Discrete Dynamics in Nature and Society

Volume 2008 (2008), Article ID 728420, 22 pages

http://dx.doi.org/10.1155/2008/728420

## A Wave-Spectrum Analysis of Urban Population Density: Entropy, Fractal, and Spatial Localization

Department of Urban and Economic Geography, College of Urban and Environmental Sciences, Peking University, Beijing 100871, China

Received 9 December 2007; Revised 15 May 2008; Accepted 16 August 2008

Academic Editor: Michael Batty

Copyright © 2008 Yanguang Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The method of spectral analysis is employed to research the spatial dynamics of urban population distribution. First of all, the negative exponential model is derived in a new way by using an entropy-maximizing idea. Then an approximate scaling relation between wave number and spectral density is derived by Fourier transform of the negative exponential model. The theoretical results suggest the locality of urban population activities. So the principle of entropy maximization can be utilized to interpret the locality and localization of urban morphology. The wave-spectrum model is applied to the city in the real world, Hangzhou, China, and spectral exponents can give the dimension values of the fractal lines of urban population profiles. The changing trend of the fractal dimension does reflect the localization of urban population growth and diffusion. This research on spatial dynamics of urban evolvement is significant for modeling spatial complexity and simulating spatial complication of city systems by cellular automata.

#### 1. Introduction

*The greatest shortcoming of the human race is our inability to understand the exponential function.*

Albert A. Bartlett

Simulating the spatial dynamics of urban population is an interesting but a difficult project. Urban population density can be modeled by two types of functions: one is the exponential function known as Clark’s law [1], the other is the power function proposed by Smeed [2]. Geographers used to employ the exponential model to characterize population density of monocentric cities. However, Smeed’s model has been favored since fractal cities came to front (see, e.g., [3–5]). In fact, if we reject Smeed’s model, we will be unable to interpret the law of allometric growth on urban area and population in theory. On the other hand, if we avoid Clark’s model, we will not be able to describe many cities’ population density empirically. Geographers are often placed in a dilemma when dealing with spatial dynamics of urban evolution.

In fact, the exponential function implies translational symmetry, while the power function denotes dilation symmetry or scaling symmetry; the exponential function implies simplicity and randomness, while the power function indicates complexity and structure [6, 7]. In fractal geometry, two exponential functions can often construct a power function, while a power function can always be decomposed into two exponential functions [8]. It is difficult for us to understand the exponential function, and it is especially difficult to understand the relation between the exponential distribution and the power-law distribution. A conjecture is that exponential law and power law represent, respectively, two modes of urban evolvement which supplement each other.

Population is one of the two central variables which can be employed to explore the dynamics of cities [9]. However, the underlying rationale of intraurban population growth and diffusion is still a question pending further discussion. Clark’s law on urban density can provide a window for us to apprehend the dynamics of urban morphology from the angle of view of population. The negative exponential distribution seems to mean nonfractal structure of urban population, but it can be associated with fractal structure by the Fourier transform. In order to probe the mysteries of fractal cities and the related spatial dynamics, we must research the essentials of negative exponential distribution.

In this paper,
the exponential model of urban density will be explored by using the wave-spectral
analysis. The significance of studying the classical model is in three aspects.
The first is to reveal the locality and localization of urban population evolution,
which is very important for simulating spatial complexity of cities through
computers. The second is to find a new approach to evaluating a kind of fractal dimension of
urban form, which differs from but can make up *box dimension* and *radial
dimension*. The third is to understand spatial
complexity of urban evolvement in the new perspective. The study of complexity
concerns emergence of fractals, localization, strange attractor, symmetry
breaking, and so on [10]. Fractal structure
and localization can be brought to light to some extent from the negative
exponential distribution by means of spectral analysis.

