For a long time, many methods are developed to make temporal signal analyses based on time series. However, for geographical systems, spatial signal analyses are as important as temporal signal analyses. Nonstationary spatial and temporal processes are associated with nonlinearity, and cannot be effectively analyzed by conventional analytical approaches. Fractal theory provides a powerful tool for exploring complexity and is helpful for spatio-temporal signal analysis. This paper is devoted to researching spatial signals of geographical systems by means of wave-spectrum scaling. The traffic networks of 10 Chinese cities are taken as cases for positive studies. Fast Fourier transform and least squares regression analysis are employed to calculate spectral exponents. The results show that the wave-spectral density distribution of all these urban traffic networks follows scaling law, and the spectral scaling exponents can be converted to fractal dimension values. Using the fractal parameters, we can make spatial analyses for the geographical signals. The analytical process can be generalized to temporal signal analyses. The wave-spectrum scaling methods can be applied to both self-similar fractal signals and self-affine fractal signals in the geographical world.
We investigate how efficient each area of urban street network is connected. Using the detour index, geographic features of street network and pairwise efficiency is studied. To do so, the detour index of 1,832,118 travel route pairs for 85 global cities are explored. We show that the detour index in urban street network is strongly dependent on the angular separation of a pair respective to the city center and that it is a unique property introduced by the intrinsic core-periphery structure of urban street networks. Lastly, a new way of mapping the street network to visualize the overview of …
Models of street networks underlie research in urban travel behavior, accessibility, design patterns, and morphology. These models are commonly defined as planar, meaning they can be represented in two dimensions without any underpasses or overpasses. However, real-world urban street networks exist in three-dimensional space and frequently feature grade separation such as bridges and tunnels: planar simplifications can be useful but they also impact the results of real-world street network analysis. This study measures the nonplanarity of drivable and walkable street networks in the centers of…
Complex systems have been widely studied by social and natural scientists in terms of their dynamics and their structure. Scholars of cities and urban planning have incorporated complexity theories from qualitative and quantitative perspectives. From a structural standpoint, the urban form may be characterized by the morphological complexity of its circulation networks - particularly their density, resilience, centrality, and connectedness. This dissertation unpacks theories of nonlinearity and complex systems, then develops a framework for assessing the complexity of urban form and street net…
We explore the relation between urban road network characteristics particularly circuitry, street orientation entropy and the city's topography on the one hand and the building's orientation entropy on the other in order to quantify their effect on the city's solar potential. These statistical measures of the road network reveal the interplay between the built environment's design and its sustainability.
Urban density, in the form of residents' and visitors' concentration, is long considered to foster diverse exchanges of interpersonal knowledge and skills, which are intrinsic to sustainable human settlements. However, with current urban studies primarily devoted to city and district-level analysis, we cannot unveil the elemental connection between urban density and diversity. Here we use an anonymized and privacy-enhanced mobile data set of 0.5 million opted-in users from three metropolitan areas in the U.S to show that at the scale of urban streets, density is not the only path to diversity.…
We study centrality in urban street patterns of different world cities represented as networks in geographical space. The results indicate that a spatial analysis based on a set of four centrality indices allows an extended visualization and characterization of the city structure. Planned and self-organized cities clearly belong to two different universality classes. In particular, self-organized cities exhibit scale-free properties similar to those found in the degree distributions of non-spatial networks.
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…
OpenStreetMap offers a valuable source of worldwide geospatial data useful to urban researchers. This study uses the OSMnx software to automatically download and analyze 27,000 US street networks from OpenStreetMap at metropolitan, municipal, and neighborhood scales - namely, every US city and town, census urbanized area, and Zillow-defined neighborhood. It presents empirical findings on US urban form and street network characteristics, emphasizing measures relevant to graph theory, transportation, urban design, and morphology such as structure, connectedness, density, centrality, and resilien…
Street networks may be planned according to clear organizing principles or they may evolve organically through accretion, but their configurations and orientations help define a city's spatial logic and order. Measures of entropy reveal a city's streets' order and disorder. Past studies have explored individual cases of orientation and entropy, but little is known about broader patterns and trends worldwide. This study examines street network orientation, configuration, and entropy in 100 cities around the world using OpenStreetMap data and OSMnx. It measures the entropy of street bearings in …
We investigate urban street networks as a whole within the frameworks of information physics and statistical physics. Urban street networks are envisaged as evolving social systems subject to a logarithmical entropic equilibrium.
