 
1
A Scaling Approach to Evaluating the Distance Exponent of 
the Urban Gravity Model 
 
Yanguang Chen, Linshan Huang 
(Department of Geography, College of Urban and Environmental Sciences, Peking University, Beijing 
100871, P.R. China. E-mail: chenyg@pku.edu.cn) 
 
Abstract: The gravity model is one of important models of social physics and human geography, 
but several basic theoretical and methodological problems remain to be solved. In particular, it is 
hard to explain and evaluate the distance exponent using the ideas from Euclidean geometry. This 
paper is devoted to exploring the distance-decay parameter of the urban gravity model. Based on 
the concepts from fractal geometry, several fractal parameter relations can be derived from scaling 
laws of self-similar hierarchies of cities. Results show that the distance exponent is just a scaling 
exponent, which equals the average fractal dimension of the size measurements of the cities within 
a geographical region. The scaling exponent can be evaluated with the product of Zipf’s exponent 
of size distributions and the fractal dimension of spatial distributions of geographical elements such 
as cities and towns. The new equations are applied to China’s cities, and the empirical results accord 
with the theoretical expectations. The findings lend further support to the suggestion that the 
geographical gravity model is a fractal model, and its distance exponent is associated with a fractal 
dimension and Zipf’s exponent. This work will help geographers understand the gravity model using 
fractal theory and estimate the distance exponent using fractal modeling. 
 
Key words: gravity model; Zipf’s law; central-place network; hierarchy of cities; fractal dimension; 
allometric scaling 
 

 
2
1 Introduction 
A set of gravity models have been applied to explain and predict various behaviors of spatial 
interactions in many social sciences. Among this family of models, the basic one is the gravity 
model of migration based on an inverse power-law distance-decay effect, which was proposed by 
analogy with Newton’s law of gravitation. It can be used to describe the strength of interaction 
between two places (Haynes and Fotheringham, 1984; Liu and Sui et al, 2014; Rodrigue et al, 2009; 
Sen and Smith, 1995). According to the model, any two places attract each another by a force that 
is directly proportional to the product of their sizes and inversely proportional to the bth power of 
the distance between them, and b is the distance-decay exponent, which is often termed distance 
exponent for short (Haggett et al, 1977). The size of a place can be measured with appropriate 
variables such as population numbers, built-up area, and gross domestic product (GDP). Recently, 
the gravity model has been employed to study the attractive effect of various new-fashioned human 
and physical activities by means of modern technology (Balcan et al, 2009; Goh et al, 2012; Kang 
et al, 2012; Kang et al, 2013; Jung et al, 2008; Lee et al, 2014; Liang, 2009; Liu and Wang et al, 
2014; Simini et al, 2012). The model is empirically effective for describing spatial interactions, but 
it is obstructed by two problems on the distance exponent, b. One is the dimensional problem, that 
is, it is impossible to interpret the distance exponent in light of Euclidean geometry (Haggett et al, 
1977; Haynes, 1975). The other is the algorithmic problem, namely, it is hard to estimate the 
numerical value of the distance exponent (Mikkonen and Luoma, 1999). 
The first problem can be readily solved by using ideas from fractals. Fractal geometry was 
developed by Mandelbrot (1982), and it is a powerful tool for spatial analysis in geographical 
research (Batty, 2005; Batty and Longley, 1994; Frankhauser, 1994; Frankhauser, 1998). It can be 
proved that the distance exponent is a fractal parameter indicating a space dimension (Chen, 2015). 
However, how to evaluate the distance exponent is still a difficult problem that remains to be solved. 
A new discovery is that the distance exponent is associated with the Zipf exponent of the rank-size 
distribution and the fractal dimension of a self-organized network of urban places (Chen, 2011). If 
we examine the spatial interaction between cities and towns in a region, the distance exponent of 
the gravity model is just the product of the Zipf exponent of the city-size distribution and the fractal 
dimension of the corresponding central-place network. The formula has been derived by combining 

 
3
central-place theory and the rank-size rule (Chen, 2011). A pending problem is how to use the 
formula to estimate the distance exponent for gravity analysis in empirical studies. 
This problem can be solved using a self-similar hierarchy with cascade structure. On the one hand, 
the rank-size distribution is mathematically equivalent to the hierarchical structure (Chen, 2012a; 
Chen, 2012b). On the other, the hierarchical structure and the network structure represent two 
different sides of the same coin (Batty and Longley, 1994). In short, the rank-size distribution and 
the network structure can be linked with one another by hierarchical structure. Thus, the Zipf 
exponent of the rank-size distribution and the fractal dimension of spatial network can be associated 
with each other through hierarchical scaling, which suggests a new approach to estimating the 
distance exponent. This study is mainly based on urban geography and is devoted to exploring the 
methods for distance exponent estimation. The rest of this article is organized as follows. In Section 
2, a scaling relation will be derived from the gravity model based on the inverse power law, and 
spatial scaling will be transformed into hierarchical scaling in terms of the inherent association 
between hierarchy and network. Three scaling approaches to estimating the distance exponent will 
be proposed. In Section 3, Chinese cities will be employed to make an empirical analysis by means 
of census data and spatial distance data. In Section 4, several related questions will be discussed. 
Finally, the discussion will be concluded by summarizing the main points in the work. 
2 Theoretical results 
2.1 Breaking-point relation 
The new method can be derived from the well-known breaking-point formula based on the basic 
gravity model. Consider three locations within a geographical region, i, j, and x, and x falls between 
i and j. If there are cities or towns at these locations, the “mass” of the settlements can be measured 
by population or other size quantity. Thus the gravity can be expressed as 
i
x
ix
b
ix
PP
I
G L

,                                  (1) 
j
x
jx
b
jx
P P
I
G L

,                                 (2) 
where Iix denotes the gravity between locations i and x, Ijx denotes the gravity between location j 

 
4
and x, Lix indicates the distance between locations i and x, Ljx indicates the distance between 
locations j and x, Pi, Pj, and Px are the size of the cities at locations i, j, and x, G refers to the gravity 
coefficient, and b to the distance exponent indicative of spatial friction. Combining equations (1) 
and (2) yields 
/
/
b
i
i
ix
b
j
j
jx
P
L
I
P
L
I

.                                 (3) 
Suppose that there exists a special location for x, where Iix=Ijx. In this case, eliminating x yields 
(
)b
i
ix
j
jx
P
L
P
L

,                                 (4) 
which is familiar to geographers. Chen (2015) demonstrated that equation (4) represents a fractal 
dimension relationship. Equation (4) is identical in form to the well-known breaking-point formula 
derived by Reilly (1931) and Converse (1949). By using the hierarchical scaling laws of urban 
systems, we can reveal the spatial meaning of the parameter b and find a new way of evaluating it. 
2.2 Distance exponent based on hierarchical scaling laws 
If a geographical region is large enough, the size distribution of cities within the region may be 
consistent with Zipf’s law. However, the laws of complex social and economic systems are not of 
spatio-temporal translational symmetry, and city-size distributions do not follow the only law (Chen, 
2016). Sometimes, Zipf’s law is replaced by other mathematical laws such as Lavalette law 
(Cerqueti and Ausloos, 2015). Zipf’s law is one of the well-known rank-size scaling law, indicating 
self-organized criticality of urban evolution (Bak, 1996; Chen and Zhou, 2008). Suppose that city 
development in a region fall into self-organized critical state. The general Zipf formula can be 
expressed as follows (Zipf, 1949) 
1
( )
q
P k
Pk


