A Scaling Approach to Evaluating the Distance Exponent of
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: [email protected])
Abstract: The gravity model is one of important models of social physics and human geography,
but several basic theoretical and methodological problems are still pending and remain to be
solved. In particular, it is hard to explain and evaluate the distance exponent using the ideas from
traditional theory. This paper is devoted to studying the distance decay parameter of the urban
gravity model. Based on the ideas from fractal geometry, several fractal parameter relations can be
derived from the scaling laws of self-similar hierarchy of cities. The 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
distribution 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 expectation. The findings lend
further support to the suggestion that the geographical gravity model is a fractal model, and its
distance exponent is associated with fractal dimension and Zipf’s exponent. This work will be
helpful for geographers to understand the gravity model using the fractal notion and to estimate
the main model parameter using fractal modeling method.
Key words: gravity model; spatial interaction; Zipf’s law; central place network; hierarchy of
cities; fractals; fractal dimension; allometric scaling
1
1 Introduction
Gravity model have been applying to explaining and predicting various behaviors of spatial
interaction in many social sciences. Among this family of models, the basic one is the gravity
model of migration based on inverse power-law 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 with 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). These
years, the gravity model is employed to study the attractive effect of various newfashioned human
activities by means of modern technology (Kang et al, 2012; Kang et al, 2013; Liang, 2009; Liu
and Wang et al, 2014). The model is empirically effective for describing spatial interaction, 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 the light of Euclidean geometry (Haggett et al,
1977; Haynes, 1975); the other is the algorithmic problem, that is, it is hard to estimate the
numerical value of the distance exponent (Mikkonen and Luoma, 1999).
Now, the first problem can be readily solved by using the ideas from fractals. Fractal geometry
is 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 fractional 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 self-organized network of places (Chen,
2011). If we examine the spatial interaction between cities and towns of a region, the distance
exponent of the gravity model is just the product of the Zipf exponent of city size distribution and
the fractal dimension of the corresponding central place network. The formula has been derived
from the combination of central place theory and the rank-size rule. The new pending problem is
2
how to use the formula to estimate the distance exponent for gravity analysis in empirical studies.
The problem can be solved using 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 the 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 through hierarchical scaling, which suggests a new approach to estimating the distance
exponent. This study is mainly based on urban geography. The rest parts of this article are
organized as follows. In Section 2, a scaling relation will be derived from the gravity model based
on the inverse power law, and the spatial scaling will be transformed into hierarchical scaling in
terms of the inherent association between hierarchy and network. Thus three scaling approaches to
estimating the distance exponent will be proposed. In Section 2, Chinese cities will be employed
to make an empirical analysis by means of census data and spatial distance data. In Section 3,
several related questions will be discussed. Finally, the paper will be concluded by summarizing
the main points in the work.
2 Models
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 comes
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 ix = G
Pi Px
,
Lbix
(1)
I jx = G
Pj Px
,
Lbjx
(2)
where I denotes the gravity, L indicates the distance between two locations, G refers to the gravity
coefficient, and b to the distance-decay parameter indicating spatial friction. Combining equations
3
(1) and (2) yields
Pi / Lbi
I
= ix .
b
Pj / L j I jx
(3)
Suppose that there exists a special location for x, where Iix=Ijx. In this case, we have
Pi
L
= ( i )b ,
Pj
Lj
(4)
which is familiar to geographers and indicates a fractal dimension relation (Chen, 2015). From
equation (4) it follows 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
Generally speaking, the size distribution of cities within a geographical region complies with
Zipf’s law (Zipf, 1949), which can formulated as follows
P(k ) = P1k −q ,
(5)
where k refers to the rank of cities in a 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 the cities with size
greater than or equal to P(k). Reducing P(k) to P and substituting k with N(P), we have
N(P) = ηP− p ,
(6)
which is mathematically equivalent to Pareto’s law. This indicates that Zipf’s law is equivalent to
Pareto’s law in mathematics, and the Zipf distribution is theoreitcally equivalent to the Pareto
distribution (Chen, 2014a). In equation (6), the power exponent p=1/q denotes the Pareto scaling
exponent, and η=P1p represents the proportionality coefficient.
It has been proved 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:
4
Nm = N1rfm−1 ,
(7)
Pm = P1rp1− m ,
(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 city number 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, 2012c; Davis, 1978). According
to central places theory propounded by Christaller (1933/1966), we have the third exponential
model such as (Chen, 2011)
Lm = L1rl1− m ,
(9)
where Lm denotes the average distance between two urban places in the mth class, L1 is the
distance between the urban places in the top class, and rl=Lm/Lm+1 is the distance ratio of cities.