In this paper, we develop a novel approach to measuring urban sprawl based on street nodes and naturally defined urban boundaries, both extracted from massive volunteered geographic information OpenStreetMap databases through some data-intensive computing processes. The street nodes are defined as street intersections and ends, while the naturally defined urban boundaries constitute what we call natural cities. We find that the street nodes are significantly correlated with population of cities. Based on this finding, we set street nodes as a proxy of population to measure urban sprawl. We fur…
We compare the structural properties of the street networks of ten different European cities using their primal representation. We investigate the properties of the geometry of the networks and a set of centrality measures highlighting differences and similarities among cases. In particular, we found that cities share structural similarities due to their quasi planarity but that there are also several distinctive geometrical proprieties. A Principal Component Analysis is also performed on the distributions of centralities and their respective moments, which is used to find distinctive characte…
Urban morphogenesis is the process of formation of its elements and the specialization of its suburbs. Street networks are the structural part of the system. Understand their formation reveals crucial information about urban transformation and dynamics behind its functioning. In this work we focus on the morphogenesis of street networks and we study it through a spatial network generator model. This latter is composed by three layers (a reaction-diffusion layer, a dynamic vector field and a spatial network) surrounded by an environment. The emerging network feeds back to its morphogenetic elem…
Cities consist of infrastructure that enables transportation, which can be considered as topology in abstract terms. Once cities are physically organized in terms of infrastructure, people interact with each other to form the values, which can be regarded as the meta-information of the cities. The topology and meta-information coevolve together as the cities are developed. In this study, we investigate the relationship between the topology and meta-information for a street network, which has aspects of both a complex network and planar graph. The degree of organization of a street structure de…
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…
The fractal dimension growth of urban form can be described with sigmoid functions such as logistic function due to squashing effect. The sigmoid curves of fractal dimension suggest a type of spatial replacement dynamics of urban evolution. How to understand the underlying rationale of the fractal dimension curves is a pending problem. This study is based on two previous findings. First, normalized fractal dimension proved to equal normalized spatial entropy; second, a sigmoid function proceeds from an urban-rural interaction model. Defining urban space-filling measurement by spatial entropy, …
Recent theoretical and empirical studies have focused on the structural properties of complex relational networks in social, biological and technological systems. Here we study the basic properties of twenty 1-square-mile samples of street patterns of different world cities. Samples are represented by spatial (planar) graphs, i.e. valued graphs defined by metric rather than topologic distance and where street intersections are turned into nodes and streets into edges. We study the distribution of nodes in the 2-dimensional plane. We then evaluate the local properties of the graphs by measuring…
The percentage of urban areas dedicated to streets and public spaces is a crucial feature of the spatial planning of cities. By 2050 urban mobility will be one of the biggest confronts for global cities. This study has been carried out to assess the ratio of public space allocated to the streets in some cities of Bangladesh. The length, width, area, and number of street crossings have been counted for the city core and its suburban area as an indicator of the form and pattern of the street layout. This exercise does not consider the rest of the public spaces, like gardens and general public sp…
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…
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…
We investigate urban street networks as a whole within the frameworks of information physics and statistical physics. Urban street networks are envisaged as evolving social systems subject to a Boltzmann-mesoscopic entropy conservation. For self-organized urban street networks, our paradigm has already allowed us to recover the effectively observed scale-free distribution of roads and to foresee the distribution of junctions. The entropy conservation is interpreted as the conservation of the surprisal of the city-dwellers for their urban street network. In view to extend our investigations to …
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 time series of fractal dimension values of urban form always take on sigmoid curves. The basic model of these curves is logistic function. From the logistic model of fractal dimension curves, we can derive the growth rate formula and acceleration formula of city development. Using the inflexions of the fractal parameter curves, we can identify the different phases of urban evolution. The main results are as follows. (1) Based on the curve of fractal dimension of urban form, urban growth can be divided into four stages: initial slow growth, accelerated fast growth, decelerated fast growth, …
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…
Cities worldwide exhibit a variety of street network patterns and configurations that shape human mobility, equity, health, and livelihoods. This study models and analyzes the street networks of every urban area in the world, using boundaries derived from the Global Human Settlement Layer. Street network data are acquired and modeled from OpenStreetMap with the open-source OSMnx software. In total, this study models over 160 million OpenStreetMap street network nodes and over 320 million edges across 8,914 urban areas in 178 countries, and attaches elevation and grade data. This article presen…
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…
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…
This paper aims to measure the efficiency of urban street networks (a kind of complex networks) from the perspective of the multidimensional chain of connectivity (or flow). More specifically, we define two quantities: flow dimension and flow capacity, to characterize structures of urban street networks. To our surprise for the topologies of urban street networks, previously confirmed as a form of small world and scale-free networks, we find that (1) the range of their flow dimension is rather wider than their random and regular counterparts, (2) their flow dimension shows a power-law distribu…
The complexity of urban street networks is well accepted to reside in the information space where roads map to nodes and junctions to links between nodes. Assuming that information networks preserve their amount of surprisal on average leads us to adapt the single-flip Metropolis algorithm to compute information networks with minimal amounts of surprisal.