,                                  (5) 
where k refers to the rank of cities in descending order, P(k) to the size of the city of rank k, q denotes 
the Zipf scaling exponent, and the proportionality coefficient P1 indicates the size of the largest city 
in theory. City size is always measured with resident population. The inverse function of equation 
(5) is k = [P(k)/P1]-1/q, in which the rank k represents the number of cities with size greater than or 
equal to P(k). Reducing P(k) to P and substituting k with N(P), we have 

 
5
( )
p
N P
P



,                                 (6) 
which can be converted into the function of Pareto’s density distribution. This implies that Zipf’s 
law is equivalent in mathematics to Pareto’s law and that the Zipf distribution is theoretically 
equivalent to the Pareto distribution (Chen, 2014a). In equation (6), the coefficient η=P1p refers to 
the proportionality constant, and the power exponent p=1/q denotes the Pareto scaling exponent, 
indicating the fractal dimension of city rank-size distributions. The reciprocal relation between p 
and q is based on pure mathematical derivation. In empirical analyses, this relation proved to be 
replaced by p=R2/q, where R denotes Pearson correlation coefficient (Chen, 2016; Chen and Zhou, 
2003; Tan and Fan, 2004). 
The rank-size distribution is actually a signature of the self-organized hierarchy with cascade 
structure. It has been demonstrated that if the city-size distribution follows Zipf’s law, the cities can 
be organized into a self-similar hierarchy (Chen, 2012a; Chen, 2012b). The hierarchy of cities 
consisting of M classes (levels) in a top-down order can be described with the discrete expressions 
of two exponential functions as below: 
1
1 n
m
m
N
N r


,                                 (7) 
1
1 p
m
m
P
Pr 

,                                  (8) 
where m=1, 2, …, M refers to the order of classes in the hierarchy of cities (M is a positive integer), 
Nm and Pm denote the number of cities and average size of the cities in the mth class, N1 and P1 are 
the number and mean size of the cities in the top class, and rn=Nm+1/Nm and rp=Pm/Pm+1 are the 
number ratio and size ratio of cities, respectively. If rf=2 as given, then the value of rp can be 
calculated; if rp=2 as given, the value of rp can be derived (Chen, 2012b; Davis, 1978). According 
to the central-place theory propounded by Christaller (1933/1966), we have the third exponential 
model such as (Chen, 2011) 
1
1 l
m
m
L
L r 

,                                  (9) 
where Lm denotes the average distance between two urban places in the mth class, L1 is the average 
distance between the urban places in the top class, and rl=Lm/Lm+1 is the distance ratio of cities (the 
subscript of L is the number 1, and the subscript of r is the letter l). Here we assume there more than 
two cities at the first level of an urban hierarchy. If there are just two cities in the first class, L1 will 

 
6
refer to the distance between the two cities; if there is only one top-level city, L1 can be substituted 
by the radius of the equivalent circle of a study area. 
The fractal model and hierarchical scaling relations can be derived from the three exponential 
functions. From equations (7) and (8), the size-number hierarchical scaling relation can be derived 
as 
p
m
m
N
P



,                                 (10) 
which is equivalent to Pareto’s law, equation (6) [see Appendix 1]. The proportionality coefficient 
is μ=N1P1p, and the scaling exponent can be redefined as 
1
n
1
p
ln(
/
)
ln
ln(
/
)
ln
m
m
m
m
N
N
r
p
P
P
r




,                           (11) 
which is regarded as the similarity dimension of city-size distributions. It is in fact a ratio of two 
fractal dimensions (Chen, 2014a). From equations (7) and (9), a fractal model of central places is 
derived as 
D
m
m
N
L



,                                  (12) 
where π=N1L1D refers to the proportionality coefficient, and D to the fractal dimension of spatial 
distribution of urban places. The similarity dimension of fractals can be given by 
1
n
1
l
ln(
/
)
ln
ln(
/
)
ln
m
m
m
m
N
N
r
D
L
L
r




,                          (13) 
which is the fractal parameter of central place networks and can be termed the network dimension. 
From equations (8) and (9), a size-range allometric relation is derived as 
m
m
P
L


,                                   (14) 
where κ=P1L1-σ refers to the proportionality coefficient and σ to the allometric scaling exponent of 
population distribution. The allometric exponent can be given by 
p
1
1
l
ln
ln(
/
)
ln(
/
)
ln
m
m
m
m
r
P
P
L
L
r





,                            (15) 
which suggests that the allometric exponent is actually a fractal dimension of urban population. 
Using the theoretical framework of urban hierarchical scaling, we can derive a new equation for 
the distance exponent of the gravity model. Based on the hierarchical structure of urban systems, 
the breaking-point relation can be rewritten as 

 
7
1
1
(
)b
m
m
m
m
P
L
P
L



,                                (16) 
which is derived from equation (4) and can be regarded as a variant of equation (4) [see Appendix 
2]. Since rp=Pm/Pm+1, and rl=Lm/Lm+1, equation (16) can be expressed in the form 
p
l
b
r
r

.                                   (17) 
Thus we have 
p
1
1
l
ln
ln(
/
)
ln(
/
)
ln
m
m
m
m
r
P
P
b
L
L
r






.                          (18) 
This suggests that the distance exponent is equal to the allometric scaling exponent of city size and 
its spatial distribution. By equations (11), (13), and (15), we can get 
p
n
n
l
l
p
ln
ln
ln
ln
ln
ln
r
r
r
b
D
p
r
r
r
q





,                         (19) 
which suggests 
b
qD

.                                   (20) 
For the classical central-place systems, D→2. Thus, if q→1, we will have b→2. The special value 
b=2 is used in empirical analyses (Jung et al, 2008). The Zipf exponent can be calculated by rank-
size scaling analysis, and the network dimension can be computed by self-similar network analysis. 
Therefore, the distance exponent can be readily estimated by the parameter relationship shown in 
equation (20). 
2.3 Three approaches for evaluating the distance exponent 
New methods can be found for estimating the distance exponent of the urban gravity model. The 
Pareto scaling exponent has been proved to be a ratio of the fractal dimension of a network of cities 
to the average dimension of city population (Chen, 2014a). Accordingly, the Zipf scaling exponent 
is the reciprocal of this dimension ratio, that is 
p
1
D
q
p
D


,                                  (21) 
where D denotes the fractal dimension of a network of cities, and Dp represents the average value 
of the fractal dimensions of the urban population distribution of different cities within the city 
network(Chen, 2014a). The reciprocal relation q=1/p has been clarified above, the numerical 

 
8
relations between p, q and D, Dp should be explained in a few words. According to the principle of 
dimension consistency (Mandelbrot, 1982; Lee, 1989; Takayasu, 1990), we have a geometric 
measure relation as follows Nm=μPm^(-D/Dp), where μ is a proportionality coefficient. Comparing 
this equation with equation (10) yields p= D/Dp. Substituting equation (21) into equation (20) yields 
P
b
D

.                                    (22) 
In fact, in terms of the general geometric measure relation (Chen, 2015; Feder, 1988; Lee, 1989; 
Mandelbrot, 1982; Takayasu, 1990), equation (14) can be rewritten as 
p
l
1/
1/
D
D
m
m
P
L

