Unraveling the Rank-Size Rule with Self-Similar Hierarchies

Transcription

1 Unraveling the Rank-Size Rule with Self-Siilar Hierarchies Yanguang Chen (Departent of Geography, College of Urban and Environental Sciences, Peking University, 0087, Beiing, China. Eail: Abstract: Many scientists are interested in but puzzled by the various inverse power laws with a negative exponent such as the rank-size rule. The rank-size rule is a very siple scaling law followed by any observations of the ubiquitous epirical patterns in physical and social systes. Where there is a rank-size distribution, there will be a hierarchy with cascade structure. However, the equivalence relation between the rank-size rule and the hierarchical scaling law reains to be atheatically deonstrated and epirically testified. In this paper, theoretical derivation, atheatical experients, and epirical analysis are eployed to show that the rank-size rule is equivalent in theory to the hierarchical scaling law (the n principle). Abstracting an ordered set of quantities in the for {, /,, /k, } fro the rank-size rule, I prove a geoetric subdivision theore of the haronic sequence (k,, 3, ). By the theore, the rank-size distribution can be transfored into a self-siilar hierarchy, thus a power law can be decoposed as a pair of exponential laws, and further the rank-size power law can be reconstructed as a hierarchical scaling law. A nuber of ubiquitous epirical observations and rules, including Zipf s law, Pareto s distribution, fractals, alloetric scaling, /f noise, can be unified into the hierarchical fraework. The self-siilar hierarchy can provide us with a new perspective of looking at the inverse power law of nature or even how nature works. Key words: Zipf s law; rank-size rule; fractal; /f noise; alloetry; hierarchy. Introduction An attractive but ysterious phenoenon is that any types of data studied in the physical and social sciences can be approxiated with the well-known Zipf distribution (Zipf, 949). The nuerical relations between rank and size generally follow Zipf s law, and the scaling exponent (q)

2 is close to q in ost cases (Krugan, 996). Another siilar phenoenon is the /f β noise, the nuerical relation between frequency and spectral density always follow the inverse power law, and the spectral exponent (β) is often close to β. The rank-size distribution and the /ƒ-like noises occur widely in nature and huan systes, and they are associated with fractals (Bak, 996; Mandelbrot, 999). A nuber of scientists are interested and puzzled by the rank-size pattern and /f noise, which appear in any coplex systes. Despite nuerous studies, the theoretical essence of these ubiquitous epirical observations has not yet been brought to light so far and reains to be explored. The rank-size rule is often regarded as the special case of generalized Zipf s law related to the Yule distribution and the Pareto distribution. The Zipf distribution is in fact one of a faily of varied scaling relationships (Bettencourt et al, 007). If the scaling exponent (q) of the general Zipf distribution equals, the rank-size relationship is known as the rank-size rule (Knox and Marston, 006). The rule describes a certain rearkable statistical regularity and fors a source of considerable interest in any fields. A great nuber of physical and social phenoena including the distribution of city sizes in a nation, sizes of business firs, wealth aong individuals, lengths of rivers, areas of islands and islets, street hierarchies, particle sizes, and frequencies of word usage, satisfy the rank-size distribution epirically (Axtell, 00; Batty, 008; Berry, 96; Blank and Soloon, 000; Jiang, 009; Konopka and Martindale, 995; Marquet, 00; Rybski et al, 009; Sion and Bonini, 958; Turcotte, 997). If we can reveal the underlying rationale of the rank-size scaling relation, the related discrete power law probability distributions as well as /f noise will becoe ore understandable. In fact, any scientists study power laws, but little knows that the siplest approach to researching a power law is a pair of exponential laws. The key is the self-siilar hierarchy. Davis (978) once ade an interesting discovery about hierarchies of cities. If we group the city sizes around the world into different classes according to the geoetric sequence with the coon ratio equal to, the city nubers in these classes also approxiately for a geoetric sequence and the coon ratio approaches. The size-nuber relation of urban hierarchies is known the n rule (n0,,, ). The n principle is a theoretical window to probe into the rank-size rule and the siilar power law distributions. This paper will discuss the following questions. First, where there is a rank-size rule, there is a n rule and vice versa. The n rule is actually an equivalent of