The rest of this paper is structured as follows. Section 2 presents a new
derivation of the negative exponential model of urban population density by the entropy-maximizing principle,
which is actually one of the fundamental reasons of fractal cities [11]. Based on the
exponential function, an approximate power-law relation between wave number and
spectral density is derived by Fourier transform. Section 3 provides an
empirical analysis, including spectral analysis, correlogram analysis,
and information entropy analysis, by applying the theoretical
models to the city of Hangzhou, China. The computations lend support to the theoretical inferences given in Section 2. In Section 4, the differences
and relationships between the negative exponential distribution and the inverse
power-law distribution are discussed to distinguish the concept of *locality* from that of *action at a distance*.

#### 2. Mathematical Models

##### 2.1. New Derivation of Clark’s Law

A power law indicating fractal structure of urban systems can be decomposed into two exponential laws [8], and the exponential laws can be derived by using the entropy-maximizing method [12]. This suggests that fractal structure can be interpreted with the principle of entropy maximization, and exponential function is an important bridge between entropy and power law. On the other hand, as complex spatial system, an urban phenomenon can be modeled with different mathematical expressions under different conditions. Urban population density can be described by a number of functions, among which the negative exponential function is always valid in empirical analysis [1, 13, 14]. Since that the exponential law can connect entropy maximization and fractal, we are naturally interested in the cause and effect of the urban exponential distribution. It will be shown that Clark’s law comes between entropy-maximizing process and special fractal structure.

As a theoretical study, this paper is focused on a monocentric city, and all the data analyses are based on the idea from statistical average. In this instance, the growth of cities is often regarded as a process of spatiotemporal diffusion [15], which can be abstracted as the following partial differential equation where denotes growth/decay coefficient or transfer coefficient, is called “diffusivity” or diffusion coefficient, and and refer to two directions of spatial diffusion. For an isotropic diffusion, one direction (say, ) has no difference from the other direction (say, ), we have , where represents the distance from the center of city (where ). In order words, we can substitute one-dimension diffusion process for two-dimension process to analyze the isotropic city systems. Now, if does not change with time, namely, if , then (2.1) reduces to the common differential equation characterizing one-dimension diffusion such as (the initial condition is , while the boundary condition ), whose solution is just the exponential function known as Clark’s law (see Appendix A). This suggests that the exponential law in fact reflects an instantaneous equilibrium of urban population diffusion.

Assuming that population density at distance from the city center declines monotonically, Clark [1] proposed an empirical model that can be written as where is a constant of proportionality which is supposed to equal the central density, that is, denotes a rate at which the effect of distance attenuates, and refers to a characteristic radius of urban population distribution. Thus we have . Clark [1] fitted the log transform of (2.1) to more than 20 cities by using linear regression. The results form the solid empirical foundation of the negative exponential law of monocentric urban density.

In the real world, urban growth is often not isotropic, but in an average sense, we can regard an anisotropic process as an isotropic process. Just based on this idea, Clark’s law is propounded. An urban population density function is actually defined in one-dimension space but it includes information of two-dimension space. Generally speaking, an exponential distribution function can be derived from the entropy maximization principle. Bussiere and Snickers [16] once showed that Clark’s model could be derived from Wilson’s [17] spatial interaction models (see also Wilson [18]), which is based on the entropy-maximizing principle. In fact, under ideal conditions, Clark’s model can be derived in a very simple way from several geographical assumptions by using entropy-maximizing methods. Now, in order to reveal the physical essence of exponential distribution of urban population, a new derivation of (2.2) is given in this subsection. The mathematical deduction is more graceful and compendious than previous derivation presented by Bussiere and Snickers [16], and the process is helpful for exploring the spatial dynamics of urban evolution.

Suppose that the total population in the urban field of a monocentric city is , and the urban growth is considered to be a continuous process in time and space. An urban field is defined as a bounding circle based on the center of the urban cluster, marked by the maximum radius which contains the whole cluster [3, page 340]. Imagining that the urban map has been digitalized with low resolution, we can “string” pixels, which may be called “cells,” by drawing a straight line or a radial from the center of the city to the boundary (see Figure 1). Further, suppose the population in the th cell along the “line” is , and the whole population along the line is . The variable has dual attributes. On the one hand, it denotes the population size within the th cell along the line, and on the other, it represents just the average population density of the th ring comprising a number of cells.

