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Title

A controllable model of a random multiplicative
process for the entire distribution of population

Author(s)Tomita, Shinji; Hayashi, Yukio

Citation

Physica A: Statistical Mechanics and its
Applications, 387: 1345‑1351

Issue Date

2008‑02‑15

Type

Journal Article

Text version

author

URL

http://hdl.handle.net/10119/7914

RightsNOTICE: This is the author's version of a work
accepted for publication by Elsevier. Shinji
Tomita, Yukio Hayashi, Physica A: Statistical
Mechanics and its Applications, 387, 2008, 1345‑1351,
http://dx.doi.org/10.1016/j.physa.2007.09.028

Description

Japan Advanced Institute of Science and Technology

A Controllable Model of Random Multiplicative Process for Entire
Distribution of Population
Shinji Tomita ∗ and Yukio Hayashi
School of Knowledge Science, Japan Advanced Institute of Science and Technology
1-1 Asahidai, Nomi-city, Ishikawa 923-1292

Abstract
A random multiplicative process (RMP) is one of the basic models which can generate a power law distribution.
Actually, the distribution generated by RMP has two parts, which are closely matched to the head of a lognormal
distribution and the tail of a power law distribution. We investigated the relation between a shape of distributions
and model variables. By changing input variables, we explained the origin of the cumulative population
distributions of municipalities and prefectures in Japan from 1980 to 2006. This controllability of RMP can beapplied to a power law distribution in various other fields.
Keywords: Random Multiplicative Process; controllability; cumulative population distribution;

1.

Introduction
The evolution of the population seems to change with time due to various factors such as economy,

environment and policy, etc. However, there is empirical regularity about the distribution of the
population. Zipf ’s law for cities is one of the most striking empirical facts. Zipf ’s law: In terms of
distribution, this means that the probability that the size of a city is greater than some S isapproximately proportional to 1/S. N(Size ≥ S) = aS -β with β ≅ 1 in general. Zipf ’s law holds acrosscountries with very different economic structures and histories such as the early and modern UnitedStates [1], France and Japan in the twentieth century [2], China in the mid-nineteenth century [3],and India in the early twentieth century [4].
Though this law seems to be robust, in previous literature on Zipf ’s law, only large cities wereconsidered [5]. This implies that most samples of small cities were ignored. However, when we think
about entire distribution of population, we cannot ignore small cities. Actually, in the high growth of
the Japanese economy after 1955 a great demographic shift happened, since the young persons
migrated to the city regions from the rural regions. Forty percent of municipalities are specified as
depopulated areas now. We can’t see the entire problem by considering only large cities. Therefore, theentire distribution of population and its generation mechanism should be studied.
The purpose of this paper is twofold. First, we explain the cumulative population distributions of(all) municipalities and prefectures in Japan. Second, we investigate the origin of those distributions
by a random multiplicative process (RMP), which is a simple model. We show that the real data of
cumulative population distribution are almost constant, and this data can be generated by controllingvariables of the RMP. The organization of this paper is as follows. In the next section, we analyze thecumulative population distribution of all municipalities and prefectures in Japan from 1980 to 2006.
In Section 3, we introduce the RMP. In Section 4, we examine the features of the distribution
generated by the RMP. In Section 5, we fit simulated results with real data. In Section 6, we∗Corresponding author.

E-mail addresses: s-tomita@jaist.ac.jp (Shinji Tomita), yhayashi@jaist.ac.jp (Yukio Hayashi)

summarize our results.
2.

Data analysis
We have a total population of 127.92 million in Japan at present. The population in 2006 was thetenth largest in the world, equivalent to 2.0 percent of the global total 1 . This population occupies over1800 municipalities and 47 prefectures. In this paper, we treat two kinds of data. The first is
population of municipalities (micro scale) and second is prefectures (medium scale). We analyze the
distribution of population in the range from micro-scale aspect and medium-scale one, and investigate
the common features. We acquired the data of municipalities 2 and prefectures 3 from the Statistics
Bureau in Japan. The main advantage of using these census data is that they cover the entire
population distribution. Municipality is a generic name for a city, a town, or a village. Most
municipalities have very few inhabitants. A number of municipalities have been eliminated after agreat merge in 2000. Table 1 shows transition of number of municipalities; the total number ofmunicipalities has decreased by 1435 in these 26 years.
Table.1 Number of Municipalities
Year
Number ofmunicipalities