The fractal model and hierarchical scaling relations can be derived from the three exponential
functions. From equations (7) and (8) it follows the size-number hierarchical scaling relation
Nm = μPm− p ,
(10)
which is equivalent to the Pareto’s law, equation (6). The proportionality coefficient is μ=N1P1p,
and the scaling exponent can be redefined as
p=−
ln(Nm / Nm +1 ) ln rn
=
,
ln(Pm / Pm +1 ) ln rp
(11)
which is regarded as the similarity dimension of city-size distributions. It is in fact a ratio of two
fractal dimension (Chen, 2014a). From equations (7) and (9) it follows a fractal model of central
places as below:
Nm = πL−mD ,
(12)
where π=N1L1D refers to the proportionality coefficient, and D to the fractal dimension of spatial
distribution of cities and towns. The fractal dimension of similarity can be given by
D=−
ln(N m / N m +1 ) ln rn
=
,
ln(Lm / Lm +1 ) ln rl
(13)
which is the fractal parameter of central place networks and can be termed network dimension.
5
From equations (8) and (9) it follows a size-range allometric relation
Pm = κLσm ,
(14)
where κ=P1L1σ refers to the proportionality coefficient, and σ to the allometric scaling exponent of
population distribution. The allometric exponent can be given by
σ=
ln(Pm / Pm +1 ) ln rp
=
,
ln(Lm / Lm +1 ) ln rl
(15)
which suggests that the allometric exponent is a fractal dimension of urban population.
Using the theoretical framework of the urban hierarchical scaling, we can derive new relation
for the distance exponent of the gravity model. Based on the hierarchical structure of urban
systems, the breaking-point relation can be rewritten as
Pm
L
= ( m )b ,
Pm +1
Lm +1
(16)
which is derived from equation (4). As rp=Pm/Pm+1, rl=Lm/Lm+1, equation (14) can be expressed in
the form
rp = rlb .
(17)
Thus we have
b=
ln(Pm / Pm +1 ) ln rp
=
=σ .
ln(Lm / Lm +1 ) ln rl
(18)
This suggests that the distance exponent is equal to the allometric scaling exponent of city size and
its spatial distribution. According to equations (11), (13), and (15), we have
D=
ln rn ln rp ln rn
b
=
= σp = ,
ln rl ln rl ln rp
q
(19)
which suggests
b = qD .
(20)
Since q→1, if D→2, we will have b→2. The Zipf exponent can be calculated by the rank-size
scaling analysis, and the network dimension can be computed with the self-similar network
analysis. Thus the distance exponent can be readily estimated by the parameter relation, equation
(20).
6
2.3 Three approaches to evaluating distance exponent
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
q=
1 Dp
,
=
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 urban population distribution of different cities within the city
network (Chen, 2014a). Substituting equation (21) into equation (20) yields
b = DP .
(22)
In fact, according to the general geometric measure relation (Chen, 2015; Takayasu, 1990),
equation (14) can be rewritten as
1 / Dp
Pm
∝ L1m/ Dl ,
(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
D / Dl
Pm = κLmp
D
= κLmp .
(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 the distance exponent of the gravity model is the
average fractal dimension of urban population of the cities inside a geographical region.
In practice, the distance exponent can be approximately 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 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
ln N (ε )
,
ε →0
ln ε
Db = − lim
(25)
where ε refers to the linear scale of boxes, N(ε) to the number of the nonempty boxes, and Db is
7
the box dimension. In practice, the box dimension can be evaluated by the following relations
N (ε ) = N1ε − Db ,
(26)
where N1 is the proportionality coefficient. Theoretically, N1=1, and empirically, N1 is close to 1.
Thus the distance exponent is given by
b = qDb ,
(27)
which is an substitute of equation (20).
Now, three scaling approaches to estimating the distance exponent can be presented as follows
(Figure 1). The first approach is the direct calculation using the size-distance scaling, equation
(24), and the fractal dimension of size measurement, Dp, is just the distance exponent (b=Dp). The
second approach is the indirect estimation using the number-size scaling and the number-distance
scaling, that is, equations (10) and (12), and the product of the fractal dimension of network, D,
and the reciprocal the scaling exponent, p, is theoretically equal to the distance exponent (b=D/p).
The third approach is the indirect estimation using Zipf’s law and the box-counting method,
including equations (14) and (26), and the product of the Zipf exponent, q, and the box dimension,
Db, is approximately equal to the distance exponent (b=qDb).