Since Clark’s law is just the solution to the one-dimension diffusion equation, we can examine one-dimension population distribution based on the idea from statistical average. The postulates of this study can be summarized as follows. (1) A monocentric city has no strict boundary because of scaling invariance of urban form. (2) Population is dense enough in urban field. The next step is to find the functional relationship between density and distance . For this purpose, the entropy-maximizing method is employed. The number of states of the population distributed in all the cells along the radial, , can be expressed as an ordered division problem We use the state in one-dimension urban space to represent the state in two-dimension urban space in average sense. Then the entropy of population distribution profile, , is given by Suppose that the entropy approaches to maximization. We can define an objective function such as According to our assumptions stated above, the objective of city evolution is subject to two constraint conditions as follows: Equation (2.6) can be understood easily, but (2.7) need be made clear. Here refers to the population number in the center of the city, and to the population number in the th ring that is measured with a circle of cells. In fact, if we measure the distance by the size of the cells, namely, take the diameter of cells as length unit, then is just the distance from the centroid of the th cell to the center of the city. That is, when the cells are very very small, the ordinal number can represent the radius of the th ring, and is the corresponding perimeter.

In this case, if we can find a mathematical expression to describe the relationship between and , the problem will be solved immediately. Thus our question can be turned into the process of finding conditional extremum because that the value of entropy depends on the density of spatial distribution of urban population. A Lagrangian function is constructed as where is the Lagrangian multiplier (LM). Theoretically and are both large enough in terms of our postulates. According to the well-known Stirling’s formula , we have an approximate relation, , where is considerably big. So, differentiating (2.8) partially with respect to yields Considering the condition of extremum , we have

In theory, we can improve the resolution of digital map unlimitedly, and thus, the cell/pixels become infinitesimal. That is to say, for simplicity, the discrete distance variable represented by can be replaced with a continuous one represented by for the time being, that is, . Inserted with (2.10), the discrete (2.7) in which is used as a substitute for can be rewritten as an integration expression where is the radius of urban field and it can be defined by , here is the Feret’s diameter [3], (see Kaye [19]). Equation (2.11) is the continuous expression replacing (2.7). In keeping with the first postulate, is large enough. Using integration by decomposition and taking l’Hospital’s rule into account, we can find the solution of (2.11) such as (see Appendix B) Substituting (2.12) into (2.10) yields If as given, then (2.13) collapses to The characteristic radius of the city, , can be defined by Inserting (2.15) into (2.14) gives Substituting (2.14) and (2.16) into (2.13) immediately yields Clark’s law, that is, (2.2). Further, inserting (2.2) into (2.6), we can derive . The maximum of entropy can be proved to be , where is the base of the natural system of logarithms, having a numerical value of 2.7183 approximately.

Entropy maximization suggests the most probable distribution on some conditions. The negative exponential distribution of urban population density is not inevitable, but it is the most probable state for at least the monocentric city. This kind of distribution suggests a special fractal profile, which can be brought to light by Fourier analysis and scaling wave-spectrum relation.

##### 2.2. Wave-Spectrum Function of Urban Density

The negative exponential model of urban density is in essence a special spatial correlation function. A power-law relation between wave number and spectrum density can be derived from the exponential function. Considering the interaction between the cells along the radial in Figure 1, we can construct a density-density correlation function as follows: where denotes the population density of cell X at distance from the city center, refers to the population density of another cell at distance from X. Given , it follows that one cell becomes the center of the city, and the spatial correlation function collapses to an exponential function If the data are so normalized that , we have , and thus (2.18) is equivalent to (2.2). In this case, Clark’s law is just a special density-density correlation function, which indicates spatial correlating action between the city center and the location at distance from the center. The distance parameter, , is relative to the spatial correlation length. A larger value of the characteristic radius suggests a longer correlation distance.