1980

1985

1990

1995

2000

2006

3257

3253

3253

3234

3229

1822

Number of eliminatedmunicipalities

-4

0

-19

-5

-1407

A prefecture consists of a larger area including municipalities (cities, towns, and villages). The
number of prefectures has been 47 for a long time. Figure 1 shows the cumulative population
distributions of municipalities (a) and of prefectures (b), respectively. From these figures, the entire
distribution seems not to have changed so much in these 26 years. The distributions seem to be divided
into upper part (upper 20%) and lower part (lower 80%), both on a micro scale (municipalities) and a
medium scale (prefectures). The upper parts of the distributions of municipalities and prefectures
correspond to over 30000 and 4000000 inhabitants. This means that the distribution deviates from a
power law in the lower 80% part. We call the lower 80% part of the distribution the “head”, and theupper 20% part the “tail”. These two parts are similar to the double Pareto distribution, which is
closely matched to the head of a lognormal distribution and the tail of a power law distribution [6]. In
the tail part, a power law can be expressed as

N ( x) = ax − β(1)where N(x) and x are the rank of a site (municipality or prefecture) and the population, respectively.

β denotes an exponent in the power law. By taking logarithm, Eq.(1) is rewritten asln N ( x) = ln a − β ln x(2)

After calculating a regression expression in the tail part by least squares method, the exponent of

http://memorva.jp/ranking/unfpa/who_2006_population.php
Data of 1980, 85, 90, 95 and 2000: http://www.stat.go.jp/data/kokusei/2000/kako/danjo/index.htm
2006: http://www.soumu.go.jp/c-gyousei/020918.htm
3 Data of 1980, 85, 90 and 95: http://www.stat.go.jp/data/jinsui/wagakuni/index.htm
2000: http://www.stat.go.jp/data/jinsui/2-3.htm
2006: http://www.soumu.go.jp/c-gyousei/020918.html
12

municipalities is 1.19 ± 0.04, and the exponent of prefectures is 2.49 ± 0.06. It can be said that these
distributions are stationary states.
(a)

(b)

104

Municipality 1980
1985
1990
1995
2000
2006

103

10

head
N(x)

N(x)

head

2

10

2

Prefecture 1980
1985
1990
1995
2000
2006

1

10

tail
101

tail
10

0

10

0

2

10

3

4

5

10

10

6

10

7

10

10

6

7

10

x

10
x

Fig. 1. Log-log plots of the rank of the site N(x) vs. the population x from 1980 to 2006. (a) municipalities and (b)prefectures. The solid lines are power laws with the exponents 1.2 (a) and 2.5 (b), respectively. Dotted lines are thediverging point of head (lower 80%) and tail (upper 20%).

3.

The controllable model
We consider a Random Multiplicative Process (RMP) proposed by Takayasu [7, 8] to generate thereal population distributions of municipalities and prefectures. The RMP is one of the basic models
which can generate a power law distribution in the upper tail. The fundamental idea of the RMP waspointed out by Champernowne [9], and the mathematical formalization is given by Kesten [10]. The
RMP is a stochastic process with multiplicative and additive noises. The model equation ischaracterized by the following stochastic time evolution equation:xi (t + 1) = bi (t ) xi (t ) + f i (t )i = 1,2,...N(3)

where xi(t) denotes the population of the i-th municipality or prefecture at time t. bi(t) > 0 is amultiplicative noise; xi(t+1) increases for bi(t) > 1 and decreases for bi(t) < 1, and fi(t) is an additivenoise. Takayasu set the distribution functions of bi(t) and fi(t) according to the Poisson distribution andthe symmetric Gaussian, respectively. However, the result of a power law distribution doesn’t dependon these or any other settings [11]. In other words, we can obtain it from any kind of distribution.
In the context of population dynamics, bi(t) and fi(t) can be interpreted as the growth rate andunexpected comings and goings, respectively. Because these noises include various factors, it isdifficult to specify the explicit distribution form. A further argument is necessary to specify thephysical meanings of noise. For the sake of simplicity, we assume bi(t) and fi(t) are generated by
uniformly random noises in a certain range, independent of time, temporarily. There are two reasonsthat we assume bi(t) and fi(t) to be uniformly random noises in a certain range. First is that growthrate bi(t) and unexpected comings and goings fi(t) will be bounded variables, not infinite. Second is thatuniform distribution may include various cases, because it does not assume specified distribution.
We should remark that random multiplicative noise bi(t) corresponds to growth rate, which does

not depend on the size of the sites . This is confirmed in the United States [12] and Japan [13]. All
cities follow similar processes; i.e., their growth processes have a common mean (equal to the meancity growth rate). This homogeneity of growth processes is often referred to as Gibrat’s law [14].
The RMP can generate a power law distribution with the specified exponent β. The relationshipbetween exponent β and multiplicative noise b(t) is given by the expectationbβ = 1(4)This relation leads to the following expression.
bmax
1b β db = 1∫
bmax − bmin bmin

