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Hierarchy of cities reflects the ubiquitous structure frequently observed in the natural world and social institutions. Where there is a hierarchy with cascade structure, there is a Zipf's rank-size distribution, and

The well-known Zipf’s law is a very basic principle for city-size distributions, and empirically, the Zipf distribution is always associated with hierarchical structure of urban systems. Hierarchy is frequently observed within the natural world as well as in social institutions, and it is a form of organization of complex systems which depend on or produce a strong differentiation in power and size between the parts of the whole [

Urban evolution takes on two prominent properties: one is the Zipf distribution at the large scale [

In this paper, Zipf’s law, allometric scaling, and fractal relations will be integrated into the same framework based on hierarchy of cities, and, then, a model of playing cards will be proposed to explain the Zipf distribution and hierarchical scaling. From this framework, we can gain an insight into cities in the new perspective. Especially, this theoretical framework and model can be generalized to physical scientific fields. The rest of this paper is organized as follows. In Section

First of all, the mathematical description of hierarchies of cities should be presented here. Grouping the cities in a large-scale region into ^{n}

Several power-law relations can be derived from the above exponential laws. Rearranging (

In theory, the size-number scaling relation, (

The hierarchy of cities reflects the cascade structure which is ubiquitous in both physical and human systems. To provide a general pattern for us to understand how the evolutive systems are self-organized, we can draw an analogy between cities, rivers, and earthquake energy distributions (Figure

The models of hierarchies of cities, rivers, and earthquakes with cascade structure. (Note: the sketch maps only show the first four classes for the top-down models, or the last four classes for the bottom-up models.)

Hierarchy of cities

Networks of rivers

Hierarchy of earthquakes

These exponential models can be employed to characterize river networks and hierarchies of the seismic activities of a region (say, Japan) over a period of time (say, 30 years). Equations (

Comparison between the exponential laws of cities and those of rivers and earthquake energy.

Exponential law | Hierarch of cities | Network of rivers | Energy of earthquake |
---|---|---|---|

The first law (number law) | |||

The second law (size law) | |||

The third law (area law) |

Despite all these similarities, there are clear differences among cities, rivers, and earthquake energy distributions as hierarchies. Actually, hierarchies can be divided into two types: one is the

Typically, Horton-Strahler’s laws are on real hierarchies, while Gutenberg-Richter’s laws on dummy hierarchies (Table

Differences between two typical types of hierarchies with cascade structure.

Type | Cascade structure | Interclass relation | Connection | Typical example |
---|---|---|---|---|

Real hierarchy | Physical structure | Geometric relation | Concrete connection | River systems |

Dummy hierarchy | Mathematical structure | Algebraic relation | Abstract connection | Earthquake energy distribution |

The theoretical regularity of city size distributions can be empirically revealed at large scale [

The hierarchy of the 452 cities in USA and the related measures (2000).

Class | City number ( | Average population size ( | Average urban area ( |
---|---|---|---|

1 | 1 | 17799861.000 | 8683.200 |

2 | 2 | 10048695.500 | 4908.995 |

3 | 4 | 4561564.500 | 3923.070 |

4 | 8 | 3335242.625 | 2828.796 |

5 | 16 | 1690796.250 | 1493.243 |

6 | 32 | 815564.656 | 899.782 |

7 | 64 | 354537.344 | 451.605 |

8 | 128 | 156158.125 | 217.896 |

9 | (197) | 69740.228 | 103.053 |

The scaling patterns for the hierarchy of the 452 cities in America (2000).

The scaling relation between urban population and city number

The scaling relation between urban area and city number

The allometric relation between urban area and population

The least squares calculations involved in the data in Table

The fractal parameters and related scaling exponents can also be estimated by the common ratios. As mentioned above, the number ratio is given

According to the mathematical relationships between different models illuminated in Section

Theoretically, the fractal parameters or scaling exponents of a hierarchy of cities from different ways, including power laws, exponential laws, and common ratios, should be the identical with each other. However, in practice, the results based on different approaches are always close to but different from one another due to the uncontrollable factors such as random noises, spatial scale, and degree of system development. The average values of the fractal dimension and allometric scaling exponent can be calculated as

Another large-scale urban system is in the People’s Republic of China (PRC). By the similar method, the 660 cities of China in 2005 can be classified by population size into 10 levels

The hierarchy of the 660 cities in PRC and the related measures (2005).

