strongly supporting the absence of single temporal scaling and further illustrating the failure of cross-sectional urban scaling in predicting temporal growth of cities. We propose a conceptual model that can clarify the essential difference and also connections between the cross-sectional scaling law and temporal trajectories of cities. Our model shows that cities have an extra growth of built-up area over time besides the supposed growth predicted by the cross-sectional scaling law. Disparities of extra growth among different-sized cities change the cross-sectional scaling exponent. Further analyses of GDP and other indicators confirm the contradiction between cross-sectional and temporal scaling relationships and the validity of the conceptual model. Our findings may open a new avenue towards the science of cities.
The anomalous Hall effect (AHE) has been studied systematically in the low-conductivity ferromagnetic oxide Fe$_{3-x}$Zn$_x$O$_4$ with $x = 0$, 0.1, and 0.5. We used (001), (110), and (111) oriented epitaxial Fe$_{3-x}$Zn$_x$O$_4$ films grown on MgO and sapphire substrates in different oxygen partial pressure to analyze the dependence of the AHE on crystallographic orientation, Zn content, strain state, and oxygen deficiency. Despite substantial differences in the magnetic properties and magnitudes of the anomalous Hall conductivity $sigma_{xy}^{rm AHE}$ and the longitudinal conductivity $sigma_{xx}$ over several orders of magnitude, a universal scaling relation $sigma_{xy}^{rm AHE} propto sigma_{xx}^{alpha}$ with $alpha = 1.69 pm 0.08$ was found for all investigated samples. Our results are in agreement with recent theoretical and experimental findings for ferromagnetic metals in the dirty limit, where transport is by metallic conduction. We find the same scaling relation for magnetite, where hopping transport prevails. The fact that this relation is independent of crystallographic orientation, Zn content, strain state, and oxygen deficiency suggests that it is universal and particularly does not depend on the nature of the transport mechanism.
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