Zipf's law is one the most conspicuous empirical facts for cities, however, there is no convincing explanation for the scaling relation between rank and size and its scaling exponent. Based on the idea from general fractals and scaling, this paper proposes a dual competition hypothesis of city develop to explain the value intervals and the special value, 1, of the power exponent. Zipf's law and Pareto's law can be mathematically transformed into one another. Based on the Pareto distribution, a frequency correlation function can be constructed. By scaling analysis and multifractals spectrum, the parameter interval of Pareto exponent is derived as (0.5, 1]; Based on the Zipf distribution, a size correlation function can be built, and it is opposite to the first one. By the second correlation function and multifractals notion, the Pareto exponent interval is derived as [1, 2). Thus the process of urban evolution falls into two effects: one is Pareto effect indicating city number increase (external complexity), and the other Zipf effect indicating city size growth (internal complexity). Because of struggle of the two effects, the scaling exponent varies from 0.5 to 2; but if the two effects reach equilibrium with each other, the scaling exponent approaches 1. A series of mathematical experiments on hierarchical correlation are employed to verify the models and a conclusion can be drawn that if cities in a given region follow Zipf's law, the frequency and size correlations will follow the scaling law. This theory can be generalized to interpret the inverse power-law distributions in various fields of physical and social sciences.
Urban population density always follows the exponential distribution and can be described with Clark's model. Because of this, the spatial distribution of urban population used to be regarded as non-fractal pattern. However, Clark's model differs from the exponential function in mathematics because that urban population is distributed on the fractal support of landform and land-use form. By using mathematical transform and empirical evidence, we argue that there are self-affine scaling relations and local power laws behind the exponential distribution of urban density. The scale parameter of C…
The conventional mathematical methods are based on characteristic length, while urban form has no characteristic length in many aspects. Urban area is a measure of scale dependence, which indicates the scale-free distribution of urban patterns. Thus, the urban description based on characteristic lengths should be replaced by urban characterization based on scaling. Fractal geometry is one powerful tool for scaling analysis of cities. Fractal parameters can be defined by entropy and correlation functions. However, how to understand city fractals is still a pending question. By means of logic de…
The scaling exponent of a hierarchy of cities used to be regarded as a fractal parameter. The Pareto exponent was treated as the fractal dimension of size distribution of cities, while the Zipf exponent was treated as the reciprocal of the fractal dimension. However, this viewpoint is not exact. In this paper, I will present a new interpretation of the scaling exponent of rank-size distributions. The ideas from fractal measure relation and the principle of dimension consistency are employed to explore the essence of Pareto's and Zipf's scaling exponents. The Pareto exponent proved to be a rati…
Fractal dimension curves of urban growth can be modeled with sigmoid functions, including logistic function and quadratic logistic function. Different types of logistic functions indicate different spatial dynamics. The fractal dimension curves of urban growth in western countries follows the common logistic function, while these curves of cities in northern China follows quadratic logistic function. Now we want to know whether all Chinese cities follow the same rules of urban evolution. This paper is devoted to exploring the fractals and fractal dimension properties of the city of Shenzhen in…
The spatial distributions of cities fall into two groups: one is the simple distribution with characteristic scale (e.g. exponential distribution), and the other is the complex distribution without characteristic scale (e.g. power-law distribution). The latter belongs to scale-free distributions, which can be modeled with fractal geometry. However, fractal dimension is not suitable for the former distribution. In contrast, spatial entropy can be used to measure any types of urban distributions. This paper is devoted to generalizing multifractal parameters by means of dual relation between Eucl…
Urban form and growth can be described with fractal dimension, which is a measurement of space filling of urban evolution. Based on empirical analyses, a discovery is made that the time series of fractal dimension of urban form can be treated as a sigmoid function of time. Among various sigmoid functions, the logistic function is the most probable selection. However, how to use the model of fractal dimension growth to explain and predict urban growth is a pending problem remaining to be solved. This paper is devoted to modeling fractal dimension evolution of different types of cities. A intere…
Urban agglomerations exhibit complex emergent features of which Zipf's law, i.e.\ a power-law size distribution, and fractality may be regarded as the most prominent ones. We propose a simplistic model for the generation of city-like structures which is solely based on the assumption that growth is more likely to take place close to inhabited space. The model involves one parameter which is an exponent determining how strongly the attraction decays with the distance. In addition, the model is run iteratively so that existing clusters can grow (together) and new ones can emerge. The model is ca…
Fractal structure of a system suggests the optimal way in which parts arranged or put together to form a whole. The ideas from fractals have a potential application to the researches on urban sustainable development. To characterize fractal cities, we need the measure of fractional dimension. However, if the fractal organization is concealed in the complex spatial distributions of geographical phenomena, the common methods of evaluating fractal parameter will be disabled. In this article, a new model is proposed to describe urban density and estimate fractal dimension of urban form. If urban d…