Note that the autocorrelation
function and the energy spectrum can be converted to each other through Fourier’s
cosine transform: where is the unit of complex number, denotes
wave number, that is, the reciprocal of the wavelength,
represents corresponding energy spectral density.
The concept of *energy
spectrum* comes from engineering mathematics. The product of Fourier
transform of a function and its conjugate bears an analogy with the mathematical
form of energy in physics [20]. In
the light of the symmetry of correlation function, the Fourier transform of (2.18) can be given in the form As is large enough, we have Thus the energy spectral density can be gained according to
energy integral such as [21] In practice, the length of sample path, , is generally limited, therefore the *wave spectrum* density is always employed to substitute for the *energy spectrum* density .
Then (2.22) can be rewritten as [22] Equation (2.23) is an approximate expression
based on ideal conditions, and it can be generalized to the following scaling
relation: where is called “*spectral
exponent*” which usually ranges from 0 to 3. When value is near 1, (2.24)
indicates what is called noise (see, e.g., [23, 24]). In fact, the spectral
exponent is associated with a fractal dimension of urban population profiles.

For a time series or spatial
series, if the relation between spectral density and frequency or wave number
follows the scaling law defined by (2.24), a
fractal structure can be revealed. It has been demonstrated that, for dimension variables, the connection between and is given by [25–27] where refers to the dimension of Euclidean space. Accordingly, , where is the fractal dimension of urban population profiles , and *H* denotes
the Hurst exponent . Further, the autocorrelation coefficients
of the rate of changes can be derived from the fractional Brownian motion as in [25] This is a special density-density
correlation function, which can be understood by means of the knowledge of time
series analysis. Many methods of analyzing times series, including
autocorrelation analysis, autoregression analysis, and spectral analysis, can
be employed to deal with spatial series [28]. If
or , then we have , and thus . In this
case, the th cells act directly on
and only on the th cells, and do not act on the th cells or more. If or ,
then we have , and
thus . In
this case, for , the th cell can act directly on the th
cells positively. If or , then we have , and thus . In
this case, the th cell will act
directly on the th
cells negatively.

For the negative exponential function of urban population
density, the expected result of spectral exponent is , thus the fractal dimension is , and the
Hurst exponent is , which gives the
autocorrelation coefficient . That suggests
a spatial locality of city systems. In physics, the *principle of locality* coming from Einstein
[29] is that distant objects cannot have
direct influence on each another. In other words, an object is influenced
directly only by its immediate surroundings. The fact of spatial
autocorrelation coefficient implies that
a population cell tends to interact only on the immediate cells.

#### 3. Empirical Analysis

##### 3.1. Study Area and Data Resource

The city of Hangzhou is taken as an example to verify the wave-spectrum relation of urban population
density and related theory. Hangzhou is the capital of Zhejiang province, China. The urban density data in
1964, 1982, 1990, and 2000 come from census, which is processed by Feng [14]. The census tract data are based on *jiedao*, which bears an analogy with the UK enumeration districts [30], or the US
subdistricts [13], while the system of *jiedao* has an analogy with the urban zonal
system in Western literature (see [3, page 325]). In the demographic sense, a *jiedao* is a census
tract. The data are processed by means of spatial weighed average. The length
of sample path is 26, and the maximum urban radius is 15.3 km. The method of processing data is illuminated
in detail by Feng [14]. Some necessary
explanations on sampling and data processing are made as follows.

(1) Sampling area. The data cover the metropolitan area (MA) of Hangzhou, which is greater than the urbanized area (UA). The spatial scopes of sampling in four years are same in order to make it sure that the parameters from 1964 through 2000 are comparable. Because of scaling invariance of urban form [3], we take no account of the borderline between the urban and rural areas.