Class ( | City number ( | Average population size ( | Average urban area ( |
---|---|---|---|

1 | 1 | 1778.420 | 819.880 |

2 | 2 | 1182.875 | 956.500 |

3 | 4 | 626.830 | 567.405 |

4 | 8 | 407.219 | 261.399 |

5 | 16 | 237.608 | 183.454 |

6 | 32 | 148.627 | 144.776 |

7 | 64 | 82.504 | 70.169 |

8 | 128 | 43.948 | 44.371 |

9 | 256 | 20.544 | 23.189 |

10 | (149) | 9.764 | 13.062 |

The scaling patterns for the hierarchy of 660 cities in China (2005). (Note: In the first two plots, the data points of the lame duck classes are treated as the outliers, which deviates from the normal scaling range because the small cities in China are of undergrowth).

The scaling relation between urban population and city number

The scaling relation between urban area and city number

The allometric relation between urban area and population.

Analogous to the US case, the least squares computations of the quantities listed in Table

The scaling exponents can also be estimated by number, size, and area ratios. The number ratio is given as

The above results imply that (

The fractal property and fractal dimension of a hierarchy of cities can be understood by analogy with the regular fractals such as Cantor set, Koch curve, and Sierpinski carpet. A fractal process is a typical hierarchy with cascade structure, and we can model it using the abovementioned exponential functions and power laws, for example, (

The collected results of the fractals parameters and scaling exponents of the hierarchies of the USA and PRC cities.

Approach | Fractal parameter or scaling exponent | |||||

USA’s cities in 2000 | PRC’s cities in 2005 | |||||

Power law | 0.974 | 1.213 | 0.793 | 1.262 | 1.435 | 0.856 |

Exponential law | 1.010 | 1.278 | 0.790 | 1.220 | 1.425 | 0.856 |

Common ratio | 0.983 | 1.217 | 0.807 | 1.184 | 1.405 | 0.842 |

Mean value | 0.989 | 1.236 | 0.797 | 1.222 | 1.422 | 0.851 |

The fractal dimensions measured by city sizes (population and area) indicate the equality of the city-size distribution. The higher fractal dimension value of a urban hierarchy suggests smaller difference between two immediate classes, while the lower dimension value suggests the larger interclass difference. For the fractal dimension measured by city population

Generally speaking, for the cities in the real world, we have

Many evidences show that urban evolution complies with some empirical laws which dominate physical systems. The economic institution, system of political organization, ideology, and history and phase of social development in PRC are different to a great extent from those in USA. However, where the statistical average is concerned, the cities in the two different countries follow the same scaling laws. Of course, the similarity at the large scale admits the differences at the small scale, thus the stability at the macrolevel can coexist with the variability at the microlevel of cities [

All in all, the hierarchy of cities can be described with three exponential models, or four power-law models including Zipf’s law. The exponential models reflect the “longitudinal” or “vertical” distribution across different classes, while the power-law models reflect “latitudinal” or “horizontal” relation between two different measurements (say, urban area and population size) (see Appendix

Urban hierarchy represents the ubiquitous structure frequently observed in physical and social systems. Studies on the cascade structure with fractal properties will be helpful for us to understand how a system is self-organized in the world. In the spatiotemporal evolution of cities in a region, there are at least two kinds of the unity of opposites. One is the global target and local action, and the other is determinate rule (at the macro level) and the random behavior (at the micro level). To interpret the mechanism of urban evolution and the emergence of rank-size patterns, a deck-shuffling theory is proposed here. A regional system (a global area) consists of many subsystems (local areas), and each subsystem can be represented by a card. The card-shuffling process symbolizes the introduction of randomicity or chance factors into evolution of regions and cities. The model of shuffling cards is only a metaphor, and the logical relation between this model and real systems of cities is not very significant.

Suppose there are many blank cards. We can play a simple “game” step by step as follows (Figure

A sketch map of shuffling cards of network of cities. (Note: The sizes of cities conform to the rank-size rule, equation (^{n}

Blank cards

Ordered network

Shuffling cards

Spatial rearrangement.

For simplicity, let the number of cards in the array be

Then draw a hierarchy of “cities” to form a regular network with cascade structure in light of (

Note that these cards are not blank and form a deck now. Unfix and mix these cards together, then riffle these cards again and again at your pleasure (Figure

Take out cards at random one by one from the deck, and place them one by one to form a

Examining these shuffled cards in array, you will find no ordered network structure of “cities” anymore. The physical structure of the network of “cities” may not follow the exponential laws and power laws yet. To reveal the hidden order, we must reconstruct the hierarchy according to certain scaling rule. Thus the physical cascade structure changes to the mathematical cascade structure, and then the regular physical hierarchy can be replaced with the dummy hierarchy (Table

Comparison of hierarchy model between the cases before and after shuffling cards.