,                                (23) 
where Dp refers to the dimension of Pm, and Dl to the dimension of Lm. The symbol “” denotes 
“be proportional to”. As Lm is a distance measurement, the dimension Dl=1. Thus equation (14) can 
be transformed into the following expression 
p
l
p
/
D
D
D
m
m
m
P
L
L




.                            (24) 
Comparing equation (24) with equation (14) shows σ=Dp. In light of equation (18), we have b=Dp, 
which is just equation (22). This suggests that the distance exponent of the gravity model is the 
average fractal dimension of urban populations of cities inside a geographical region. In empirical 
studies, it is hard to calculate Dp, but it is easy to evaluate q and D, and thus the distance exponent 
can be estimated by equation (20).  
In practice, the distance exponent can be estimated by the Minkowski–Bouligand dimension of a 
spatial distribution and the Zipf exponent of the corresponding rank-size distribution. The 
Minkowski–Bouligand dimension is often termed the box-counting dimension or box dimension by 
usage (Schroeder, 1991). The Zipf exponent can be given by the two-parameter Zipf model, equation 
(5). The box dimension is always defined as below 
b
0
ln
( )
lim
ln
N
D





,                              (25) 
where ε refers to the linear scale of boxes, N(ε) to the number of the nonempty boxes, and Db is the 
box dimension. Therefore, the box dimension can be evaluated by the following relations 
b
1
( )
D
N
N




,                               (26) 
where N1 is the proportionality coefficient. Theoretically, N1=1, and empirically, N1 is close to 1. 

 
9
Thus the distance exponent is given by 
b
b
qD

,                                  (27) 
which is a substituted form of equation (20).  
It is necessary to clarify the similarities and differences between these methods. Three scaling 
approaches for estimating the distance exponent can be illustrated as follows (Figure 1). The first 
approach is direct calculation using size-distance scaling, equation (24), and the fractal dimension 
of size measurement, Dp, is just the distance exponent (b=Dp). The second approach is indirect 
estimation using number-size scaling and number-distance scaling, that is, equations (10) and (12). 
These combine into equation (20), and the product of the fractal dimension of network, D, and the 
reciprocal of the scaling exponent, p, is theoretically equal to the distance exponent (b=D/p). The 
third approach is indirect estimation using Zipf’s law and the box-counting method including 
equations (5) and (26), which combine into equation (24). The product of the Zipf exponent, q, and 
the box dimension, Db, is approximately equal to the distance exponent (b=qDb).  
 
 
Figure 1 Three approaches based on spatial and hierarchical scaling for estimating the distance 
exponent of the gravity model 
 
Spatial scaling 
Hierarchical scaling 
Size-distance 
scaling (basic 
approach) 
Number-size scaling, 
Number-distance 
scaling 
Zipf’s law, 
Box-counting 
method 
b=Dp 
b=D/p 
b=qDb 
Space and size data 
processing 

 
10
3 Materials and methods 
3.1 Study area, materials and methods 
Chinese cities can be employed to test the theoretical results and to show how to estimate the 
distance exponent of the gravity model. The study area is the whole mainland China. Two datasets 
of city sizes are available including the observations of the fifth census (2000) and the sixth census 
(2010). The numbers of cities are 666 in 2000 and 654 in 2010. The 2000-year dataset was processed 
and organized by Zhou and Yu (2004a; 2004b). The analytical procedure is as follows. Step 1: 
examine city rank-size patterns. The rank-size distribution of cities indicates an urban hierarchy 
with cascade structure (Chen, 2012a; Chen, 2012b). Generally speaking, only the cities with size 
larger than a threshold value comply with the rank-size rule. Smaller cities can be regarded as 
outliers beyond the scale-free range (Chen, 2015). By means of a double logarithmic plot and 
residual outlier analysis, we can determine a scaling range for the rank-size distribution. The scaling 
range appears as a straight line segment on the log-log plot. Step 2: reconstruct order space. 
Hierarchical structure suggests an order-space of urban systems, which corresponds to the real space 
of network structure (Chen, 2014a). If a system of cities follows Zipf’s law, it can be converted into 
a self-similar hierarchy of cities with a number of levels. The rank will be replaced by city number, 
and the city size will be replaced by average size of each level. Thus the order-space based on 
hierarchical structure will substitute for the rank-size order space based on the rank-size distribution. 
Step 3: measure the nearest neighboring distance (NND). In each level of an urban hierarchy, 
each city possesses coordinate cities (Ye et al, 2000). The distance between a city and its nearest 
coordinate city is termed the nearest neighboring distance (Clark and Evans, 1954; Rayner et al, 
1971), which is a basic measurement for studies on central places and self-organized networks of 
cities (Chen, 2011). Step 4: build models and estimate model parameters. Using the datasets of 
hierarchies of cities, we can make fractal models, allometric scaling models, and hierarchical scaling 
models based on the hierarchical order-space. The ordinary least-squares (OLS) method can be 
adopted to evaluate the fractal dimension and the related scaling exponents because this algorithm 
has its significant merits (Chen, 2015). Sometimes however, the OLS algorithm is not the best 
approach for evaluating a power exponent. A new method based on the maximum likelihood 
estimation (MLE) is proposed by Clauset et al (2009) to identify various power-law distributions 

 
11 
and to estimate scaling exponents. This algorithm combines the methods of maximum-likelihood 
fitting with goodness-of-fit tests based on the Kolmogorov-Smirnov statistic and likelihood ratios. 
Unfortunately, the MLE-based approach cannot be applied to our datasets in this study for the 
following reasons. First, the MLE-based method is suitable for binned data, while this study is on 
pairs of cascade sequences. Second, the MLE-based method is developed for power-law frequency 
distributions, while this study is on fractal measure relations. 
3.2 Calculations 
The calculation procedure comprises four steps (Two Supplementary Materials are provided to 
show how to deal with the data and estimate the parameters, see Files S1 and S2). The first step is 
to investigate rank-size patterns. The common two-parameter Zipf model cannot be well fitted to 
the datasets of China’s city sizes. In fact, the rank-size distribution of Chinese cities should be 
modeled with the three-parameter Zipf’s model (Chen, 2016; Gabaix, 1999; Gell-Mann, 1994; 
Mandelbrot, 1982; Winiwarter, 1983). The model can be expressed as 
( )
(
) q
P k
C k



,                             (28) 
where k refers to city rank (k=1, 2, 3,…), and P(k) to the size of the kth city. As for the parameters, 
C is a proportionality coefficient, q denotes the Zipf scaling exponent, and ς represents a scale-
translational factor of city rank. The largest 550 cities are inside the scaling ranges, and the three-
parameter rank-size patterns are displayed in Figure 2. The first four or five ranks are removed due 
to the incomplete structure of Chinese hierarchies of cities (Chen, 2016). What is more, the cut-off 
point of scaling range is identified by combining eye observation, goodness of fit with residual 
outlier analysis (Chen, 2015). 
The second step is to reconstruct the order-space of the urban system. The three-parameter 
Zipf model suggests an absence of the leading cities in one or more top levels of a hierarchy of cities 
(Chen, 2016). For 2000-year cities, the scale-translational parameter ς =5, which suggests the first 
and second levels are absent. For the 2010-year cities, the adjusting parameter ς =4; this also 
suggests the absence of the first and second levels in the hierarchy. If the number ratio of different 
levels of cities is rn=2, cities can be organized into urban hierarchies by year (Table 1). Each 
hierarchy of cities comprises 8 levels, which reflects the order-space of the corresponding network 
of cities (m=3, 4, …, 10). The first and second levels are vacant, and the last level is a lame-duck 