(2) Calculation method. A series of concentric
rings are drawn around city center in proportional spacing (see Figure 2). The ratio of
each partial zone to the whole area between two rings is taken as the weight of
computing urban population density. A region between two adjacent rings
can be named a circular belt, which will be numbered as , where is the number of circular belt. The zones
can be numbered as , where is the number of zones. Let be the common area of the th
circular belt and the th zone, that
is, where represents the th belt, denotes the th zone, both of them are measured with area;
therefore, is the area of the intersection of and . Defining a weighted coefficient as we have in which is the population density of the th zone (*jiedao*), which can be known from the census datum. Thus where refers to the radius of the th ring.
It is evident that denotes the average density of the th circular belt. The weighted arithmetic average can lessen the influence
of zone’s scope on the estimated results of population density as much as
possible.

(3) Spatial scale. The radius difference between rings is 0.6 km, less than , that is, the average distance of urban population activity. The parameter values of can be estimated with Clark’s model, namely, (2.2).

##### 3.2. Data Processing Method and Results

The population density of Hangzhou city will be analyzed from three angles of view: spectral analysis, correlogram analysis, and information entropy analysis. Accordingly, we will compute wave-spectral density, autocorrelation function, and information entropy. The procedure of wave spectrum analysis based on fast Fourier transform (FFT) can be summed up as five steps.

*Step 1 (sample path extension). *The
symmetrical rule of the FFT’s recursive algorithm requires the length of time
series to be an integer power of 2, that is, . However, there are 26 data points in our
spatial sample path .
A process called “zero-padding” can be used to bring the number up to the next
power of 2. In this case, the best way is to add 6 zeros at the end of the data
series to bring the number to 32 (i.e., .

*Step 2 (FFT of spatial series). *Performing the FFT on the extended population density data of Hangzhou city yields a
complex data series . The processing method is so accessible that MS Excel can give the
results conveniently.

*Step 3 (spectrum density calculation). *The formula is such as . It is not difficult for us to compute
the spectral density based on the FFT results (see Table 1). The spectral density is just the product
of FFT result and its conjugate divided by the extended sample path length ().

*Step 4 (making wave-spectra plots). *As soon as the population density is transformed into spectral density, a plot
reflecting the relation between wave number and spectral density can be given
easily (see Figure 3). Let the circular belts be
numbered as , where .
Thus the wave number will be defined by .

*Step 5 (modeling the wave-spectra relations). *If the wave-spectra plots displayed in Figure 3 show some attenuation trend, we can fit the data of Table 1 to (2.24). A least square computation will give the spectral
exponent , from which we can
estimate the fractal dimension and Hurst
exponent by means of (2.25).

All the population density data of Hangzhou city in four years satisfy the power law in the form stipulated by (2.24) to a great extent. A least squares calculation utilizing the data of Table 1 gives four spectral exponent values. The fractal dimension , Hurst exponent , and the autocorrelation coefficient can be evaluated consequently (see Table 2). From 1964 to 2000, the spectral exponent values become closer and closer to , the fractal dimension values become closer and closer to , and the Hurst exponent values become closer and closer to . All of these suggest a phenomenon of localization of urban population evolution: a population cell is inclined towards acting directly on the immediate cells, and not on the alternate cells, that is, nonimmediate cells. What is more, the wave-spectrum relations and spectral exponent values indicating noise [23] remind us of the self-organized criticality (SOC) of urban evolution [31–33].

It should be made clear that the fractal dimension used
here is different from those employed to characterize two-dimension urban form
such as *box dimension* and *radial dimension* ([34], White and Engelen [35]). Generally speaking, we need three kinds of fractal
dimensions at least to characterize the city form with fractal structure. The
first is the *box dimension*, which can be estimated by
the box-counting method [36]; the second is *radial
dimension*, which is
defined by the area-radius scaling [3, 5]; and the
third is the dimension of fractal lines [26, 27], the author of this paper calls
it *profile dimension* when it is applied to
urban morphology. The third type of dimension can be estimated easily through
the wave-spectrum relation (see Appendix C).

