The size and number for the hierarchy of German cities based on the 2^{n} principle.
Abstract
A fractal is in essence a hierarchy with cascade structure, which can be described with a set of exponential functions. From these exponential functions, a set of power laws indicative of scaling can be derived. Hierarchy structure and spatial network proved to be associated with one another. This paper is devoted to exploring the theory of fractal analysis of complex systems by means of hierarchical scaling. Two research methods are utilized to make this study, including logic analysis method and empirical analysis method. The main results are as follows. First, a fractal system such as Cantor set is described from the hierarchical angle of view; based on hierarchical structure, three approaches are proposed to estimate fractal dimension. Second, the hierarchical scaling can be generalized to describe multifractals, fractal complementary sets, and self‐similar curve such as logarithmic spiral. Third, complex systems such as urban systems are demonstrated to be a selfsimilar hierarchy. The human settlements in Germany and the population of different languages in the world are taken as two examples to conduct empirical analyses. This study may reveal the association of fractal analysis with other types of scaling analysis of complex systems, and spatial optimization theory may be developed in future by combining the theories of fractals, allometry, and hierarchy.
Keywords
 fractal
 multifractals
 hierarchical scaling
 ranksize rule
 systems of human settlements
 language
1. Introduction
In recent years, the hierarchical systems with cascade structure have attracted attention of many scientists. Representing a form of organization of complex systems, hierarchy is frequently observed within the natural world and in social institutions [1]. A fractal can be treated as a self‐similar hierarchy because a fractal object bears many levels, which are systematically arranged according to scaling laws [2, 3–5]. Fractal phenomena can be described with power laws, and a power law can be decomposed into two exponential laws by means of hierarchical structure. Generally speaking, it is difficult to solve an equation based on power laws or spatial network because of dimensional problems, but it is easy to deal with the problem based on exponential models or hierarchies. Using self‐similar hierarchy, we can transform fractal scaling into
In scientific research, three factors increase the difficulty of mathematical modeling, that is,
Hierarchical scaling suggests a new perspective to examine the simple rules hiding behind the complex systems. Many types of physical and social phenomena satisfy the well‐known rank‐size distribution and thus follow Zipf’s law [7, 11, 18]. Today, Zipf’s law has been used to describe the discrete power law probability distributions in various natural and human systems [3, 19]. However, despite a large amount of research, the underlying rationale of the Zipf distribution is not yet very clear. On the other hand, many types of data associated with Zipf’s law in the physical and social sciences can be arranged in good order to form a hierarchy with cascade structure. There are lots of evidences showing that the Zipf distribution is inherently related to the self‐similar hierarchical structure, but the profound mystery has not yet to be unraveled for our understanding of natural laws. The Zipf distribution is associated with fractal structure and bears an analogy with the 1/f fluctuation [6]. Fractals, 1/f noise, and the Zipf distribution represent the observation of the ubiquitous empirical patterns in nature [19]. This article provides scientists with a new way of looking at the relations between these ubiquitous empirical patterns and the complex evolution processes in physical and social systems, and thus to understand how nature works.
A scientific research actually includes two elements of methodology, namely description and understanding. Science should proceed first by describing how a system works and then by understanding why [20]. The description process is by means of mathematics and measurement, while the understanding process is by means of observation, experience, or even artificially constructed experiments [21]. This work is devoted to exploring fractal modeling and spatial analysis based on hierarchy with cascade structure. First of all, we try to describe and understand hierarchy itself; later, we try to use hierarchical scaling to describe and understand complex systems. Two research methods are utilized in this works. One is
2. Models
2.1. Three approaches to estimating fractal dimension
A regular fractal is a typical hierarchy with cascade structure. Let’s take the well‐known Cantor set as an example to show how to describe the hierarchical structure and how to calculate its fractal dimension (Figure 1). We can use two measurements, the length (
where
According to the definitions of
From Eqs. (1) and (2), we can derive a power law in the form
in which
Based on the exponential models, the fractal dimension is
Based on the common ratios, the fractal dimension is
In theory, Eqs. (8)–(10) are equivalent to one another. Actually, by recurrence, Eq. (7) can be rewritten as
Taking logarithms on both sides of Eq. (11) yields
For the Cantor set,
This suggests that, for the regular fractal hierarchy, three approaches lead to the same result. The fractal dimension can be computed by using exponential functions, power function, or common ratios, and all these values are equal to one another. However, in practice, there are subtle differences between the results from different approaches because of random noise in observational data. Certainly, the differences are not significant and thus can be negligible.
The mathematical description and fractal dimension calculation of the Cantor set can be generalized to other regular fractals such as Koch snowflake and Sierpinski gasket or even to the route from bifurcation to chaos. As a simple fractal, the Cantor set fails to follow the rank‐size law. However, if we substitute the multifractal structure for the monofractal structure, the multiscaling Cantor set will comply with the rank‐size rule empirically.
2.2. Multifractal characterization of hierarchies
Monofractals (unifractals) represent the scale‐free systems of homogeneity, while multifractals represent the scale‐free systems of heterogeneity. In fact, as Stanley and Meakin (1988) [23] pointed out, “
where
From Eqs. (13) and (14), it follows a scaling relation as below:
which is identical in form to Eq. (7), and the capacity dimension
Two sets of parameters are always employed to characterize a multifractal system. One is the set of
where
in which
where
where
Using the above equations, we can describe multifractal Cantor set. For example, if the length of one line segment in the generator is
2.3. Hierarchical scaling in social systems
Fractal hierarchical scaling can be generalized to model general hierarchical systems with cascade structure. Suppose the elements (e.g., cities) in a large‐scale system (e.g., a regional system) are divided into
where
where
in which
where
where
where
which suggests an approximate allometric relation. If
where
3. Empirical analysis
3.1. A case of Germany “natural cities”
3.1.1. Material and data
First of all, the hierarchy of German cities is employed to illustrate hierarchical‐scaling method. Recently, Bin Jiang and his coworkers have proposed a concept of “natural city” and developed a novel approach to measure objective city sizes based on street nodes or blocks and thus urban boundaries can be naturally identified [18, 25]. The street nodes are defined as street intersections and ends, while the naturally defined urban boundaries constitute the region of what is called
3.1.2. Method and results
The analytical method is based on the theoretical models shown above. For the natural cities, the population size measurement (
where
The models of fractals and allometry can be built for German hierarchies of cities as follows. Taking number ratio
Total block ( 
Total area ( 
Average size ( 
Average area ( 
Number ( 