 
12
class, in which the real number of cities is smaller than the expected number (Davis, 1978).  
Table 1 Numbers, average sizes, and average least-distances of China’s cities 
Order 
 m 
2000 
2010 
Number Nm 
Size Pm Distance Lm Number Nm 
Size Pm Distance Lm 
1 
1 
-- 
-- 
1 
-- 
-- 
2 
2 
-- 
-- 
2 
-- 
-- 
3 
4 
890.8895  
931.7267  
4 
1319.619  
981.3669  
4 
8 
446.1617  
563.9267  
8 
697.352  
606.1919  
5 
16 
233.6337  
534.0708  
16 
340.611  
494.4720  
6 
32 
123.1711  
212.2135  
32 
170.434  
254.0088  
7 
64 
68.3960  
153.5515  
64 
81.168  
167.1989  
8 
128 
35.1952  
132.9058  
128 
44.626  
129.5111  
9 
256 
17.8514  
98.7091  
256 
22.160  
99.6486  
10 
158 
8.6543  
133.8155  
146 
9.900  
135.3284  
Note: The distance unit is kilometer (km). The two top levels (m=1, 2) are absent according to the three-parameter 
Zipf model. The last level (m=10) represents a lame-duck class because of undergrowth of cities. The principle and 
procedure of creating the hierarchy of cities in this table are illuminated by Chen (2016).  
 
 
      a. 2000                                 b. 2010 
Figure 2 The rank-size distributions of China’s cities based on the three-parameter Zipf’s model 
[Note: A three-parameter Zipf distribution corresponds to an incomplete hierarchy of cities, on the top 
of which one or more levels have not developed.] 
 
The third step is to measure the nearest neighboring distance. For a given level in an urban 
hierarchy, each city has a nearest neighbor. Based on the hierarchy with 8 levels, the distances 
between different cities can be measured by Arc GIS. For each level, the distances can be arranged 
P(k) = 6E+07(k+5)-0.9702
R²= 0.9977
10000
100000
1000000
10000000
100000000
1
10
100
1000
Size P(k)
Rank k
P(k) = 8E+07(k+4)-1.0056
R²= 0.9953
10000
100000
1000000
10000000
100000000
1
10
100
1000
Size P(k)
Rank k

 
13
as a distance matrix, from which we can extract vectors of the nearest neighboring distances. The 
average values of the elements in the mth vector represent the nearest neighboring distance of the 
mth level (m=3, 4, …, 10). In fact, for medium-sized and small cities, the average value of 
coordination city numbers is close to 6 (Haggett, 1969; Niu, 1992; Ye et al, 2000). In order to lessen 
the negative effect of random fluctuation of city distributions, we can take the next nearest neighbor 
of a city into account and measure the second nearest neighboring distance (SNND). After extract 
the NND data, we can further extract the SNND from each distance matrix. The mean of NND and 
SNND of each level can be termed dual average neighboring distance (DAND) (Table 2). In this 
study, the DAND values are employed to carry out the spatial modeling and analysis. 
 
Table 2 Average distances of China’s cities to nearest neighbors and second nearest neighbors 
m 
2000 
2010 
NND 
SNND 
DAND 
NNDs 
SNND 
DAND 
3 
807.4446  
1056.0087  
931.7267  
591.5387  
1371.1950  
981.3669  
4 
336.6311  
791.2222  
563.9267  
347.4975  
864.8864  
606.1919  
5 
444.4505  
623.6911  
534.0708  
412.1159  
576.8281  
494.4720  
6 
154.5868  
269.8403  
212.2135  
190.2816  
317.7361  
254.0088  
7 
129.6058  
177.4973  
153.5515  
131.7646  
202.6332  
167.1989  
8 
95.9974  
169.8142  
132.9058  
99.5365  
159.4856  
129.5111  
9 
81.4986  
115.9197  
98.7091  
82.0976  
117.1997  
99.6486  
10 
103.4494  
164.1817  
133.8155  
110.0906  
160.5662  
135.3284  
Note: The distance unit is kilometer (km). NND refers to the distance of a city to its nearest neighbor, SNND to the 
distance of a city to its second nearest neighbor. In this table, both NND and SNND represent the mean of a level of 
cities, and DAND denotes the dual average value of NND and SNND.  
 
The fourth step is to build models using the hierarchical datasets. Except for the lame-duck 
class, the city numbers, average city sizes, and the average distances follow exponential laws. City 
numbers increase exponentially, and average city sizes and average distances decrease exponentially 
over order m. From the three exponential laws, a set of power laws follows, including the distance-
number scaling relation, the distance-size scaling relation, and the number-size scaling relation. The 
hierarchical scaling relationship between city numbers and average city sizes of different levels is 
equivalent to the three-parameter Zipf model, and the scaling exponent q is theoretically equal to 
the Zipf exponent (Figure 3). The hierarchical scaling relationship between average distances and 
city numbers is a fractal model of an urban network, and the fractal dimension D is just the fractional 

 
14
dimension of a central-place system (Figure 4). The hierarchical scaling relationship between 
average distances and average city sizes is equivalent to the allometric scaling relationship between 
city sizes and the area of service regions of cities, and the scaling exponent σ is just the average 
fractional dimension of city population distributions (Figure 5). As indicated above, in theory, the 
parameter σ can be estimated by the product of q and D (Table 3). 
 
 
      a. 2000                                b. 2010 
Figure 3 The hierarchical scaling relationships between the numbers of China’s cities and the 
corresponding average city sizes [Note: Figure 3 corresponds to Figure 2. The small circles represent 
outliers indicative of the lame-duck classes.] 
 
 
      a. 2000                                b. 2010 
Figure 4 The hierarchical scaling relationships between numbers of China’s cities and the 
Pm = 3E+07Nm
-0.9294
R²= 0.9997
10000
100000
1000000
10000000
100000000
1
10
100
1000
Average size Pm
City number Nm
Pm = 5E+07Nm
-0.9889
R²= 0.9996
10000
100000
1000000
10000000
100000000
1
10
100
1000
Average size Pm
City number Nm
Nm = 616070.5468 Lm
-1.7486
R²= 0.9525
1
10
100
1000
10
100
1000
Number Nm
Distance Lm
Nm = 684119.7502 Lm
-1.7571
R²= 0.9822
1
10
100
1000
10
100
1000
Number Nm
Distance Lm

 
15
corresponding minimum average distances [Note: The small circles represent outliers indicative of 
the lame-duck classes.] 
 
 
      a. 2000                                  b. 2010 
Figure 5 The hierarchical scaling relationships between the minimum distances of China’s cities 
and the corresponding average city sizes [Note: The small circles represent outliers indicative of the 
lame-duck classes.] 
 
3.3 Analysis 
Two approaches have been used to estimate the distance exponent of the gravity model of China’s 
cities. One is to use the distance-size hierarchical scaling, and the other is to utilize both the distance-
number hierarchical scaling and the number-size hierarchical scaling. If we don’t remove the bottom 
level (lame-duck class), the error between the direct result (b=σ value) and the indirect result (b≈qD 
value) will be large. If we treat the bottom level as an outlier and eliminate it, the directly estimated 
value will be close to the indirectly estimated value, that is, qD≈σ (Table 3). This suggests that the 
outlier of a dataset can cause significant deviation in parameter estimation. Intuitionally, the distance 
exponent value should have decreased because of development of traffic technology from 2000 to 
2010 year. However, the distance exponent actually increased during this decade due to intervening 
opportunities resulting from city development (as for intervening opportunity, see Stouffer, 1940). 
 