Item | Before shuffling cards | After shuffling cards |
---|---|---|

Mathematical cascade structure | Exist | Keep |

Physical cascade structure | Exist | Fade away |

Fractal property | Regular fractal | Random fractal |

Zipf distribution | Exist | Keep |

Network type | Real hierarchy | Dummy hierarchy |

Spatial disaggregation and network growth (the first four steps) (by referring to [

Two (2^{1})

Four (2^{2})

Eight (2^{3})

Sixteen (2^{4})

After shuffling “cards,” the regularity of network structure will be lost, but the rank-size pattern will keep and never fade away. In this sense, Zipf’s law is in fact a signature of hierarchical structure. This can be verified by the empirical cases. Since the scaling relation of size distributions often breaks down when the scale is too large or too small [

The rank-size patterns of the US cities in 2000 and the PRC cities in 2005.

The idea from shuffling cards can be employed to interpret urban phenomena such as the relationship between central place models and spatial distribution of human settlements in the real world. The central place models suggest the ideal hierarchies of human settlements with cascade structure [

A schematic diagram of symmetry breaking and reconstruction of network of cities.

The process of shuffling cards is a metaphor of symmetry breaking of

A hierarch with cascade structure can be treated as a “mathematical transform” from real cities to the regular cities (Figure

Hierarchical structure as a knowledge link between the apriori-ordered network and the random distribution of actual cities (the first four classes). (a)Random distribution. (b) Hierarchy. (c) Ordered network.

In urban studies, Zipf’s law includes three forms: the first is the one-parameter Zipf’s law, that is, the pure form of Zipf’s law; the second is the two-parameter Zipf’s law, that is, the general form of Zipf’s law; the third is the three-parameter Zipf’s law, that is, the more general form of Zipf’s law [

Zipf’s law used to be considered to contradict the hierarchy with cascade structure. Many people think that the inverse power law implies a continuous distribution, while the hierarchical structure seems to suggest a discontinuous distribution. In urban geography, the rank-size distribution of cities takes on a continuous frequency curve, which is not consistent with the hierarchical step-like frequency distribution of cities predicted by central-place theory [

Therefore, the hierarchical models are mainly based on the idea of statistical average rather than reality or observations. In terms of statistical average, the rank-size distribution can always be transformed into a hierarchy with cascade structure. However, the traditional hierarchical structure predicted by central place theory cannot be transformed into the rank-size distribution. On the other hand, the size distributions in the real world support Zipf’s law and the hierarchical model based on statistical average instead of the step-like hierarchical distribution. Consequently, a conclusion can be drawn that the absolute hierarchy should be substituted by the statistical hierarchy associated with the rank-size distribution. Precisely based on this concept, the metaphor of shuffling cards is proposed to interpret the urban evolution coming between chaos and order.

To sum up, Zipf’s law is a simple rule reflecting the ubiquitous general empirical observations in both physical and human fields, but the underlying rationale of the Zipf distribution has not yet been revealed. This paper tries to develop a model to illuminate the theoretical essence of the rank-size distribution: the invariable patterns of evolutive network or hierarchy. The hierarchy with cascade structure provides us with a new way of looking at the rank-size distribution. The hierarchy can be characterized by both exponential laws and power laws from two different perspectives. The exponential models (e.g., the generalized

The theory of shuffling cards is not an underlying rationale, or an ultimate principle. As indicated above, it is a useful metaphor. The idea from cards shuffling is revelatory for us to find new windows, through which we can research the mechanism of the unity of opposites such as chaos and order, randomicity and certainty, and complexity and simplicity. A conjecture or hypothesis is that complex physical and social systems are organized by the principle of dualistic structure. One is the mathematical structure with regularity, and the other is the physical structure with irregularity or randomicity. The mathematical structure represents the

The longitudinal relations are the associations across different classes, while the latitudinal relations are the correspondences between different measures such as city population size and urban area. These relations can be illustrated with the following figure (Figure

A schematic diagram on the longitudinal relations and latitudinal relations of urban hierarchy (the first four classes).

If the size distribution of cities follows Zipf’s law, it will always conform to the Leimkuhler’s version of Bradford’s “law of scattering” [

This research was sponsored by the National Natural Science Foundation of China (Grant No. 41171129). The support is gratefully acknowledged. Many thanks to the anonymous reviewers whose interesting comments were helpful in improving the quality of this paper.