1  28,866  402657796.2  28866.0  402657796.2  1 
2  50,709  731271674.1  25354.5  365635837.1  2 
3  77,576  1030661786.8  19394.0  257665446.7  4 
4  86,071  973558025.6  10758.9  121694753.2  8 
5  82,700  999267240.9  5168.8  62454202.6  16 
6  80,912  940916731.4  2528.5  29403647.9  32 
7  72,397  986813213.3  1131.2  15418956.5  64 
8  75,375  1070810188.5  588.9  8365704.6  128 
9  79,299  1165806475.4  309.8  4553931.5  256 
10  84,327  1271861134.1  164.7  2484103.8  512 
11  84,599  1310854103.8  82.6  1280131.0  1024 
12  75,214  1138100595.1  36.7  555713.2  2048 
13  21,820  197476690.8  20.5  185424.1  1065^{*} 
The goodness of fit is about
The goodness of fit is about
The goodness of fit is around
3.2. A case of language hierarchy in the world
The hierarchical scaling can be used to model the rank‐size distribution of languages by population. Where population size is concerned, there are 107 top languages in the world such as Chinese, English, and Spanish. In data processing, the population size is rescaled by dividing it with 1,000,000 for simplicity. Gleich et al. (2000) [27] gave a list of the 15 languages by number of native speakers (Table 2). The rank‐size model of the 107 languages is as below:
Level  Number  Language and population  Total population  Average population  Size ratio  

1  1  Chinese  885  885  885  
2  2  English  470  Spanish  332  802  401  2.207 
3  4  Bengali  189  Portuguese  170  711  177.75  2.256 
Indic  182  Russian  170  
4  8  Japanese  125  Korean  75  657  82.125  2.164 
German  98  French  72  
Wu‐Chinese  77  Vietnamese  68  
Javanese  76  Telugu  66 
where
Using the hierarchical scaling, we can estimate the fractal dimension of the size distribution of languages in the better way. According to the 2^{n} rule, the 107 languages fall into eight classes by size (Table 3). In the top level, one language, that is, Chinese, and the total of Chinese‐speaking population is 885 million; in the second level, two languages, English and Spanish, with total population 654 million, and so on. The number ratio is defined as
Level ( 
Total population ( 
Number ( 
Average size ( 
Size ratio ( 