Table 3 The estimated values of model parameters of the hierarchies of China’s cities based on all 
Pm = 0.0176Lm
1.5791
R²= 0.9503
1
10
100
1000
10000
10
100
1000
Average size Pm
Distance Lm
Pm = 0.0107Lm
1.7109
R²= 0.9843
1
10
100
1000
10000
10
100
1000
Average size Pm
Distance Lm

 
16
data points and scaling ranges 
Scale 
Parameter 
2000 
2010 
Estimated value 
R2 
Estimated value 
R2 
All data points 
(Including 
bottom level) 
q 
1.0355 
0.9403 
1.1047 
0.9314 
D 
1.7486 
0.9525 
1.7571 
0.9822 
σ 
1.7774 
0.8631 
1.9199 
0.8950 
qD 
1.8106 
-- 
1.9411 
-- 
Scaling range 
(Eliminating 
bottom level) 
q 
0.9294 
0.9997 
0.9889 
0.9996 
D 
1.7008 
0.9527 
1.7280 
0.9823 
σ 
1.5791 
0.9503 
1.7109 
0.9843 
qD 
1.5808 
-- 
1.7088 
-- 
 
Another factor that impacts the estimated value of the scaling exponent is the spatial pattern of 
cities. The spatial distribution of urban populations is often a self-affine pattern rather than a self-
similar pattern. In other words, the urban population distribution is based on anisotropic growth 
instead of isotropic growth. As a result, the points on a double logarithmic scatter plot form two 
trend lines rather than one. The scaling break used to be treated as bi-fractals (White and Engelen, 
1993; White and Engelen, 1994). In many cases, bi-fractals proceed from self-affine distributions. 
The spatial distributions of Chinese cities bear significant self-affinity (Figure 6). However, they 
evolve from self-affine patterns into self-similar patterns because the difference between the fractal 
dimension values of the two scaling ranges became smaller from 2000 to 2010 (Table 4). In order 
to avoid the influence of spatial self-affinity, the fractal dimension can be estimated by the box-
counting method, which yields what is called box dimension mentioned above. The box-counting 
method has been employed to estimate the fractal dimension of urban forms (Benguigui et al, 2000; 
Feng and Chen, 2010; Shen, 2002). For simplicity, the classical box-counting method can be 
replaced by the functional box-counting method (Chen, 1995; Lovejoy et al, 1987). This method is 
simple and effective. For example, for the cities of Henan province of China in 2000, the Zipf 
exponent of the city-size distribution was about q=0.968 (Jiang and Yao, 2010), and the box 
dimension of the central-place network was about Db=1.8585(Chen, 2014b). Thus the distance 
exponent of the gravity model of Henan’s urban system was about b=0.968*1.8585=1.799 in 2000. 
As far as mainland China is concerned, in 2000 the box dimension of the urban network is about Db 
=1.8068, the Zipf exponent is q=0.9702, thus b=0.9702*1.8068=1.7530; in 2010 Db=1.8193, 
q=1.0056, and we have b=1.0056*1.8193=1.8295 [see Chen (2014b) and Feng and Chen (2000) for 

 
17
the box counting method]. 
Table 4 Fractal dimension values of population distributions based on bi-scaling ranges 
Item 
2000 
2010 
Parameter 
R2 
Parameter 
R2 
Scaling range 1 (Lm≤ L6) 
2.5915 
0.9762 
2.1669 
0.9898 
Scaling range 2 (Lm≥ L6) 
1.2944 
0.8814 
1.5474 
0.9597 
Intersection (L6) 
254.0088 
212.2135 
Average of two parameters 
1.9430 
1.8572 
Difference of two parameters 
1.2971 
0.6195 
 
 
      a. 2000                                  b. 2010 
Figure 6 Self-affine patterns and the bi-scaling relationships between the minimum distances of 
China’s cities and the corresponding average city sizes [Note: The small circles represent the 
intersections of the two scaling ranges] 
4 Discussion 
4.1 Evaluation of different methods 
The research results support the theoretical inference that the rank-size scaling and fractal 
modeling can be employed to evaluate the distance exponent of the gravity model. On the basis of 
the empirical analyses, a preliminary evaluation of the three approaches to estimating the distance 
exponent can be made. The first approach based on size-distance scaling is simple and direct, but it 
is sometimes hindered by self-affine spatial distributions. The second approach based on the 
Pm = 0.1066Lm
1.2944
R²= 0.8814
Pm = 0.0001Lm
2.5915
R²= 0.9762
1
10
100
1000
10000
10
100
1000
Average size Pm
Distance Lm
Pm = 0.0299Lm
1.5474
R²= 0.9597
Pm = 0.0011Lm
2.1669
R²= 0.9898
1
10
100
1000
10000
10
100
1000
Average size Pm
Distance Lm

 
18
combination of number-distance scaling and number-size scaling is effective, but it is indirect and 
sometimes influenced by spatial self-affinity. The third approach, the combination of Zipf’s law and 
the box-counting method, can avoid the negative influence of self-affine patterns, but it is difficult 
to match the sample of a size distribution with that of the corresponding spatial distribution. Nothing 
is perfect. Each method has its merits and defects. A comparison of advantages and disadvantages 
between these approaches is displayed in Table 5. 
 
Table 5 A comparison of three approaches for estimating the distance exponent 
Method 
Model 
Formula 
Merit 
Defect 
The first 
approach 
p
D
m
m
P
L


 
P
b
D

 
Direct and 
simple 
Sensitive to self-
affine patterns 
The second 
approach 
D
m
m
N
L



, 
p
m
m
N
P



 
2 /
q
R
p

,
b
qD

 
Easy to 
understand 
Indirect and 
influenced by self-
affinity 
The third 
approach 
( )
(
) q
P k
C k



,
b
1
( )
D
N
N




 
b
b
qD

 
Easy to 
calculate 
Indirect and 
influenced by 
sample difference 
 
The gravity model has been employed to research spatial interactions in both natural and human 
geographical phenomena. In empirical studies, there are two basic distance-decay functions for 
modeling gravity (Haggett et al, 1977): one is the inverse power law (Kang et al, 2012; Jung et al, 
2008), and the other, the negative exponential law (Balcan et al, 2009; Wilson, 2000). If we examine 
the local or partial spatial interactions between different places, the gravity models take on 
generalized production functions such as (Lee et al, 2014; Machay, 1958) 
i
j
ij
ij
P P
I
K
L




,                                 (29) 
which is based on the power-law decay, and (Balcan et al, 2009) 
ij
i
j
ij
L
P P
I
K
e




,                                 (30) 
which is based on exponential decay. In these formulae, K, α, β, and γ are parameters. However, if 
all the spatial relationships in a network are taken into consideration, equations (29) and (30) can be 
transformed into the standard forms as follows (Chen, 2015) 