Spectral analysis and correlation analysis represent different sides of the same coin in theory, while empirically correlation analysis and spectral analysis supplement each other. Therefore, a correlogram analysis of Hangzhou urban density should be made to consolidate the results of wave-spectra analysis. A spatial autocorrelation function can be based on the relationship between and , where refers to displacement analogous to time lag in time series analysis. Part of the autocorrelation function (ACF) and partial autocorrelation function (PACF) values is listed in Table 3, and the results in 2000 are shown in Figure 4. The ACF attenuates gradually when the displacement becomes long and displays some damped oscillation. What interests us is the PACF, which cuts off at a displacement of 1. That is, partial autocorrelation coefficients (PACCs) are not significantly different from 0 when the displacement (see Table 3 and Figure 4(b)). The cutoff of PACF at a displacement of 1 suggests a possible locality in spatial activities of urban population: a population cell acts directly on and only on the proximate population cells, not on alternate cells. Evidently, the correlogram analysis lends further support to the conclusion drawn from the wave-spectrum analysis.

There exists a
mathematical relation between fractal dimension and information entropy. In a
sense, Hausdorff dimension can be proved to be equivalent to Shannon’s
information entropy [37]. It is hard
to evaluate the population distribution entropy by using (2.4), we can only estimate the maximum entropy by
using the formula ,
which is based on one-dimensional *continuous* measure (see Table 2). However, it is easy to calculate
the one-dimensional *discrete* information entropy of population profile along the radial (see Figure 1). Defining a probability such as
where variables and fulfill the same
roles as in (2.3) or (2.4), then we have an
information entropy
in which refers to the Shannon’s entropy. The results of spatial entropy for Hangzhou’s population distribution in four
years are as follows: nat in 1964, nat
in 1982, nat in
1990, and nat in
2000. The maximum information entropy based on discrete measure is nat. The
redundancy measuring the ratio of
actual entropy to the maximum entropy and subtracting this ratio from 1 can be
applied to spatial entropy statistics [38]. Using the formula , the redundancy is
computed as follows: , , , . The information
entropy values become larger and larger, and the redundancy values approach the
minimum value 0. This trend gives further weight to the viewpoint that the
dynamics of urban population evolvement in Hangzhou
is actually a process of entropy
maximization.

As stated above, information entropy maximization implies negative exponential distribution of urban population density, and the exponential distribution denotes spectral exponent and thus fractal dimension . On the other hand, the PACF based on the exponential function shows a cutoff at a displacement of 1. All of these suggest a localization tendency of urban population distribution in Hangzhou. The reasoning process from entropy maximization to localization of spatial distributions of urban population is illustrated as Figure 5. In physics, localization is a phenomenon according to which the stationary quantum states of electrons in an extended system are localized due to disorder [39]. As for cities, localization can be defined as follows: a system of nonlocality changes gradually to that of locality.

##### 4. Questions and Discussion

As indicated above, urban population density can be modeled by different functions under different conditions. The diversity or variability of urban models suggests asymmetry or symmetry breaking of geographical systems, which thus suggests spatial complexity of city systems and complication of urban evolution. Besides the negative exponential function, the inverse power function is also very important in modeling urban form. The relations between the exponential function and the power function were expounded by Batty and Kim [40]. Two questions will be discussed and answered here. The first is the difference between the negative exponential distribution and the inverse power-law distribution where the spatial dynamics is concerned, and the second is the locality and localization of urban population evolution.

The negative exponential function known as Clark’s model and the inverse power function known as Smeed’s model are two types of special spatial correlation functions. Exponential correlation function implies simple structure, while power-law correlation function suggests complex dynamics. If and only if a system falls into the self-organized critical state, the spatial correlation will follow a power law. Otherwise it follows an exponential law [22, 23]. In urban studies, the exponential function is always used to characterize urban population density, while the power function can be used to model urban land use density. Research into the relation between exponential law and power law is instructive for us to explore deeply the spatial dynamics of urban morphology.