1  885,000,000  1  885000000.0  
2  654,000,000  2  327000000.0  2.706 
3  711,000,000  4  177750000.0  1.840 
4  656,687,800  8  82085975.0  2.165 
5  751,058,000  16  46941125.0  1.749 
6  668,446,000  32  20888937.5  2.247 
7  433,020,412  44  9841373.0  2.123 
The squared correlation coefficient is
4. Questions and discussion
4.1. Hierarchical scaling: a universal law
A complex system is always associated with hierarchy with cascade structure, which indicates self‐similarity. A self‐similar hierarchy such as cities as systems and systems of cities can be described with three types of scaling laws:
Next, hierarchical scaling is generalized to describe fractal complementary sets and quasi‐fractal structure, which represent two typical cases of hierarchical description besides fractals. The basic property of fractals is self‐similarity. For convenience of expression and reasoning, the concept of self‐similarity point should be defined. A fractal construction starts from an initiator by way of generator. If a fractal’s generator has two parts indicative of two fractal units, the fractal bears two self‐similarity points; if a fractal’s generator has three parts, the fractal possesses three self‐similarity points, and so on. For example, Cantor set has two self‐similarity points, Sierpinski gasket has three self‐similarity points, Koch curve has four self‐similarity points, and the box growing fractal has five self‐similarity points. The number of self‐similarity points is equal to the number ratio, that is, the common ratio of fractal units at different levels. A real fractal bears at least two self‐similarity points, this suggests cross‐similarity of a fractal besides the self‐similarity. Self‐similarity indicates dilation symmetry, where cross‐similarity implies translation symmetry. However, if and only if a system possesses more than one self‐similarity point, the system can be treated as a real fractal system, and this system can be characterized by fractal geometry. A fractal bears both dilation and translation symmetry. The systems with only one self‐similarity point such as logarithmic spiral can be described with hierarchical scaling. However, it cannot be characterized by fractal geometry. In this case, we can supplement fractal analysis by means of hierarchical scaling.
4.2. Hierarchies of fractal complementary sets
A fractal set and its complementary set represent two different sides of the same coin. The dimension of a fractal is always a fractional value, coming between the topological dimension and the Euclidean dimension of its embedding space. Certainly, the similarity dimension is of exception and may be greater than its embedding dimension. The dimension of the corresponding complement, however, is equal to the Euclidean dimension of the embedding space. Anyway, the Lebesgue measure of a fractal set is zero; by contrast, the Lebesgue measure of the fractal complement is greater than zero. Let us see the following patterns. Figure 9(a) shows the generator (i.e., the second step) of Vicsek’s growing fractal set [37], which bears an analogy with urban growth; Figure 9(b) illustrates the complementary set of the fractal set (the second step). It is easy to prove that the dimension of a fractal’s complement is a Euclidean dimension. If we use box‐counting method to measure the complement of a fractal defined in a two‐dimension space, the extreme of the nonempty box number is
where
which is equal to the Euclidean dimension of the embedding space.
However, a fractal set and its complement are of unity of opposites. A thin fractal is characterized with the fractal parameter, and the value of a fractal dimension is determined by both the fractal set and its complement. Without fractal dimension, we will know little about a fractal; without fractal complement, a fractal will degenerate to a Euclidean geometrical object. This suggests that the fractal dimension of a fractal can be inferred by its complement by means of hierarchical scaling. For example, in fractal urban studies, an urban space includes two parts: one is fractal set and the other fractal complement. If we define a fractal city in a two‐dimensional space, the form of urban growth can be represented by a built‐up pattern, which comprises varied patches in a digital map. Further, if we define an urban region using a circular area or a square area, the blank space in the urban region can be treated as a fractal complement of a city. Certainly, a self‐organized system such as cities in the real world is more complicated than the regular fractals in the mathematical world. The differences between fractal cities and real fractals can be reflected by the models and parameters in the computational world.
A set of exponential functions and power laws can be employed to characterize the hierarchical structure of fractal complementary sets. Suppose the number of fractal units in a generator is
where the parameter
in which
Level  Cantor set  Koch curve  