 
19
i
j
ij
b
ij
PP
I
G L

,                                 (31) 
ij
i
j
ij
bL
PP
I
G
e

,                                 (32) 
where the notation is the same as equations (1) and (2). The parameter b in equation (32) is a 
distance-decay coefficient. Different types of gravity models have different spheres of application. 
Generally speaking, the exponential-based model is more applicable to smaller regions or simpler 
systems while the power-based model is more suitable for larger regions or more complex systems 
(Chen, 2015). If the distance-decay functions are appropriately selected, the gravity models can be 
well fitted to the observational data of spatial flows (e.g., Balcan et al, 2009; Batty and Karmeshu, 
1983; Kang et al, 2012; Goh et al, 2012; Jung et al, 2008; Lee et al, 2014). From gravity modeling, 
we can gain new insights into the dynamic process of spatial interactions. 
4.2 Further thinking 
Using the mathematical relations derived above, we can develop the gravity theory of human 
geography and solve the difficult problems which puzzle geographers for a long time. One problem 
is the question of the gravity constant. During the quantitative revolution of geography (1953-1976), 
geographers tried in vain to find constant values for the parameters of the gravity model, G and b, 
by means of mathematical derivation and statistical analysis (Harvey, 1969). Since then, many 
geographers cast doubt on the theoretical basis of the gravity model. Today, we know that human 
geographical systems differ from classical physical systems. The natural laws of human 
geographical systems don’t comply with the rule of spatio-temporal translational symmetry (Chen, 
2015). Thus we cannot find constant values for the model parameters of geographical gravity. As 
indicated above, the distance exponent of the urban gravity model is the average fractal dimension 
of urban population of the cities within a geographical region. The rest may be deduced by analogy. 
However, one basic property of geographical systems is regional differentiation. The condition of 
one geographical region differs from those of other geographical regions. Thus the average fractal 
dimension of urban population of one region is different from that of another region. 
Another problem is the dimension dilemma. According to dimensional analysis, the theoretical 
value of the distance exponent of the gravity model should equal 1, 2, or 3, which are dimension 

 
20
values of Euclidean geometry. However, the observed values from empirical studies often deviated 
from theoretical values significantly and took on fractional numbers (Haynes, 1975; Haggett et al, 
1977; Rybski et al, 2013). Today, it is easy to solve the difficult problem of dimensional analysis. 
The distance exponent is, in essence, a fractal dimension of urban population (Chen, 2015). A fractal 
dimension always exhibits a fractional number. So it is not surprising that empirical results of 
distance exponent estimation fail to equal 1, 2, or 3. If we define our study area within a 2-
dimensional space, the distance exponent will range from 1 to 2; if we definite our study area inside 
a 3-dimensional space, the distance exponent will vary from 1 to 3. Generally speaking, the distance 
exponent falls between 1 and 3 in empirical studies. This lends further support to the suggestion that 
fractal geometry, allometric analysis, and network theory can be integrated into a new theoretical 
framework to reinterpret many of our theories in physical and human geography (Batty, 1992; Batty, 
2008; Chen, 2015). 
It has not been reported in the previous literature that Zipf’s law and the fractal dimension of 
urban network can be combined to estimate the distance exponent of the gravity model. This is just 
the novelty of this work. However, this research has two main shortcomings. First, the case study is 
only based on the census data of Chinese cities. We have no data for cities in other countries for the 
time being. In this case, we cannot carry out a cross-sectional comparative analysis. Second, the 
methods are developed for self-similar fractal structures. However, many urban systems are random 
self-affine fractals. We have no effective approaches for dealing with the self-affine spatial 
distribution of cities. Anyway, the value of a paper rests with its inspiration rather than its perfection.  
5 Conclusions 
By the empirical study based on Chinese cities, we can reconsider the geographical gravity model 
and its parameters. New findings lie in three aspects, that is, the fractality and scaling behind the 
gravity model, the fractal dimension property of the distance exponent, and the effective methods 
of parameter estimation. All these are based on the hierarchy of cities with cascade structure. The 
main conclusions can be reached as follows. First, the geographical gravity model is a fractal 
model of spatial interaction associated with a self-similar hierarchy. Before the advent of fractal 
geometry, the gravity model was explained by ideas from Euclidean geometry. However, 

 
21
geographical systems are fractal systems with fractional dimensions. Many difficult problems such 
as the dimensional dilemma and gravity constants resulted from traditional explanations based on 
Euclidean geometry. Today, many theoretical problems relating to the gravity model and its 
parameters can be easily solved using ideas from fractals, scaling, and allometry based on a 
hierarchical structure. Second, the distance exponent of the gravity model is a kind of fractal 
dimension indicating a given geographical quantity of matter. The property of fractal dimension 
depends on size measurements such as population size, urban area, and economic product. If we use 
population sizes of cities to determine the attraction power, the distance exponent will represent the 
average fractal dimension of the urban populations of cities within a geographical region. Similarly, 
if we employ urban areas to define the attraction power, the distance exponent will reflect the 
average fractal dimension of urbanized areas of cities in a geographical region. The rest may be 
deduced by analogy. Third, the distance exponent of urban gravity can be estimated by Zipf’s 
law and the box-counting method. The distance exponent equals the product of the hierarchical 
scaling exponent of size distributions of cities and the fractal dimension of a network of cities in a 
study area. It is easy to evaluate the scaling exponent of a rank-size distribution (indicative of a 
hierarchy) and corresponding fractal dimension of an urban system (indicative of a network). The 
rank-size exponents can be computed by Zipf’s law, and the fractal dimensions of urban networks 
can be calculated using the box-counting method. Thus the distance exponent can be estimated by 
using fractal modeling of size distributions and spatial distributions of cities. 
Acknowledgements 
This research was sponsored by the National Natural Science Foundation of China (Grant No. 41671167). 
The supports are gratefully acknowledged.  
Appendixes 
Appendix 1 The relationships between Zipf’s law, Pareto distribution, and 
hierarchical scaling law 
From Zipf’s law, we can derive Pareto distribution and hierarchical scaling law. For simplicity, 
let’s see the rank-size distribution based on pure Zipf’s law, P(k)=1/k. Using this equation, we can 

 
22
generate a harmonic sequence such as 1, 1/2, 1/3, …, 1/k, …. According to the generalized Davis’s 
2n rule, the harmonic sequence can be organized into a hierarchy with cascade structure (Table I). 
From the hierarchy, we can derive equation (6) and equation (10). The city number in each class is 
Nm=1, 2, 4, …, 2m-1, where m=1,2,3,…. This sequence can be formulated as 
1
1
1 n
2m
m
m
N
N r




,                              (A1) 
in which N1=1 and rn=2. Let the size threshold be P=1, 1/2, 1/4,…, 1/2m-1. Clearly, the number of 
cities with population size greater than or equal to P is N(P) =1,2,4,…, 2m-1. Thus we have 
1
1
( )
2m
p
N P
P
P







,                          (A2) 
where η=1 and p=1. Equation (A2) is equivalent to the standard Pareto distribution. Comparing 
equation (A1) with equation (A2) shows Nm=N(p). On the other hand, it can be proved that the total 
size of cities at each level is Sm→ln(2)=0.6931 (Chen, 2012a). The average city size of each level is 
1
1 p
1
ln(2)
2
m
m
m
m
m
S
P
Pr
N





,                           (A3) 
in which P1=ln(2) and rp=2. From equation (A3) it follows 
1
ln(2)
p
m
m
m
m
m
S
N
P
P
P






,                         (A4) 
where μ→ln(2) and p→1. Equation (A4) is equivalent to the pure Zipf’s law. The scaling exponent 
p is asymptotically close to 1. This mathematical process can be generalized to common Zipf’s law 
and Pareto distribution (A Supplementary Material is provided to show how to transform the 
harmonic sequence into a hierarchy, see File S3). 
 