The negative exponential distribution indicates locality in theory, while the power law implies action at a distance. This viewpoint can be verified by the correlogram analysis based on simple simulation computation. Both the ACF and PACF can provide a summary of a time or spatial series’ dynamics [41]. The ACF based on the standard exponential distribution displays a gradual one-sided damping (see Figure 6(a)), while the PACF of the exponential distribution cuts off at a displacement of 1—the partial autocorrelations drop abruptly to 0 beyond displacement 1 (see Figure 6(b)). The PACF seems to suggest a property of locality associated with the exponential distribution of urban density. As for Hangzhou city, the PACF is consistent with the result based on the standard exponential distribution, but the ACF differs in the damping way just because that the exponential distribution in the real world is not often very standard. In other words, the urban population dynamics of Hangzhou from 1964 to 2000 is only gradual localization without proper locality.

It is revealing to compare the correlogram of the exponential distribution with that of the power-law distribution. The ACF and PACF based on the standard power-law function differ from those based on the standard exponential function in an important way. The ACF of the power-law distribution displays a slow one-sided damping (see Figure 7(a)), while the corresponding PACF displays rapid one-sided damping without cutoff (see Figure 7(b)). In short, both the ACF and PACF of the power-law distribution are trailing, and this phenomenon reminds us of the action at a distance of spatial activities.

The differences of ACF and PACF between the exponential distribution and the power-law distribution are obvious and interesting. The ACF of the power-law distribution decays more slowly than that of the exponential distribution. In particular, the PACF of the power-law distribution is trailing, while the PACF of the exponential distribution cuts it off at the displacement of 1. The former suggests an action at a distance, while the latter reminds us of locality of spatial interaction (see Figure 8). The similarities and differences between the correlograms of the exponential distribution and that of the power-law distribution are tabulated as follows (see Table 4). The correlograms of population density distributions of Hangzhou are more similar to those of the exponential distribution than those of the power-law distribution.

As indicated above, the exponential distribution has a characteristic length, , which indicates simple geometrical patterns, while the power-law distribution has no characteristic length, which indicates complex patterns associated with fractal form and structure. Revealing the relationship between locality and action at a distance of urban evolution is very important for modeling spatial complexity by using cellular automata (CA). The original CA model possesses locality. In urban simulation, the CA’s locality is gradually replaced by action at a distance [35, 42, 43]. For urban-land dynamics, the CA model with action at a distance is suitable, but for urban population dynamics, the things may be more complicated because that urban population models are not one and only.

The scaling wave-spectrum relation and fractal properties of urban density suggest a dual character of urban evolution. On the one hand, the growth of cities look like particle motion, which can be simulated by means of CA technique, including diffusion-limited aggregation (DLA) and dielectric breakdown model (DBM), and so forth [3]. On the other hand, the statistical average of urban population distribution reminds us of the wave motion, or a ripple spreading from the center to the periphery. A city seems to be a set of dynamic particles indicating chaos or disorder distributed on the ripple indicative of order. In fact, intuitively, the spatial complexity displayed by city seems to express a struggle between order and chaos. An urban model of ripple-particle duality should be proposed to address temporal-spatial evolution of cities. As space is limited, the related questions will be made clear in the future work.

##### 5. Conclusions

The study of this paper may be of revelation for modeling
spatial complexity and simulating the urban growth and form. Geographers used
to rely heavily on the rules associated with *action at a distance*, but neglect the rules based on the *locality* of urban population activity. However,
urban spatial dynamics seems to be the unity of opposites of *locality* and *action at a distance*. The keys of
comprehending this paper rest with three aspects. (1) Density is a zero-dimension
measure, but urban density function is defined in one-dimension space, from which
we can learn the information of two-dimension space. (2) Urban density models
are in essence spatial correlation function, which can be converted into *energy spectrum* by Fourier transform and
vice versa. Energy spectral
density divided by sample path length is the wave spectral density. (3) If the
relation between wave-spectrum
density and wave number shows scaling invariance, fractal dimension can be
estimated indirectly through the spectral exponent. The main points
of the paper can be summarized as follows.