Urban form has been empirically demonstrated to be of scaling invariance and can be described with fractal geometry. However, the rational range of fractal dimension value and the relationships between various fractal indicators of cities are not yet revealed in theory. By mathematical deduction and transformation (e.g. Fourier transform), I find that scaling analysis, spectral analysis, and spatial correlation analysis are all associated with fractal concepts and can be integrated into a new approach to fractal analysis of cities. This method can be termed 3S analyses of urban form. Using the…
Just as natural river networks are known to be globally self-similar, recent research has shown that human-built urban networks, such as road networks, are also functionally self-similar, and have fractal topology with power-law node-degree distributions (p(k) = a k). Here we show, for the first time, that other urban infrastructure networks (sanitary and storm-water sewers), which sustain flows of critical services for urban citizens, also show scale-free functional topologies. For roads and drainage networks, we compared functional topological metrics, derived from high-resolution data (70,0…
The things surrounding us vary dramatically, which implies that there are far more small things than large ones, e.g., far more small cities than large ones in the world. This dramatic variation is often referred to as fractal or scaling. To better reveal the fractal or scaling structure, a new classification scheme, namely head/tail breaks, has been developed to recursively derive different classes or hierarchical levels. The head/tail breaks works as such: divide things into a few large ones in the head (those above the average) and many small ones (those below the average) in the tail, and …
The improved city clustering algorithm can be used to identify urban boundaries on a digital map, and the results are a set of isolines. The relationships between the urban measurements within the variable boundaries follow allometric scaling law, which indicates spatial allometry of cities. This paper is devoted to exploring the fractal dimension proceeding from urban spatial allometry. By theoretical reasoning and empirical analysis of urban traffic network, we can derive a set of fractal dimension from the spatial allometry and reveal the basic property of the new fractal parameters. The fi…
Urban scaling laws relate socio-economic, behavioral, and physical variables to the population size of cities and allow for a new paradigm of city planning, and an understanding of urban resilience and economies. Independently of culture and climate, almost all cities exhibit two fundamental scaling exponents, one sub-linear and one super-linear that are related. Here we show that based on fundamental fractal geometric relations of cities we derive both exponents and their relation. Sub-linear scaling arises as the ratio of the fractal dimensions of the road network and the distribution of the…
Unveiling the relationships between crime and socioeconomic factors is crucial for modeling and preventing these illegal activities. Recently, a significant advance has been made in understanding the influence of urban metrics on the levels of crime in different urban systems. In this chapter, we show how the dynamics of crime growth rate and the number of crime in cities are related to cities' size. We also discuss the role of urban metrics in crime modeling within the framework of the urban scaling hypothesis, where a data-driven approach is proposed for modeling crime. This model provides s…
This paper proposes a new method to analyze the spatial structure of urban systems using ideas from fractals. Regarding a system of cities as a set of "particles" distributed randomly on a triangular lattice, we construct a spatial correlation function of cities. Suppose that the spatial correlation follows the power law. It can be proved that the correlation exponent is the second order generalized dimension. The spatial correlation model is applied to the system of cities in China. The results show that the Chinese urban system can be described by the correlation dimension ranging from 1.3 t…
Entropy is one of physical bases for fractal dimension definition, and the generalized fractal dimension was defined by Renyi entropy. Using fractal dimension, we can describe urban growth and form and characterize spatial complexity. A number of fractal models and measurements have been proposed for urban studies. However, the precondition for fractal dimension application is to find scaling relations in cities. In absence of scaling property, we can make use of entropy function and measurements. This paper is devoted to researching how to describe urban growth by using spatial entropy. By an…
Simulation is a powerful tool to study the behavior of physical, environmental, and social systems under different conditions. Evacuation simulation can be used to estimate the required time for people to exit a building or evacuate disaster exposed regions. While building evacuation simulation has seen significant study, city evacuation simulation is less developed. For evacuation simulations using Agent-Based Models, the characteristics of the underlying navigation algorithms are important in the overall efficiency of the simulation. In some disasters, e.g. earthquakes, evacuation takes plac…
The growth curves of fractal dimension of urban form take on squashing effect and can be described by sigmoid functions. The fractal dimension growth of urban form in western countries can be modeled by Boltzmann's equation and logistic function. However, these models cannot be well applied to the fractal dimension growth curve of Beijing city, the national capital of China. In this paper, the experimental method is employed to find parametric models for the growth curves of fractal dimension of Chinese urban form. By statistical analysis, numerical analysis, and comparative analysis, we find …
The law of allometric growth is one of basic rules for understanding urban evolution. The general form of this law is allometric scaling law. However, the deep meaning and underlying rationale of the scaling exponents remain to be brought to light. In this paper, the theories of linear algebra and regression analysis are employed to reveal the mathematical and statistic essence of allometric scaling exponents. Suppose that the geometric measure relations between a set of elements in an urban system follow the allometric growth law. An allometric scaling exponent is proved to equal in theory to…
The capability of traffic-information systems to sense the movement of millions of users and offer trip plans through mobile phones has enabled a new way of optimizing city traffic dynamics, turning transportation big data into insights and actions in a closed-loop and evaluating this approach in the real world. Existing research has applied dynamic Bayesian networks and deep neural networks to make traffic predictions from floating car data, utilized dynamic programming and simulation approaches to identify how people normally travel with dynamic traffic assignment for policy research, and in…