3 the rank-size rule. Second, the n rule can be generalized to the 3 n rule, the 4 n rule, and so on. Third, the hierarchies following the n rule possess fractal structure and can be used to investigate any observations of the ubiquitous epirical patterns such as fractals and /f noise. Hierarchy is frequently observed within the natural living world as well as in social institutions, representing a for of organization of coplex systes without characteristic scales (Puan, 006). An interesting discovery of this study is that the self-siilar hierarchy is a agic fraework, the acro property of which is independent of the eleents at the icro level. The reainder of this article will be structured as follows. First, the rank-size rule will be abstracted as a haronic sequence, fro which the n rule, 3 n rule, 4 n rule, and generally, the n rule ( refers to natural nuber) will be atheatically derived, and thus the geoetric subdivision theore of haronic sequence is proved. In this process, the scaling relation between rank and size are decoposed as two exponential laws, and then reconstructed as a hierarchical power law. Then, three typical atheatical experients and an epirical case are eployed to confir the theoretical derivation and hypothesis. Finally, the hierarchy odels are generalized to encopass a nuber of ubiquitous general epirical observations. Matheatical derivation (odel). Rearrangeent of rank-size distribution One of the typical rank-size distribution phenoena is cities, the things failiar to us. In urban studies, if the population size of a city is ultiplied by its rank, the product will probably equal the population of the highest ranked city. If so, we will say the cities follow the rank-size rule. Matheatically, the rule states that the population of a city ranked k will be /kth of the size of the largest city P (refer to Oxford Dictionary of Geography). The rule can be expressed as P. () k P k For siplicity, let P unit. Then the rank-size rule can be abstracted as a haronic sequence such as {/k}, where k,, 3,, and equation () suggests a special /k distribution. ow, group the haronic sequence into different classes in ters of geoetric sequence. Thus, a hierarchy of cities will be constructed by taking a priate city ( 0 ), generating next order cities, then, and 3, and so on ( is a positive integer). For exaple, for 3, the hierarchy of nubers indicating 3

4 city sizes is illustrated in Figure, which only shows the first four classes as a scheatic diagra. Class Class / /3 /4 Class 3 /5 /6 /7 /8 /9 /0 / / /3 Class 4 /4 /5/6/7/8/9/0 ///3/4/5/6/7/8/9/30/3/3/33/34/35 /36/37/38/39 /40 Figure The self-siilar hierarchy of cities based on the haronic sequence fro the ideal rank-size distribution (The first 4 classes for 3) The city nubers in the hierarchy can be expressed as an exponential function. According to the ode of sequence subdivision, the nuber of order, f, can be defined as f, () frf where f. Obviously, the interclass nuber ratio, r f, equals the coon ratio,, that is r f f f. (3) What interests us is the average value of each class. If we can prove that the su of nubers in each level (S ) approaches a constant asyptotically, that is S f P constant, then the average size of different classes (P ) will decay exponentially, i.e., P constant/ -. This suggests that the rank-size rule is atheatically equivalent to the n rule. In this instance, the rank-size distribution will be converted into a hierarchy with cascade structure and fractal diension.. The geoetric subdivision theore of the haronic sequence The above question can be abstracted as a atheatical proble to be proved, and the proof is very siple. According to the rule of subdivision of haronic sequence based on geoetric proportion, the suation of nubers in each class can be expressed as 4

5 5 0 ) ( ) ( ) ( ) ( S L, (4) where S is the nuber su of order,,, 3, M,, 3, 4, 0,,,, - -. Here M is the largest nuber of class order. Matheatical experients show that the su of nubers in each level will approach ln as M (see Subsection 3.). If we can prove S li ln, (5) then the above proble will be resolved, and the conclusion can be obtained that the rank-size rule can be theoretically led to the n rule. By equation (4), the nuber su of each class can be rewritten in the for 0 ) ( ) ( 0 S L. (6) Suppose the function f(x)/(x) is defined within the interval [0, -]. Using the fractions ) (,, 0, 0 L to divide the closed interval [0, -] into - - equal parts with a length of (-)/( - -). Further defining x, (7) we have Δ x. (8) Thus the su of the th class can be expressed as Δ 0 0 li li li x x S, (9) Apparently, the final result of suation is