Firstly, one of the important physical mechanisms of urban growth and population diffusion is information entropy maximization indicating spatial optimization. From the viewpoint of statistical average, urban population density distributions of monocentric cities always satisfy the negative exponential function, which can be derived by using entropy-maximizing methods. Entropy maximization actually implies minimum cost when benefit is certain, or maximum benefit when cost is determinate. In other words, entropy maximizing in human systems suggests a process of optimization. Urban population density tends to evolve into an optimum distribution through self-organization.

Secondly, the negative exponential distribution implies locality or localization of urban population activities. Entropy maximization interprets the negative exponential distribution, and the scaling wave-spectrum relation coming from the negative exponential function predicts a locality of urban population activities in theory. In terms of the empirical evidences, the wave spectral analysis shows a localization process of urban population evolution, while the spatial autocorrelation analysis associated with wave spectral analysis demonstrates a locality of spatial interaction of population cells.

Thirdly, urban evolution seems to possess a dual nature of locality and action at a distance. The concept of locality should be as important as the idea of action at a distance for urban modeling and simulation. Locality is to urban population what action at a distance is to urban land use. The former relates to the negative exponential distribution, while the latter to the inverse power-law distribution. The power law indicates fractal structure, and the exponential law can be connected with fractals by Fourier transform. A conjecture is that if a city evolves into a self-organized critical state, the negative exponential distribution may change to the inverse power distribution.

#### Appendices

#### A. How to Derive (2.2) from (2.1)

Substituting polar coordinates for Cartesian coordinates, we can also derive the negative exponential function from the diffusion model. Let us consider a Laplacian equation such as where is the Laplacian operator, other notations fulfill the same roles as in (2.1). For the anisotropic diffusion in two-dimension space, the relation between Cartesian coordinates and polar coordinate is and . Thus (A.1) can be converted into in which refers to polar radius and to polar angle. However, if we examine the isotropic diffusion in one-dimension space, we will have , then , , thus (A.1) in which is of inexistence can be changed to The initial condition is , while the boundary condition is . A special solution to (A.3) is just Clark’s model, namely (2.2) in the text.

#### B. How to Derive (2.12) from (2.11)

We can derive (2.12) from (2.11) as follows. According to the l’Hospital’s rule, when , we have Using integration by decomposition yields Please note that the in (B.2). Therefore, we get which is just equivalent to (2.12) in the text.

#### C. Box Dimension, Radial Dimension, and Profile Dimension

Suppose that a three-axis coordinate system is constructed by (latitude), (longitude), and (altitude). We use the three-axis coordinate to describe the Euclidean space in which a city exists. Then the box dimension and the radial dimension are defined in the space described by axes and , while the profile dimension is defined in the space described by axes and , or by axes and . This paper is mainly involved with the profile dimension , which is derived from the fractional Brownian motion (fBm) and dimensional analysis. In the course of urban development, the values of box dimension and radial dimension always increase over time. However, the profile dimension values of urban density decreases with the lapse of time, approaching to 1.5.

Actually, radial
dimension can reflect
the information of the three-dimension space in the sense of average. The
author has derived a relation between the radial dimension and profile
dimension of urban morphology by using Fourier transform. The result is , where refers to the *radial dimension*,
and to the *profile dimension*. According to the
fractal dimension equation, the radial dimension of Hangzhou’s population, , can be estimated as follows: in 1964, in 1982, in 1990, and in 2000. This kind of fractal dimension value increases
with the passage of time and approaches to .
The related problems will be discussed in detail in the companion paper “Exploring fractal parameters of urban growth and form with
wave-spectrum analysis” (forthcoming).

#### Acknowledgments

This research was supported financially by the National Natural Science Foundation of China (Grant no. 40771061). Comments by the anonymous referees have been very helpful in preparing the final version of this paper. The author would like to thank Dr. Yuwang Hu at Xinyang Normal University for assistance in mathematics, Dr. Shiguo Jiang at Ohio State University for his editorial helps, and Dr. Jian Feng at the Peking University for providing essential material on the urban density of Hangzhou.

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