Table I A hierarchy of 2047 cities with cascade structure coming from pure Zipf’s distribution 
Level 
m 
City number 
Nm, N(P) 
Hierarchy (The first number at each level 
acts as the size threshold, P) 
Sum 
Sm  
Average 
size Pm 
1 
1 
1 
 
 
 
 
1 
1 
2 
2 
1/2 
1/3 
 
 
 
0.8333 
0.4167 
3 
4 
1/4 
1/5 
1/6 
1/7 
 
0.7595 
0.1899 
4 
8 
1/8 
1/9 
1/10 
1/11 
…… 
0.7254 
0.0907 
5 
16 
1/16 
1/17 
1/18 
1/19 
…… 
0.7090 
0.0443 
6 
32 
1/32 
1/33 
1/34 
1/35 
…… 
0.7010 
0.0219 
7 
64 
1/64 
1/65 
1/66 
1/67 
…… 
0.6971 
0.0109 
8 
128 
1/128 
1/129 
1/130 
1/131 
…… 
0.6951 
0.0054 

 
23
9 
256 
1/256 
1/257 
1/258 
1/259 
…… 
0.6941 
0.0027 
10 
512 
1/512 
1/513 
1/514 
1/515 
…… 
0.6936 
0.0014 
11 
1024 
1/1024 
1/1025 
1/1026 
1/1027 
…… 
0.6934 
0.0007 
Note: Starting from the fourth level, only the first four numbers are displayed due to the limited space. If the city 
number is not equal to 2m-1, the final level will be a lame-duck class. 
Appendix 2 The principle of dimension consistency and fractal measure relation 
As early as the ancient Greek times, mathematicians found the principle of dimension consistency. 
That is, a measure is in proportion to another measure if and only if the two measure have the same 
dimension. For length L, area A, volume V, and arbitrary measure M, we have a proportion relation 
L1/1=kaA1/2=kvV1/3=kmM1/d, where ka, kv, km are proportionality coefficients, and d is the dimension 
of M (Lee, 1989; Feder, 1988). This is well-known geometric measure relation that can generalized 
to fractal measure relation (Feder, 1989; Mandelbrot, 1982; Takayasu, 1990). Equation (16) can be 
obtained by the dimensional analysis based on the geometric measure relation. It is easy to derive 
equation (14) from equations (8) and (9). Replacing m by m+1 in equation (14) yields 
1
1
m
m
P
L




,                                (B1) 
Combining equation (B1) with equation (14) yields 
1
1
(
)
m
m
m
m
P
L
P
L




,                              (B2) 
which shares the same mathematical structure with equation (4). According to the principle of 
dimension consistency, Pm, Pm+1, Pi, and Pj have the same space dimension Dp, and Lm, Lm+1, Lix, 
and Ljx bear the same space dimension Dl. Thus we have two geometric measure relations as follows 
p
l
/
1
1
(
)
D
D
m
m
m
m
P
L
P
L



,                            (B3) 
p
l
/
(
)
D
D
i
ix
j
jx
P
L
P
L

.                              (B4) 
Comparing equations (B3) and (B4) with equations (B2) and (4) yields 
p
l
D
b
D


.                                 (B5) 
Substituting equation (B5) into equation (B2) yields equation (16), immediately. This suggests that, 
although Pm and Pm+1 differ from Pi and Pj and Lm and Lm+1 differ from Lix and Ljx, the scaling 

 
24
exponent based on Pm/Pm+1 and Lm/Lm+1 is equivalent to that based on Pi/Pj and Lix/Ljx. 
Legends of Supplementary Materials for on-line publication only 
S1 Data processing for Chinese hierarchy of cities in 2000 (XLSX) 
Based on the 2000 year urban population census data of China, the nearest neighboring distance 
(NND) and second nearest neighbor distance (SNND) were calculated, and the rank-size series of 
urban population was transformed into a hierarchy with cascade structure according to the 
generalized 2n rule. The distance matrixes were extracted by ArcGIS, and the other processes were 
fulfilled by MS Excel. 
S2 Data processing for Chinese hierarchy of cities in 2000 (XLSX) 
Based on the 2010 year urban population census data, the NND and SNND were calculated, and the 
rank-size series of urban population was transformed into a self-similar hierarchy. Comparing the 
data processing for 2000 with that for 2010, the reader will understand the whole process of data 
processing in this paper. Then, readers can make computation by writing computer programs. 
S3 The process of transforming Zipf's distribution into self-similar hierarchy 
(XLSX) 
A harmonic sequence consisting of 2047 data points was generated using the pure Zipf model. The 
sequence representing standard Zipf distribution was converted into a hierarchy with cascade 
structure. 
References 
Bak P (1996). How Nature Works: the Science of Self-organized Criticality. New York: Springer 
Balcan D, Colizza V, Gonçalves B, Hu H, Ramasco JJ, Vespignani A (2009).Multiscale mobility 
networks and the spatial spreading of infectious diseases. PNAS, 106(51):21484 -21489 
Batty M (1992). The fractal nature of geography. Geographical Magazine, 64(5): 34-36 
Batty M (2005). Cities and Complexity: Understanding Cities with Cellular Automata, Agent-Based 

 
25
Models, and Fractals. Cambridge, MA: The MIT Press 
Batty M (2008). The size, scale, and shape of cities. Science, 319: 769-771 
Batty M, Karmeshu (1983). A strategy for generating and testing models of migration and urban growth. 
Regional Studies, 17(4): 223-236 
Batty M, Longley PA (1994). Fractal Cities: A Geometry of Form and Function. London: Academic 
Press 
Benguigui L, Czamanski D, Marinov M, Portugali J (2000). When and where is a city fractal? 
Environment and Planning B: Planning and Design, 27(4): 507–519 
Cerqueti R, Ausloos M (2015). Evidence of economic regularities and disparities of Italian regions from 
aggregated tax income size data. Physica A, 421(1): 187-207 
Chen T (1995). Studies on Fractal Systems of Cities and Towns in the Central Plains of China. 
Changchun: Department of Geography, Northeast Normal University [Master's Degree Thesis in 
Chinese] 
Chen YG (2011). Fractal systems of central places based on intermittency of space-filling. Chaos, 
Solitons & Fractals, 44(8): 619-632 
Chen YG (2012a). The mathematical relationship between Zipf's law and the hierarchical scaling law. 
Physica A, 391(11): 3285-3299 
Chen YG (2012b). Zipf's law, hierarchical structure, and cards-shuffling model for urban development. 
Discrete Dynamics in Nature and Society, Volume 2012, Article ID 480196, 21 pages 
Chen YG (2014a). The spatial meaning of Pareto’s scaling exponent of city-size distributions. Fractals, 
22(1-2):1450001 
Chen YG (2014b). Multifractals of central place systems: models, dimension spectrums, and empirical 
analysis. Physica A, 402: 266-282 
Chen YG (2015). The distance-decay function of geographical gravity model: power law or exponential 
law? Chaos, Solitons & Fractals, 77: 174-189 
Chen YG (2016). The evolution of Zipf’s law indicative of city development. Physica A, 443: 555-567 
Chen YG, Zhou YX (2003). The rank-size rule and fractal hierarchies of cities: mathematical models and 
empirical analyses. Environment and Planning B: Planning and Design, 30(6): 799–818 
Chen YG, Zhou YX (2008). Scaling laws and indications of self-organized criticality in urban systems. 
Chaos, Solitons & Fractals, 35(1): 85-98 