Space and size data
processing
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
Figure 1 The three approaches based on spatial and hierarchical scaling to estimating the
distance exponent of the gravity model
8
3 Empirical analysis
3.1 Study area, materials and methods
Chinese cities can be employed to testify the theoretical results and to show how to estimate the
distance exponent of the gravity model. The study area contains the mainland China. Two datasets
of city sizes are available for this research, including the observations of the fifth census (2000)
and the sixth census (2010). The numbers of cities are 666 in 2000 and 654 cities in 2010. The
2000-year dataset was processed and put to rights 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 is a signature of an urban hierarchy with cascade structure (Chen, 2012). Generally speaking,
only the cities with size larger than a threshold value comply with the rank-size rule. The smaller
cities can be regarded as outlier beyond the scale-free range (Chen, 2015). By means of a double
logarithmic plot, we can determine a scaling range for the rank-size distribution. The scaling range
takes on a straight line segment on the log-log plot. Step 2: reconstruct order space. Hierarchical
structure suggests order space of urban systems, which corresponds to the real space of network
structure (Chen, 2014a). If a network 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 rank-size order space
based on the rank-size distribution will be substituted with the order space based on hierarchical
structure. Step 3: measure the nearest neighboring distance (NND). In a level of an urban
hierarchy, each city possesses its 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 hierarhical order space. The ordinary least square (OLS)
method can be adopted to evaluate the fractal dimension and the related scaling exponents because
this algorithm is very suitable for estimating power exponents (Chen, 2015).
9
3.2 Calculations
First of all, the rank-size patterns should be investigated. 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 and Zhou, 2003;
Gabaix, 1999; Gell-Mann, 1994; Mandelbrot, 1982; Winiwarter, 1983). The model can be
expressed as
P(r ) = C(r + k )−q ,
(28)
where r refers to city rank (r=1, 2, 3,…), and P(r) to the size of the rth city. As for the parameters,
C is a proportionality coefficient, q denotes the Zipf scaling exponent, and k 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 as below (Figure 2).
100000000
100000000
P(r) = 6E+07(r+5)-0.9702
R² = 0.9977
10000000
Size P(r)
Size P(r)
10000000
P(r) = 8E+07(r+4)-1.0056
R² = 0.9953
1000000
100000
1000000
100000
10000
10000
1
10
100
1000
1
Rank r
10
100
1000
Rank r
a. 2000
b. 2010
Figure 2 The rank-size distributions of China’s cities based on the three-parameter Zipf’s model
The next step is to reconstruct order space of the urban system. Sometimes, the three-parameter
Zipf model suggests an absence of the leading cities in one or more top levels of a hierarchy of
cities. For the 2000-year cities, the scale-translational parameter k=5, and which suggests the first
and second levels are absent. For the 2010-year cities, the adjusting parameter k=4, and this also
suggests the absence of the first and second levels in the hierarchy. Let the number ratio of
different levels of cities be rn=2, the cities can be organized into urban hierarchies by year (Table
10
1). Each hierarchy of cities comprises 8 levels, which reflect the order space of the corresponding
network of cities (m=3, 4, …, 10). The top two levels are vacant, and the last level is a lame-duck
class, in which the real number of cities is smaller than the expected number (Davis, 1978).
Table 1 The 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
2
3
4
5
6
7
8
9
1
2
4
8
16
32
64
128
256
--890.8895
446.1617
233.6337
123.1711
68.3960
35.1952
17.8514
--931.7267
563.9267
534.0708
212.2135
153.5515
132.9058
98.7091
1
2
4
8
16
32
64
128
256
--1319.619
697.352
340.611
170.434
81.168
44.626
22.160
--981.3669
606.1919
494.4720
254.0088
167.1989
129.5111
99.6486
10
158
8.6543
133.8155
146
9.900
135.3284
Note: 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 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 come into
being a distance matrix, from which we can extract a vector of the nearest neighboring distances.
The average values of the elements in the mth vector represents 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 make spatial modeling and analysis.
Table 2 The average distances of China’s cities to the nearest neighbors and the second nearest
11
neighbors
m
3
4
5
6
7
8
9
10
2000
2010
NND
SNND
DAND
NNDs
SNND
DAND
807.4446
336.6311
444.4505
154.5868
129.6058
95.9974
81.4986
103.4494
1056.0087
791.2222
623.6911
269.8403
177.4973
169.8142
115.9197
164.1817
931.7267
563.9267
534.0708
212.2135
153.5515
132.9058
98.7091
133.8155
591.5387
347.4975
412.1159
190.2816
131.7646
99.5365
82.0976
110.0906
1371.1950
864.8864
576.8281
317.7361
202.6332
159.4856
117.1997
160.5662
981.3669
606.1919
494.4720
254.0088
167.1989
129.5111
99.6486
135.3284
Note: 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.