6 li S li 0 f ( x ) Δx 0 dx x [ ln( x) ] ln 0. (0) This indicates S is asyptotically close to ln as, and the proof of equation (5) is coplete. The atheatical proof gives the geoetric subdivision theore of the haronic sequence: If we group a haronic sequence {/k} into different classes, and the aount of nubers in each class for a geoetric sequence such as 0,,,,, then the su of nuerical value in each class asyptotically approaches a constant ln. An exponential law of average size can be derived fro the geoetric subdivision theore. Defining the average size in the th class as P S /f yields S f P. According to equation (5), under the liit condition, we have f P ln. () This iplies a special inverse power law f (ln)p -. In the light of equation (), the average size ratio of adacent classes is r P f li, () P f p in which the coon ratio r p denotes the size ratio. The rank-size relationship suggests a fractal structure (Mandelbrot, 983; Chen and Zhou, 006), and the siilarity diension of the hierarchy is ln r D ln r f p ln f ln P / f / P ln ln, (3) An exponential odel can be deduced fro equation () by recursion, and the result is P P / P rp. (4) So far, the inverse power law, equation (), has been decoposed as a pair of exponential laws: one is the nuber law, equation (), and the other is size law, equation (4). If the forer is defined, we will have the latter, and vice versa. The exponential functions can be tered the n rule of hierarchies. When, the exponential laws is ust the n rule (Davis, 978). Fro equations () and (4) follows that f P D μ, (5) 6

7 where μf P refers to the proportional coefficient, and Dlnr f /lnr p to the fractal paraeter defined by equation (3). This suggests that the rank-size scaling law of size distributions has been transfored into the size-nuber scaling law of hierarchical structure. In theory, if and D as given, then μ f P ln. However, coparing equation (5) with equation () shows that μs ln. This iplies that if the haronic sequence is rearranged by a geoetric sequence, the fractal structure of the hierarchical systes coes into being gradually. Fractal diension is a paraeter under liit condition, and the first class or the largest city always akes an exception. This is consistent with the definition of fractals (Mandelbrot, 983). In practice, if we apply the scaling law to the real hierarchies of cities, the data points of size vs nuber ay depart fro the scaling range on the log-log plot when is very sall (esp. ). ow, the geoetric subdivision theore of the haronic sequence can be re-expressed as follows. If a haronic sequence {/k} is arranged into a hierarchy with cascade structure in ters of a geoetric sequence, 0,,,,, then the average values in different classes will decay in a negative exponential way. This theore indicates one of the agic properties of the haronic sequence. 3 Matheatical experients and epirical evidence (result) 3. Matheatical experients A nuber of atheatical experients can be perfored to verify the geoetric subdivision theore of the haronic sequence, and thus to support the equivalent relationship between the rank-size distribution and the self-siilar hierarchical structure. The atheatical experient is so siple that we can carry out it with MS Excel. Let s take city-size distribution as an exaple, and let P unit to represent the size of the largest city. Under the ideal condition, the city sizes for an haronic sequence such as [, /, /3,, /k, ] in ters of equation (). If as given, then the hierarchy of cities is as follows: []; [/, /3]; [/4, /5, /6, /7]; ; [/( - ), /( - ),,/( -)];. The coon ratio of city nubers in different classes is r f f /f, where,,, M, and the positive integer M approaches infinity in theory. In these experients, we can take M0, which is enough to show the hierarchical regularity. 7

8 ow, let s see the su of city sizes in each class. For the first class, the su is ; for the second class, the su is // ; for the third class, the su is /4/5/6/ , and so on. When the class order becoes larger and larger, the su is closer and closer to ln() The average size of each class P approaches ln()/ -, so the average size ratio r p P /P approaches r f f /f. Thus the fractal diension goes near D. For the first 0 classes, the results are listed in Table and displayed in Figure (a). The estiated value of the fractal diension is about D If M is large enough, or the first class is reoved as an outlier in the least squares coputation, the diension value will be adequately close to. Table The results of atheatical experients by converting the haronic sequence based on the rank-size rule into the geoetric sequences (exaples) Coon ratio (r f ) 3 Class () City nuber (f ) Total size (f P ) Average size (P ) Size ratio (r p ) M M- ln() ln()/( M- ) M 3 M- ln(3) ln(3)/(3 M- ) 3 8