A type of fractal dimension definition is based on the generalized entropy function. Both entropy and fractal dimension can be employed to characterize complex spatial systems such as cities and regions. Despite the inherent connect between entropy and fractal dimension, they have different application scopes and directions in urban studies. This paper focuses on exploring how to convert entropy measurement into fractal dimension for the spatial analysis of scale-free urban phenomena using ideas from scaling. Urban systems proved to be random prefractal and multifractals systems. The entropy o…
In previous research we defined a methodology for mapping centrality in urban networks. Such methodology, named Multiple Centrality Assessment (MCA), makes it possible to ascertain how each street is structurally central in a city according to several different notions of centrality, as well as different scales of "being central". In this study we investigate the case of Bologna, northern Italy, about how much higher street centrality statistically "determines" a higher presence of activities (shops and services). Our work develops a methodology, based on a kernel density evaluation, that enha…
Spatial patterns and processes of cities can be described with various entropy functions. However, spatial entropy always depends on the scale of measurement, and it is difficult to find a characteristic value for it. In contrast, fractal parameters can be employed to characterize scale-free phenomena. This paper is devoted to exploring the similarities and differences between spatial entropy and fractal dimension in urban description. Drawing an analogy between cities and growing fractals, we illustrate the definitions of fractal dimension based on different entropy concepts. Three representa…
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…
Many complex networks demonstrate a phenomenon of striking degree correlations, i.e., a node tends to link to other nodes with similar (or dissimilar) degrees. From the perspective of degree correlations, this paper attempts to characterize topological structures of urban street networks. We adopted six urban street networks (three European and three North American), and converted them into network topologies in which nodes and edges respectively represent individual streets and street intersections, and compared the network topologies to three reference network topologies (biological, technol…
Urban street networks of unplanned or self-organized cities typically exhibit astonishing scale-free patterns. This scale-freeness can be shown, within the maximum entropy formalism (MaxEnt), as the manifestation of a fluctuating system that preserves on average some amount of information. Monte Carlo methods that can further this perspective are cruelly missing. Here we adapt to self-organized urban street networks the Metropolis algorithm. The "coming to equilibrium" distribution is established with MaxEnt by taking scale-freeness as prior hypothesis along with symmetry-conservation argument…
Circuity, the ratio of network distances to straight-line distances, is an important measure of urban street network structure and transportation efficiency. Circuity results from a circulation network's configuration, planning, and underlying terrain. In turn, it impacts how humans use urban space for settlement and travel. Although past research has examined overall street network circuity, researchers have not studied the relative circuity of walkable versus drivable circulation networks. This study uses OpenStreetMap data to explore relative network circuity. We download walkable and driva…
Computational notebooks offer researchers, practitioners, students, and educators the ability to interactively conduct analytics and disseminate reproducible workflows that weave together code, visuals, and narratives. This article explores the potential of computational notebooks in urban analytics and planning, demonstrating their utility through a case study of OSMnx and its tutorials repository. OSMnx is a Python package for working with OpenStreetMap data and modeling, analyzing, and visualizing street networks anywhere in the world. Its official demos and tutorials are distributed as ope…
Urban streets patterns form planar networks whose empirical properties cannot be accounted for by simple models such as regular grids or Voronoi tesselations. Striking statistical regularities across different cities have been recently empirically found, suggesting that a general and details-independent mechanism may be in action. We propose a simple model based on a local optimization process combined with ideas previously proposed in studies of leaf pattern formation. The statistical properties of this model are in good agreement with the observed empirical patterns. Our results thus suggest…
This paper presents a new perspective of looking at the relation between fractals and chaos by means of cities. Especially, a principle of space filling and spatial replacement is proposed to explain the fractal dimension of urban form. The fractal dimension evolution of urban growth can be empirically modeled with Boltzmann's equation. For the normalized data, Boltzmann's equation is equivalent to the logistic function. The logistic equation can be transformed into the well-known 1-dimensional logistic map, which is based on a 2-dimensional map suggesting spatial replacement dynamics of city …
Urban scholars have studied street networks in various ways, but there are data availability and consistency limitations to the current urban planning/street network analysis literature. To address these challenges, this article presents OSMnx, a new tool to make the collection of data and creation and analysis of street networks simple, consistent, automatable and sound from the perspectives of graph theory, transportation, and urban design. OSMnx contributes five significant capabilities for researchers and practitioners: first, the automated downloading of political boundaries and building …