Traffic networks have been proved to be fractal systems. However, previous studies mainly focused on monofractal networks, while complex systems are of multifractal structure. This paper is devoted to exploring the general regularities of multifractal scaling processes in the street network of 12 Chinese cities. The city clustering algorithm is employed to identify urban boundaries for defining comparable study areas; box-counting method and the direct determination method are utilized to extract spatial data; the least squares calculation is employed to estimate the global and local multifrac…
Cities are complex systems, their complexity manifests itself through fractality of their spatial structures and by power law distributions (scaling) of multiple urban attributes. Here we report on the previously unreported manifestation of urban complexity -- scaling in patterns of residential racial segregation. A high-resolution racial grid of a city is segmented into racial enclaves which are patches of stationary racial composition. Empirical PDFs of patch areas and population counts in 41 US cities were analyzed to reveal that these variables have distributions which are either power law…
The gravity model is one of important models of social physics and human geography, but several basic theoretical and methodological problems remain to be solved. In particular, it is hard to explain and evaluate the distance exponent using the ideas from Euclidean geometry. This paper is devoted to exploring the distance-decay parameter of the urban gravity model. Based on the concepts from fractal geometry, several fractal parameter relations can be derived from scaling laws of self-similar hierarchies of cities. Results show that the distance exponent is just a scaling exponent, which equal…
Empirical evidence suggest that most urban systems experience a transition from a monocentric to a polycentric organisation 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 unstability of the monocentric regime, and that the number of subcenters and the total commuting distance within a city scale sublinearly with its population, predictions which are in agreement with data gathered for around 9000 US cities between 1994 and 2010.
Urban areas are a high-stake target of climate change mitigation and adaptation measures. To understand, predict and improve the energy performance of cities, the scientific community develops numerical models that describe how they interact with the atmosphere through heat and moisture exchanges at all scales. In this review, we present recent advances that are at the origin of last decade's revolution in computer graphics, and recent breakthroughs in statistical physics that extend well established path-integral formulations to non-linear coupled models. We argue that this rare conjunction o…
I propose and briefly define the concept of Urban Isobenefit Lines by using functions as easy as efficient, whose results can offer a rich tool to use into spatial equilibrium analysis involving cities. They are line joining urban points with equal level of positional advantage from city amenities. The results which one obtain by implementing a chosen function, gave specific scenarios: numerically described by indicators and graphically visualized by efficient city matrix views. This is also a theoretical concept for the Urban Economics theory and Spatial Equilibrium analysis in cities.
We use mobile device data to construct empirical interpersonal physical contact networks in the city of Portland, Oregon, both before and after social distancing measures were enacted during the COVID-19 pandemic. These networks reveal how social distancing measures and the public's reaction to the incipient pandemic affected the connectivity patterns within the city. We find that as the pandemic developed there was a substantial decrease in the number of individuals with many contacts. We further study the impact of these different network topologies on the spread of COVID-19 by simulating an…
We introduce a predictive algorithm for the smart growth of cities with populations upward of 100,000, allowing for extensive simulations of growth plans and their effects upon an urban populous. A smart growth metric is calculated to evaluate the progress of a city at each phase of its adaptation of the growth plan, which is measured using a weighted entropy method. The predictive algorithm itself is built from a unique differential model, which calculates the growth of a city from smart growth proposals that are individually assessed by a logistic weight model. These proposals are then sorte…
Since the very first days of announcing Covid 19 as a global pandemic, researchers around the world from different backgrounds and organizations including national governments and the World Health Organization (WHO) have extensively begun using Modeling and Simulation (M&S) techniques in deciding on the optimal strategies in order to mitigate the public health and economical effects of Covid-19. In this paper, we present our results from applying M&S on the city of Casablanca, Morocco to comprehend (i): How it can be relatively easy or difficult for such pandemic to penetrate to the studied re…
The taxi business has been overly regulated for many decades. Regulations are supposed to ensure safety and fairness within a controlled competitive environment. By influencing both drivers and riders choices and behaviors, emerging e-hailing services (e.g., Uber and Lyft) have been reshaping the existing competition in the last few years. This study investigates the existing competition between the mainstream hailing services (i.e., Yellow and Green Cabs) and e-hailing services (i.e., Uber) in New York City. Their competition is investigated in terms of market segmentation, emerging demands, …
We review and introduce a generalized reaction-diffusion approach to epidemic spreading in a metapopulation modeled as a complex network. The metapopulation consists of susceptible and infected individuals that are grouped in subpopulations symbolising cities and villages that are coupled by human travel in a transportation network. By analytic methods and numerical simulations we calculate the fraction of infected people in the metaopoluation in the long time limit, as well as the relevant parameters characterising the epidemic threshold that separates an epidemic from a non-epidemic phase. W…