9 M 4 M- ln(4) ln(4)/(4 M- ) 4 The second typical experient is for 3, and the hierarchy is []; [/, /3, /4]; [/5, /6,, /3];. The organized ode and hierarchical structure are illustrated in Figure. For the first class, the su is ; for the second class, the su is //3/4.0833; for the third class, the su is /5/6 /3.0968, and so on. The su approaches ln(3).0986 rapidly, and the average size runs to P ln(3)/3 - swiftly. As a result, the average size ratio r p P /P tends to r f f /f 3. For the first 0 classes, the sequences are listed in Table and displayed in Figure (b). The estiated value of the fractal diension is about D.005, very close to the expected value. The third experient is for 4, and the hierarchy is []; [/, /3, /4, /5]; [/6, /7,, /];. The process can be understood by the siilar approach to the above ones, and the results are listed in Table and displayed in Figure (c). The estiated value of the fractal diension is about D.055, near. The rest, i.e., for 5, 6, 7,, ay be deduced by analogy. Theoretically, is an arbitrary positive integer. Of course, all these experient results are approxiate since only the first 0 classes are taken into account. The hierarch of cities can be described with the size-nuber inverse power law. For exaple, for 3, a least squares calculation based on the first 0 classes yields f ˆ P The goodness of fit is R (Figure (b)). The power law can be decoposed as two exponential laws. The first odel is ad hoc given as f 3 -, while the second one based on the first 0 classes is such as 9

10 ˆ.098. P 3.579e The goodness of fit is R , and the fractal diension is estiated as D ln(3)/ln(.987).0053, very close to the value fro the power function. a. b. 3 c. 4 0

11 Figure The scaling relations between city nubers and average sizes of different classes in the ideal self-siilar hierarchy of cities 3. Epirical evidence Aong various cases, cities in a given region are the typical phenoena which follow the rank-size scaling law (Batty, 008; Berry, 96; Carroll, 98; Gabaix, 999). If the region studied is large enough to cover the territorial scope influenced by the largest city, the city size distribution will coply with Zipf s law. If the scaling exponent is about q, we have a rank-size pattern. The cities of the United States of Aerica (USA) will be eployed to testify the odels given above. A question is that the data of city population do not refer to units defined in strictly coparable ters. In urban geography, there three basic concepts used to define urban areas and populations, naely, city proper (CP) without suburb, urban aggloeration (UA) with suburb, and etropolitan area (MA) ore suburbs (Davis, 978). o atter what kind of urban area is considered, the city sizes satisfy the Zipf distribution in the case of very large scale. However, only the size distribution of the population defined within UA bears a scaling exponent near q, and on the whole, UA corresponds to concept of urbanized area. Figure 3 The rank-size pattern of the US cities by the population within urbanized area (000) The data of this case study coe fro the US Census Bureau, but only the 45 cities with population in urbanized area are over 50,000 in 000 and the data are available at internet

12 ( The Zipf odel of these cities is such as ˆ.5 P k k, The goodness of fit is about R 0.989, the scaling exponent is about q.5, and the corresponding fractal diension can be estiated as D/q 0.9. The shortcoing of the US case lies in two aspects: one is that the saple is not large enough (only 45 cities), the other is that the rank-size pattern is not so satisfying (Figure 3). Despite these flaws, the exaple is enough for us to illustrate the equivalence relation between the rank-size rule and the n rule on self-siilar hierarchical structure. Corresponding to the atheatical experients, three kinds of self-siilar hierarchies will be constructed. Taking nuber ratio r f, 3, 4, we can group the cities into different classes according to the n rule, 3 n rule, and 4 n rule. The results, including city nuber, total population, and average population size in each class and size ratio between any two iediate classes, are listed in Table. In each hierarchy, two classes, i.e., top class and botto class, are special and can be considered to be exceptional values. As can be seen in Tables and, the first class always gets departure fro the scaling range; the last class is always a lae duck class due to undergrowth or absence of data of sall cities (Davis, 978). For exaple, for, the city nuber in the 9th class is expected to be f , but only 97 city data are available. In this instance, the first and last classes can be treated as outliers and should be reoved fro the least squares calculations. The data points of the edian part for a scaling range on each log-log plot. Table The results of epirical analysis by grouping the US city sizes (by urbanized area) in 000 into different classes according as the n, 3 n, and 4 n rules Coon ratio (r f ) Class () City nuber (f ) Total population (f P ) Average size (P ) Size ratio (r p ) *