100000000
100000000
Pm = 5E+07Nm-0.9889
R² = 0.9996
Pm = 3E+07Nm-0.9294
R² = 0.9997
10000000
Average size Pm
Average size Pm
10000000
1000000
100000
1000000
100000
10000
10000
1
10
100
1000
1
10
100
1000
City number Nm
City number Nm
a. 2000
b. 2010
Figure 3 The hierarchical scaling relationships between the numbers of China’s cities and the
corresponding average city sizes [Note: The small circles represent outliers indicative of the
lame-duck classes]
The fourth step is to build models using the hierarchical datasets. Except the lame-duck class,
the city numbers, average city sizes, the average distances follow exponential laws. City numbers
take on exponential increase, and average city sizes and the average distances take on exponential
decrease over order m. From the three exponential laws it follows a set of power laws, including
the distance-number scaling relation, the distance-size scaling relation, and the number-size
12
scaling relation. The hierarchical scaling relationship between city numbers and average city sizes
of different levels is equivalent 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 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).
1000
100
100
Number Nm
Number Nm
1000
10
10
Nm = 616070.5468 Lm-1.7486
R² = 0.9525
Nm = 684119.7502 Lm-1.7571
R² = 0.9822
1
1
10
100
1000
10
Distance Lm
100
1000
Distance Lm
a. 2000
b. 2010
Figure 4 The hierarchical scaling relationships between numbers of China’s cities and the
corresponding minimum average distances [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 estimate the exponent using the distance-size hierarchical scaling, and the
other is estimate the parameter using 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
13
direct result (σ value) and the indirect result (qD value) will be large. If we eliminate the bottom
level as an outlier, 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 a significant deviation of
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 the intervening opportunities resulting from city
development (See Stouffer (1940) for the concept of intervening opportunity).
Table 3 The estimated values of model parameters of the hierarchies of China’s cities based on all
data points and scaling ranges
Scale
Parameter
2000
2010
Estimated value
R
2
Estimated value
R2
All data points
(Including
bottom level)
q
D
σ
qD
1.0355
1.7486
1.7774
1.8106
0.9403
0.9525
0.8631
--
1.1047
1.7571
1.9199
1.9411
0.9314
0.9822
0.8950
--
Scaling range
(Eliminating
bottom level)
q
D
σ
qD
0.9294
1.7008
1.5791
1.5808
0.9997
0.9527
0.9503
--
0.9889
1.7280
1.7109
1.7088
0.9996
0.9823
0.9843
--
10000
10000
Pm = 0.0107Lm1.7109
R² = 0.9843
Pm = 0.0176Lm1.5791
R² = 0.9503
1000
Average size Pm
Average size Pm
1000
100
10
100
10
1
1
10
100
1000
10
Distance Lm
100
1000
Distance Lm
a. 2000
b. 2010
Figure 5 The hierarchical scaling relationships between the minimum distances of China’s cities
and the corresponding average city size [Note: The small circles represent outliers indicative of the
lame-duck classes]
14
Another factor that impacts the estimated value of the scaling exponent is spatial patterns of
cities. The spatial distribution of urban population is often of self-affine pattern rather than
self-similar patterns. In other words, urban population distribution is of anisotropy instead of
isotropy. As a result, the scatterpoints on a double logarithmic plot form two trendlines rather than
one trendline. 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 distribution. The spatial
distributions of Chinese cities bear significant self-affinity (Figure 6). However, it seems to evolve
from self-affine patterns into self-similar patterns because the difference between the fractal
dimension values of the two scaling ranges become 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 above-mentioned. The
box-counting method has been employed to estimate the fractal dimension of urban form
(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 is about q=0.968 (Jiang
and Yao, 2010), and the box dimension of the central place network is about D=1.8585(Chen,
2014b), thus the distance exponent of the gravity model of Henan’s urban system is about
b=0.968*1.8585=1.799 in 2000.