Scale ( 
Number ( 
Scale ( 
Number ( 

Fractal  Complement  Fractal  Complement  
1  1/3^{0}  2^{0}  (2^{−1})  1/3^{0}  4^{0}  (4^{−1}) 
2  1/3^{1}  2^{1}  2^{0}  1/3^{1}  4^{1}  4^{0} 
3  1/3^{2}  2^{2}  2^{1}  1/3^{2}  4^{2}  4^{1} 
4  1/3^{3}  2^{3}  2^{2}  1/3^{3}  4^{3}  4^{2} 
5  1/3^{4}  2^{4}  2^{3}  1/3^{4}  4^{4}  4^{3} 
6  1/3^{5}  2^{5}  2^{4}  1/3^{5}  4^{5}  4^{4} 
7  1/3^{6}  2^{6}  2^{5}  1/3^{6}  4^{6}  4^{5} 
8  1/3^{7}  2^{7}  2^{6}  1/3^{7}  4^{7}  4^{6} 
9  1/3^{8}  2^{8}  2^{7}  1/3^{8}  4^{8}  4^{7} 
10  1/3^{9}  2^{9}  2^{8}  1/3^{9}  4^{9}  4^{8} 
Level  Sierpinski gasket  Vicsek snowflake  

Scale ( 
Number ( 
Scale ( 
Number ( 

Fractal  Complement  Fractal  Complement  
1  1/2^{0}  3^{0}  (3^{−1})  1/3^{0}  5^{0}  (4 × 5^{−1}) 
2  1/2^{1}  3^{1}  3^{0}  1/3^{1}  5^{1}  4 × 5^{0} 
3  1/2^{2}  3^{2}  3^{1}  1/3^{2}  5^{2}  4 × 5^{1} 
4  1/2^{3}  3^{3}  3^{2}  1/3^{3}  5^{3}  4 × 5^{2} 
5  1/2^{4}  3^{4}  3^{3}  1/3^{4}  5^{4}  4 × 5^{3} 
6  1/2^{5}  3^{5}  3^{4}  1/3^{5}  5^{5}  4 × 5^{4} 
7  1/2^{6}  3^{6}  3^{5}  1/3^{6}  5^{6}  4 × 5^{5} 
8  1/2^{7}  3^{7}  3^{6}  1/3^{7}  5^{7}  4 × 5^{6} 
9  1/2^{8}  3^{8}  3^{7}  1/3^{8}  5^{8}  4 × 5^{7} 
10  1/2^{9}  3^{9}  3^{8}  1/3^{9}  5^{9}  4 × 5^{8} 
Studies on fractal complement hierarchies are useful in urban and rural geography. In many cases, special land uses such as vacant land, water areas, and green belts can be attributed to a fractal complement rather than a fractal set [38]. However, this treatment is not necessary. Sometimes, we specially evaluate the fractal parameter of vacant land, water areas, green belts, and so on. In particular, the spatial state of a settlement may be reversed: the fractal structure evolves into fractal complementary structure and
4.3. Logarithmic spiral and hierarchical scaling
The logarithmic spiral is also termed equiangular spiral or growth spiral, which is treated as a self‐similar spiral curve in the literature and is often associated with fractal such as the Mandelbrot set. The logarithmic spiral was first described by René Descartes in 1638 and later deeply researched by Jacob Bernoulli, who was so fascinated by the
Though a logarithmic spiral is not a fractal, this curve bears the similar mathematical model to simple fractals. A logarithmic spiral can be expressed as below:
where
where
in which
This suggests that the two common ratios are equal to one another, that is,
where
This result suggests a special allometric relation between the two measurements of the logarithmic spiral. The above mathematical process shows that the logarithmic spiral as a quasi‐fractal curve can be strictly described by hierarchical scaling.
In urban studies, the logarithmic spiral study is helpful for us to understand the central place theory about human settlement systems and the rank‐size distribution of cities. Central place systems are composed of triangular lattice of points and regular hexagon area [9]. From the regular hexagonal networks, we can derive logarithmic spiral [41]. On the other hand, the mathematical models of hierarchical structure of the logarithmic spiral based on the systems of golden rectangles are similar to the models of urban hierarchies based on the rank‐size distribution. The logarithmic spiral suggests a latent link between Zipf’s law indicating hierarchical structure and Christaller’s central place models indicative of both spatial and hierarchical structure. Maybe, we can find new spatial analytical approach or spatial optimization theory by exploring the hierarchical scaling in the logarithmic spiral.
5. Conclusions
The conventional mathematical modeling is based on the idea of characteristic scales. If and only if a characteristic length is found in a system, the system can be effectively described with traditional mathematical methods. However, complex systems are principally scale‐free systems, and it is hard to find characteristic lengths from a complex system. Thus, mathematical modeling is often ineffectual. Fractal geometry provides a powerful tool for scaling analysis, which can be applied to exploring complexity associated with
Acknowledgments
This research belongs to the Key Technology R&D Program of the National Ministry of Science and Technology of China (Grant No. 2014BAL01B02). The financial support is gratefully acknowledged.
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