13 * * Source: The original data coe fro the US Census Bureau, available at: * ote: The last class of each hierarchy is actually a lae-duck class of Davis (978). a. b. 3 3

14 c. 4 Figure 4 The scaling relations between city nubers and average sizes in different classes of hierarchies of USA cities by the population in urbanized area in 000 (exaples) By eans of regression analysis, three power law odels can be built (Figure 4). For, the scaling relation is f ˆ P The goodness of fit is about R 0.989, the fractal diension is estiated as D.009. The average size ratio within the scaling range is about r p.03, which is very close to r f. For 3, the scaling relation is f ˆ P The goodness of fit is about R 0.988, the fractal diension is about D.030. The average size ratio within the scaling range is estiated as r p.933, near r f 3. For 4, the scaling relation is f ˆ P The goodness of fit is about R 0.986, the fractal diension is estiated as D.055. The average size ratio within the scaling range is about r p 3.79, close to r f 4. Obviously, if we change the nuber ratio r f, the average size ratio r p will change with it. However, the fractal diension keep around D, which suggests the scaling exponent q. If the saple were large enough, we would take 5, 6, 7,. But for out epirical analysis, this exaple is sufficient to support the hierarchical scaling odels. As expected, the hierarchies can 4

15 be described with three pairs of exponential laws. For, the nuber law is known as f (/)e ln(), the size law can be estiated as Pˆ e The goodness of fit is about R 0.99, and the fractal diension can be indirectly estiated as D ln()/ For 3, the nuber law is f (/3)e ln(3). If we reove the first class as an outlier, the size law can be estiated as Pˆ e The goodness of fit is about R 0.993, and the fractal diension is about D ln(3)/ For 4, the nuber law is f (/4)e ln(4). After reoving the first class as an exceptional value, the size law can be estiated as Pˆ e The goodness of fit is about R 0.99, and the fractal diension is about D ln(4)/ These exponential odels and the corresponding power odels can be transfored into one another. Fro these exponential functions, the fractal paraeters and the scaling exponents of the hierarchies can be evaluated approxiated. All these paraeter values of the self-siilar hierarchical structure are close to the corresponding paraeters values of the rank-size distributions. This suggests that the hierarchy with cascade structure can be described with exponential laws and power laws fro different angle of views respectively. The n rule can be applied to the cities in other countries such as Britain, France, and Geran (ore epirical evidence is shown in an attached aterial). 4 Theoretical explanation and generalization (discussion) The atheatical experients and epirical analysis lend support to the inference that the rank-size rule is asyptotically equivalent to the n rule as is large enough. But what is the significance of this study? The first is physical explanation, and the second the theoretical generalization. There are various explanations but no convincing explanation for the rank-size rule, especially in urban studies, despite the great frequency with which it has been observed. These years, soe interesting explanations appeared in different fields (Bettencourt et al, 007; Ferrer i 5