Table 4 The fractal dimension values of population distributions based on bi-scaling ranges
Item
2000
2010
2
Parameter
R2
2.1669
1.5474
0.9898
0.9597
Parameter
R
Scaling range 1 (Lm≤ L6)
Scaling range 2 (Lm≥ L6)
2.5915
1.2944
0.9762
0.8814
Intersection (L6)
254.0088
212.2135
Average of two parameters
Difference of two parameters
1.9430
1.2971
1.8572
0.6195
15
10000
10000
Pm = 0.1066Lm1.2944
R² = 0.8814
1000
Average size Pm
Average size Pm
1000
Pm = 0.0299Lm1.5474
R² = 0.9597
100
10
100
10
0.0001Lm2.5915
Pm = 0.0011Lm2.1669
R² = 0.9898
Pm =
R² = 0.9762
1
1
10
100
1000
10
100
Distance Lm
1000
Distance Lm
a. 2000
b. 2010
Figure 6 The self-affine patterns and the bi-scaling relationships between the minimum distances
of China’s cities and the corresponding average city size [Note: The small circles represent the
intersections of the two scaling ranges]
4 Discussion
4.1 Evaluation of different methods
On the basis of the empirical studies, a preliminary evaluation on the three approaches to
estimating the distance exponent can be made as follows. The first approach, the size-distance
scaling, is simple and direct, but it is sometimes hindered by self-affine spatial distributions. The
second approach, the 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 tabulated as below
(Table 5).
Table 5 A comparison between three approaches to estimating the distance exponent
Method
Model
Formula
16
Merit
Defect
The first
approach
The second
approach
The third
approach
D
Pm = κLmp
Nm = πL−mD ,
Nm = μPm− p
Direct and
simple
Sensitive to
self-affine patterns
q = R2 / p ,
Easy to
understand
Indirect influenced
by self-affinity
Easy to
calculate
Indirect and
influenced by
sample difference
b = qD
P(r ) = C(r + k )−q ,
N (ε ) = N1ε
b = DP
− Db
b = qDb
4.2 Pending questions and their answers
Using the mathematical relations derived above, we can develop the gravity theory of human
geography and solve the difficult problems which are pendent for a long time. One problem is the
question of gravity constant. During the quantitative revolution of geography (1953-1976),
geographers tried in vain to find the 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 the 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 the 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 of the basic natures of geographical systems is regional difference. The
condition of one geographical region differs to another geographical region. Thus the average
fractal dimension of urban population of one region is different from that of another region.
Another problems remaining to be solved is the dimension dilemma. According to dimensional
analysis, the expected value of the distance exponent of the gravity model in theory should equal 1,
2, or 3, which is dimension value of Euclidean geometry. However, the observed values of
empirical researches often deviated the theoretical values significantly, and take 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
17
of urban population (Chen, 2015). A fractal dimension always exhibits a fractional number. So it is
not surprised that the empirical results of the distance exponent estimation fail to equal 1, 2, or 3.
If we definite our study area within a 2-dimensional space, the distance exponent will range 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 comes 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).
5 Conclusions
By this study based on cities, we can reconsider the geographical gravity model and its
parameter. New findings rest with the fractality and scaling behind the model, the fractal
dimension property of distance exponent, and the effective method of parameter estimation. All
these are based on the hierarchy 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 self-similar hierarchy in essence. Before the advent of the fractal geometry, the gravity
model is explained by the idea from Euclidean geometry. However, geographical systems are
fractal systems with fractional dimension. Many difficult problems such as dimensional dilemma
and gravity constants resulted from the traditional explanation in geography. Today, many
theoretical problems relating to the gravity model and its parameters can be easily solved using the
ideas from fractals, scaling, and allometry based on hierarchical structure.
Second, the distance exponent of the gravity model is a kind of fractal dimension
indicating given geographical quantity of matter. The property of fractal dimension depends on
the size measurements such as population size, urban area, and economic product. If we use the
population sizes of cities to determine the attraction power, the distance exponent will represent
the average fractal dimension of urban population of the 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 area of the cities in a geographical region. The
18
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 rank-size 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 the 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.
41171129). The supports are gratefully acknowledged.
References
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
Models, and Fractals. Cambridge, MA: The MIT Press
Batty M (2008). The size, scale, and shape of cities. Science, 319: 769-771
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
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
19
Chen YG (2012a). The rank-size scaling law and entropy-maximizing principle. Physica A: Statistical
Mechanics and its Applications, 391(3): 767-778
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 (2012c). Zipf's law, 1/f noise, and fractal hierarchy. Chaos, Solitons & Fractals, 45 (1): 63-73
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: Statistical Mechanics and its Applications, 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, Zhou YX (2006). Reinterpreting central place networks using ideas from fractals and
self-organized criticality. Environment and Planning B: Planning and Design, 33(3): 345-364
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
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
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
20
York, NY: W.H. Freeman
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: Springer-Verlag
Kang CG, Ma XJ, Tong DQ, Liu Y (2012). Intra-urban human mobility patterns: An urban morphology
perspective. Physica A: Statistical Mechanics and its Applications, 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
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
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
21
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
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
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
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
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
22