Fractal geometry provides a powerful tool for scale-free spatial analysis of cities, but the fractal dimension calculation results always depend on methods and scopes of study area. This phenomenon has been puzzling many researchers. This paper is devoted to discussing the problem of uncertainty of fractal dimension estimation and the potential solutions to it. Using regular fractals as archetypes, we can reveal the causes and effects of the diversity of fractal dimension estimation results by analogy. The main factors influencing fractal dimension values of cities include prefractal structure…
The area-perimeter allometric scaling is a basic and important approach for researching fractal cities and has been studied for a long time. However, the boundary dimension of a city is always numerically overestimated by the traditional formula. An adjusting formula has been derived to revise the overestimated boundary dimension and estimate the form dimension, but the association between the global and local fractal parameters is not clear. This paper is devoted to describing the urban evolution by using the improved fractal parameters based on the area-perimeter measure relation. A system o…
A modified version of the Intelligent Driver Model was used to simulate traffic in the district of Afogados, in the city of Recife, Brazil, with the objective to verify whether the complexity of the underlying street grid, with multiple lane streets, crossings, and semaphores, is capable of exhibiting the effect of critical density: appearance of a maximum in the vehicle flux versus density curve. Numerical simulations demonstrate that this effect indeed is observed on individual avenues, while the phase offset among the avenues results in damping of this effect for the region as a whole.
The area-perimeter scaling can be employed to evaluate the fractal dimension of urban boundaries. However, the formula in common use seems to be not correct. By means of mathematical method, a new formula of calculating the boundary dimension of cities is derived from the idea of box-counting measurement and the principle of dimensional consistency in this paper. Thus, several practical results are obtained as follows. First, I derive the hyperbolic relation between the boundary dimension and form dimension of cities. Using the relation, we can estimate the form dimension through the boundary …
The network metaphor in the analysis of urban and territorial cases has a long tradition especially in transportation/land-use planning and economic geography. More recently, urban design has brought its contribution by means of the "space syntax" methodology. All these approaches, though under different terms like accessibility, proximity, integration,connectivity, cost or effort, focus on the idea that some places (or streets) are more important than others because they are more central. The study of centrality in complex systems,however, originated in other scientific areas, namely in struc…
Congestion in transport networks is a topic of theoretical interest and practical importance. In this paper we study the flow of vehicles in urban street networks. In particular, we use a cellular automata model to simulate the motion of vehicles along streets, coupled with a congestion-aware routing at street crossings. Such routing makes use of the knowledge of agents about traffic in nearby roads and allows the vehicles to dynamically update the routes towards their destinations. By implementing the model in real urban street patterns of various cities, we show that it is possible to achiev…
Easy and intuitive navigability is of central importance in cities. The actual scale-free networking of urban street networks in their topological space, where navigation information is encoded by mapping roads to nodes and junctions to links between nodes, has still no simple explanation. Emphasizing the road-junction hierarchy in a holistic and systematic way leads us to envisage urban street networks as evolving social systems subject to a Boltzmann-mesoscopic entropy conservation. This conservation, which we may interpret in terms of surprisal, ensures the passage from the road-junction hi…
The access to an ever increasing amount of information in the modern world gave rise to the development of many quantitative indicators about urban regions in the globe. Therefore, there is a growing need for a precise definition of how to delimit urban regions, so as to allow proper respective characterization and modeling. Here we present a straightforward methodology to automatically detect urban region borders around a single seed point. The method is based on a diffusion process having street crossings and terminations as source points. We exemplify the potential of the methodology by cha…
We investigate the growth dynamics of Greater London defined by the administrative boundary of the Greater London Authority, based on the evolution of its street network during the last two centuries. This is done by employing a unique dataset, consisting of the planar graph representation of nine time slices of Greater London's road network spanning 224 years, from 1786 to 2010. Within this time-frame, we address the concept of the metropolitan area or city in physical terms, in that urban evolution reveals observable transitions in the distribution of relevant geometrical properties. Given t…
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 …
Urban morphology has presented significant intellectual challenges to mathematicians and physicists ever since the eighteenth century, when Euler first explored the famous Konigsberg bridges problem. Many important regularities and scaling laws have been observed in urban studies, including Zipf's law and Gibrat's law, rendering cities attractive systems for analysis within statistical physics. Nevertheless, a broad consensus on how cities and their boundaries are defined is still lacking. Applying an elementary clustering technique to the street intersection space, we show that growth curves …