16 Cancho et al, 005; Ferrer i Cancho and Solé, 003). The rank-size distribution sees to be associated with axiu entropy odels (Mora, 00). In fact, the self-siilar hierarchy is a siple link between the axiu entropy principle and the rank-size rule. First, equations () and (4) can be derived using the entropy-axiizing ethod (Chen and Liu, 00; Curry, 964). Second, cobining equation () with equation (4) yields equation (5). Third, according to the geoetric subdivision theore of haronic sequence, equation (5) is equivalent to equation (). So, this suggests that the rank-size pattern coes fro the process of entropy axiization, which is very iportant in urban odeling (Wilson, 968; Wilson, 97). The notion of entropy axiization of huan systes is different fro the concept of entropy increase in therodynaics. A preliinary study suggests that the axiu entropy iplies a coproise between the equity for individuals and the efficiency of the whole where social and econoic systes are concerned. The hierarchical structure can be eployed to unify any different scaling rules and phenoena, and thus for an entire logic fraework for us to understand nature. The rank-size rule suggests a agic sequence, which thus suggests a agic fraework a hierarchy with cascade structure. In a sense, the self-siilar hierarchy can be copared to the hat of a agician, by which nature produces any rules and patterns such as the rank-size rule, the n rule, the 3 n rule, Pareto distribution, Zipf s law, fractals, alloetric growth, and /f noise (Figure 5). The relationships between siplicity and coplexity of physical and social systes can been seen fro these principles and patterns. Aong these, fractals, /f noise, Zipf s law, and scale-free network are regarded as the ubiquitous general epirical observations (Bak, 996; Barabási, 00). Another agic property of this sequence is the latent Pareto distribution. Defining a size scale as ξp k / - (k - ;, 3, 4, ;,, 3, ), the nuber of the cities with size greater than or equal to ξ is ust (P ξ) -. Apparently, we have an inverse power-law relation such as ( P ξ ) ξ. (6) This suggests a special Pareto distribution with scaling exponent equal to. The scaling relation also deonstrates the fractal nature of the hierarchical structure and the rank-size distribution. 6

17 Zipf s law (P k P /k q ) q q The rank-size distribution The n rule (, 3, 4, 5, 6 ) The Zipf distribution The generalized n rule The Pareto distribution Alloetric scaling Hierarchical scaling relations with fractal diension /f β noise (β) Exponential laws Entropy axiization Figure 5 The scaling relations of the hierarchy with cascade structure and related ubiquitous general epirical observations [ote: The /f noise is the special case of /f β noise, and β is the spectral exponent.] The rank-size rule is the special case of the generalized Zipf s law, which can be expressed as P k q P k, (7) where q refers to the Zipf exponent. An interesting discovery is as follows. If q, Zipf s law will be equivalent to and only to the generalized n rule with r f and r p q (This will be deonstrated in a copanion paper). If q, Zipf s law can be reduced to the rank-size rule, which thus can be decoposed as two exponential laws with r f r p (, 3, 4, ). Furtherore, an alloetric scaling relation can be derived fro equation (4). For exaple, city size can be easured not only by population but also by urban area. Substituting average area of cities in the th class A for the urban population size P in equation (4) suggests the law of alloetric scaling such as b A ap, (8) where a refers to the proportionality coefficient, and b to the scaling exponent (Chen, 00). 7

18 The scaling decoposition and reconstruction of the rank-size rule can be generalized to the /f noise, which is defined by the following frequency-spectru relation S f, (9) f where f denotes frequency, and S f is the spectral density, the sybol eans is proportional to. The frequency can be written as f k/t, where k,, 3,, T refers to the length of saple path. This suggests that equation (9) shares the identical for with equation (). Thus equation (9) can be decoposed as two exponential laws for the hierarchy of spectral densities. The nuber law is the sae as equation (), while the density law is E E r e, (0) where E is the average spectral density in the th class, E is the proportionality constant, and r e E /E denotes the average density ratio. Based on equations () and (0), the scaling law can be reconstructed in the for f E, () η σ where ηf E is the proportional coefficient, and σlnr f /lnr e is the scaling exponent. A hierarchy is always associated with a network in space (Batty and Longley, 994). Let s take cities as exaple to illustrate it. Coparing the experient results listed in Table and displayed in Figure shows that when 3, the subdivision effect of the rank-size distribution is the best for the liited classes. The hierarchy of 3 corresponds to the central place network with K3 (Christaller, 966). A central place syste is a hierarchy as well as a network following the scaling laws given above. A central place network can be characterized by the nuber law, equation (), the size law, equation (4), and the hierarchical scaling law, equation (5) (Chen and Zhou, 006). Maybe the n rule provides us with a new angle of view to understand the notion of rank clocks (Batty, 006). At the icro-level, the rank clocks show cities rising and falling in size at any ties and on any scales. However, at the acro-level, the hierarchical structure of national city-size distributions is always very stable and change little. If we regard different size classes as different energy levels of cities, change of status of an individual city is siilar to transition of the city s energy level. However, when the Zipf exponent q, the th level often has no essential difference fro the ()th level. In particular, no atter how individual cities change their 8