 
26
Christaller W (1933). Central Places in Southern Germany (trans. C. W. Baskin, 1966). Englewood Cliffs, 
NJ: Prentice Hall 
Clark PJ, Evans FC (1954). Distance to nearest neighbour as a measure of spatial relationships in 
populations. Ecology, 35: 445-453 
Clauset A, Shalizi CR, Newman MEJ (2009). Power-law distributions in empirical data. Siam Review, 
51(4): 661-703 
Converse PD (1949). New laws of retail gravitation. The Journal of Marketing, 14:379- 384 
Davis K (1978). World urbanization: 1950-1970. In: Systems of Cities. Eds. I.S. Bourne and J.W. Simons. 
New York: Oxford University Press, pp92-100 
Feder J (1988). Fractals. New York: Plenum Press 
Feng J, Chen YG (2010). Spatiotemporal evolution of urban form and land use structure in Hangzhou, 
China: evidence from fractals. Environment and Planning B: Planning and Design, 37(5): 838-856 
Frankhauser P (1994). La Fractalité des Structures Urbaines (The Fractal Aspects of Urban Structures). 
Paris: Economica 
Frankhauser P (1998). The fractal approach: A new tool for the spatial analysis of urban agglomerations. 
Population: An English Selection, 10(1): 205-240 
Gabaix X (1999). Zipf's law for cities: an explanation. Quarterly Journal of Economics, 114 (3): 739–
767 
Gell-Mann M (1994). The Quark and the Jaguar: Adventures in the Simple and the Complex. New York, 
NY: W.H. Freeman 
Goh S, Lee K, Park J-S, Choi MY (2012). Modification of the gravity model and application to the 
metropolitan Seoul subway system. Physical Review E, 86: 026102 
Haggett P, Cliff AD, Frey A (1977). Locational Analysis in Human Geography (2nd edition). 
London: Edward Arnold 
Harvey D (1969). Explanation in Geography. London: Edward Arnold 
Haynes AH (1975). Dimensional analysis: some applications in human geography. Geographical 
Analysis, 7(1): 51-68 
Haynes KE, Fotheringham AS (1984). Gravity and Spatial Interaction Models. London: SAGE 
Publications 
Jiang B, Yao X (2000). Geospatial Analysis and Modeling of Urban Structure and Dynamics. New York: 

 
27
Springer 
Jung W-S, Wang FZ, Stanley HE (2008). Gravity model in the Korean highway. Europhysics Letters, 81: 
48005 
Kang CG, Ma XJ, Tong DQ, Liu Y (2012). Intra-urban human mobility patterns: An urban morphology 
perspective. Physica A, 391(4), 1702-1717 
Kang CG, Zhang Y, Ma XJ, Liu Y (2013). Inferring properties and revealing geographical impacts of 
intercity mobile communication network of China using a subnet data set. International Journal of 
Geographical Information Science, 27(3), 431-448 
Lee S-H, Ffrancon R, Abrams DM, Kim B-J, Porter MA (2014). Matchmaker, matchmaker, make me a 
match: migration of populations via marriages in the past. Physical Review X, 4: 041009 
Lee Y (1989). An allmetric analysis of the US urban system: 1960–80. Environment and Planning A, 
21(4): 463–476 
Liang SM (2009). Research on the urban influence domains in China. International Journal of 
Geographical Information Science, 23(12): 1527-1539 
Liu Y, Sui ZW, Kang CG, Gao Y (2014). Uncovering patterns of inter-urban trip and spatial interaction 
from social media check-in data. PLoS ONE, 9(1): e86026 
Liu Y, Wang FH, Kang CG, Gao Y, Lu YM (2014). Analyzing relatedness by toponym co-occurrences 
on web pages. Transactions in GIS, 18(1), 89-107 
Lovejoy S, Schertzer D, Tsonis AA (1987). Functional box-counting and multiple elliptical dimensions 
in rain. Science, 235: 1036-1038 
Mackay JR (1958). The interactance hypothesis and boundaries in Canada: a preliminary study. The 
Canadian Geographer, 3(11):1-8 
Mandelbrot BB (1982). The Fractal Geometry of Nature. New York: W.H. Freeman and Company 
Mikkonen K, Luoma M (1999). The parameters of the gravity model are changing -- how and why? 
Journal of Transport Geography, 7(4): 277-283 
Rayner JN, Golledge RG, Collins Jr. SS (1971). Spectral analysis of settlement patterns. In: Final Report, 
NSF Grant No. GS-2781. Ohio State University Research Foundation, Columbus, Ohio, pp 60-84 
Reilly WJ (1931). The Law of Retail Gravitation. New York: The Knickerbocker Press 
Rodrigue JP, Comtois C, Slack B (2009). The Geography of Transport Systems. New York: Routledge 
Rybski D, Ros AGC, Kropp JP (2013). Distance-weighted city growth. Physical Review E, 87(4): 042114 

 
28
Schroeder M (1991). Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. New York: W. 
H. Freeman 
Sen A, Smith TE (1995). Gravity Models of Spatial Interaction Behavior: Advances in Spatial and 
Network Economics. Berlin: Springer-Verlag 
Shen G (2002). Fractal dimension and fractal growth of urbanized areas. International Journal of 
Geographical Information Science, 16(5): 419-437 
Simini F, González MC, Maritan A, Barabási AL (2012). A universal model for mobility and migration 
patterns. Nature, 484 (7392): 96-100 
Stouffer SA (1940). Intervening opportunities: a theory relating mobility and distance. American 
Sociological Review, 5(6): 845-867 
Takayasu H (1990). Fractals in the Physical Sciences. Manchester: Manchester University Press 
Tan MH, Fan CH (2004). Relationship between Zipf dimension and fractal dimension of city-size 
distribution. Geographical Research, 23(2): 243-248 [In Chinese] 
White R, Engelen G (1993). Cellular automata and fractal urban form: a cellular modeling approach to 
the evolution of urban land-use patterns. Environment and Planning A, 25(8): 1175-1199 
White R, Engelen G (1994). Urban systems dynamics and cellular automata: fractal structures between 
order and chaos. Chaos, Solitons & Fractals, 4(4): 563-583 
Wilson AG (2000). Complex Spatial Systems: The Modelling Foundations of Urban and Regional 
Analysis. Singapore: Pearson Education Asia 
Winiwarter P (1983). The genesis model—Part II: Frequency distributions of elements in selforganized 
systems. Speculations in Science and Technology, 6(2): 103-112 
Ye DN, Xu WD, He W, Li Z (2001). Symmetry distribution of cities in China. Science in China D: Earth 
Sciences, 44(8): 716-725 
Zhou YX, Yu HB (2004a). Reconstruction city population size hierarchy of China based on the fifth 
population census (I). Urban Planning, 28(6): 49-55 [In Chinese] 
Zhou YX, Yu HB (2004b). Reconstruction city population size hierarchy of China based on the fifth 
population census (II). Urban Planning, 28(8): 38-42 [In Chinese] 
Zipf GK (1949). Human Behavior and the Principle of Least Effort. Reading, MA: Addison-Wesley 