19 positions, the total energy in each class is conservative and tries to keep constant. This suggests that hierarchical structure is independent of coponents, and a syste is ore iportant than its eleents. On the other hand, no atter what nuber ratio (, 3, 4, ) is taken, the nuber one never leave the top class, and the nubers two and three never leave the second class. Therefore, the first one or ore cities always ake an exception, and the top class often represents an extraordinary level. 5 Conclusions This paper is involved with the ethod of applied atheatics, ideas fro physics, and epirical evidences of geography. I a not sure to which classification it should belong, but I know it is iportant because this work shows a new approach to exploring varied scaling laws, which appear in any fields and have caused a broad research interest in the scientific circle. Especially, I proved the geoetric subdivision theore of haronic sequence. The significance of this theore is as follows. Firstly, the rank-size rule is proved to be equivalent in theory to the n principle. This suggests that the rank-size pattern ight be a signature of hierarchical structure. The rank-size distribution with scaling exponent equal to can be transfored into a self-siilar hierarchy with fractal diension D according to the n rule, 3 n rule, 4 n rule, and so on. Generally, we have the n rule for the rank-size distributions. In this case, the scaling relation between rank and size can be decoposed as a pair of exponential laws and then reconstructed as the hierarchical scaling relation between size and nuber. Thus we can study a power law through a pair of exponential laws, which are sipler than a power law. What is ore, we can get inforation of the rank-size distribution which can not be directly obtained through Zipf s law. Secondly, the rank-size pattern suggests a special syetrical structure. The rank-size rule is the special case of Zipf s law with scaling exponent varying fro 0 to infinity. The rank-size rule is equivalent to the n rule only for the rank-size distribution with negative exponent. If the scaling exponent gets departure fro, the size distribution can only be converted into the hierarchy following the n principle. This is revealing for us to understand the special scaling exponent q. The n rule indicates that the hierarchy has no characteristic coon ratio, and the 9

20 scaling exponent iplies an arbitrary positive integer as coon ratios for the hierarchy. Thus, different size levels can share the equal political status in a syste except the first three ones. Thirdly, the self-siilar hierarchy is a agic fraework of nature with any singular properties. The first is that it can act as a link between exponential laws and power laws, the second, any basic rules can be derived fro this structure, and the third, it can be associated with a nuber of ubiquitous epirical observations. Especially, both the /k distribution and /f noise suggest the n principle in the self-organized systes. A aority of the conclusion drawn fro the rank-size rule can be generalized to the /f noise. In short, the hierarchical fraework provides us with a revealing angle of view to understand how nature works. Moreover, the agic property of the haronic sequence reains and is worthy to be atheatically studied in future. Acknowledgeents: This research was sponsored by the ational atural Science Foundation of China (Grant o ). The support is gratefully acknowledged. I would like to thank y atheatician friend, Juwang Hu, for assistance in atheatical deonstration. References Axtell RL (00). Zipf distribution of U.S. fir sizes. Science, 93: Bak P (996). How ature Works: the Science of Self-organized Criticality. ew York: Springer-Verlag Barabási A-L (00). Linked: The ew Science of etwork. Massachusetts: Persus Publishing Batty M (006). Rank clocks. ature, 444: Batty M (008). The size, scale, and shape of cities. Science, 39: Batty M, Longley PA (994). Fractal Cities: A Geoetry of For and Function. London: Acadeic Press Berry BJL (96). City size distributions and econoic developent. Econoic Developent and Cultural Change, 9(4): Bettencourt LMA, Lobo J, Helbing D, Kühnert C, West GB (007). Growth, innovation, scaling, and the pace of life in cities. PAS, 04: Blank A, Soloon S (000). Power laws in cities population, financial arkets and internet sites: 0

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