(5)

where bmax and bmin are the upper and lower limits of multiplicative noise b(t), respectively. Figure 2
illustrates the method of deciding the range of multiplicative noise b(t). The areas of ① and ② areequal. For examples, we obtain bmin = 0.24 and baverage = 0.92 by calculation from Eq.(5) after setting β =2.0 and bmax = 1.6. If bmax decreases, range of b(t) narrows and baverage increases. Conversely, if bmax
increases, range of b(t) widens and baverage decreases. From these relations between range of b(t) andbaverage, after this, we focus on baverage.
4
3.5
3

bβ

2.5
2
1.5

②

1

①

0.5
0
0

bmin

0.5

1

1.5 bmax

2bFig. 2. Method of deciding range of multiplicative noise b(t). The random variable b(t) is generated by a uniform
random number within range [bmin ,bmax].

4.

The relationship between variables and distribution
Before fitting the model with the real data, we examined the features of the distribution generated

by the RMP. First, we investigated the relationship between noise and shift of distribution. We set β =2.5, baverage = 0.99, frange = 1 and number of sites = 1000, respectively. Figure 3 shows the relationshipbetween additive noise f(t) and shift of distribution. We can shift the distribution to the right byincreasing either faverage or baverage. Similarly, we can shift it to the left by decreasing either faverage orbaverage.

10

3

10

2

N(x)

4000
20000
40000

101

10

0

10

4

10

5

10
x

6

10

7

8

10

Fig. 3. The relationship between additive noise f (t) and shift of distribution. Each symbol stands for faverage;(○) = 4000, (☓) = 20000, (□) = 40000

The relationship between noise and shift of distribution is summarized in Table 2. When we fix

baverage and want to shift the distribution to the left, we should decrease faverage. Conversely, when we fix
baverage and want to shift the distribution to the right, we should increase faverage. Decreasing baveragecorresponds to widening brange. Increasing baverage corresponds to narrowing brange. When we fix faverage
and want to shift the distribution to the left, we should decrease baverage. Conversely, when we fix faverage
and want to shift the distribution to the right, we should increase baverage. We confirmed that thestationary distribution does not depend on the value of frange.

Table 2. The relationship between noise and shift of distribution
Distribution

Noise

b

f

Shift to left

Shift to right

Average

Decrease

Increase

Range

Wide

Narrow

Average

Decrease

IncreaseRange

No Influence

Next, we investigated the relationship between the number of sites and the distribution range.
Figure 4 shows the cumulative distributions according to the number of sites. We can expand the
distribution by increasing the number of sites. Especially, head and tail are expanded by increasing
the number of sites. The distribution range of municipalities is wide, because the number ofmunicipalities is very large (over 1600). On the other hand, the distribution range of prefectures isnarrow, because the number of prefectures is very small (47).

10

4

100
1000
10000

N(x)

103

10

2

10

1

10

0

10

5

10

6

10
x

7

10

8

9

10

Fig. 4. The relationship between the number of sites and the distribution range. Log-log plots for the cumulative
population distribution. Each symbol stands for number of sites; (○) = 100, (☓) = 1000, (□) = 10000.

From examining the features of the RMP, we can control the shift of distribution, which depends onthe values of additive noise f(t) and multiplicative noise b(t), and we can control the distribution range,
which depends on the number of sites.

5.

The data fitting
To fit the model with the real data, we search appropriate variables systematically by applying abinary search. Though specific distribution can be generated just by controlling one noise, additivenoise f(t) or multiplicative noise b(t), here, we set β, baverage and number of sites respectively, andsearch for faverage. As a procedure, we repeat RMP → Compare sum of the mean squared errors (MSE)of real data and simulated result → RMP…, until we arrive at an appropriate faverage. Figure 5 shows

sum of the MSE

procedure of search for faverage.

①

④

⑤

③

②

f average

Fig. 5. Procedure of search for faverage. When faverage approaches the appropriate value, sum of MSE decreases.

Procedure of search for faverage is as follows,

Step 1. Chose an initial very small faverage ① as the minimum initial value, and calculate a regression
expression in the tail part of simulated result. Calculate the following sum of the MSE
between the real data and simulated result on the approximation line. Sum of the MSE is asfollows,S e = ∑ ( yreal − ysim )2(6)where yreal and ysim are rank of the site on approximation lines of real data and simulated
result, respectively.
Step 2. Chose very large faverage ② as the maximum initial value and calculate the sum of the MSE,
the same as in Step 1.
Step 3. Chose middle point ③ of ① and ②.
Step 4. Compare sum of the MSE ① and ②,
If ①’s error is smaller than ②’s, chose middle point ④ of ① and ③.
Step 5. Compare sum of the MSE ① and ③,
If ③’s error is smaller ①’s, chose middle point ⑤ of ③ and ④…
Step 6. When the shift value of faverage becomes less than 1, these processes are stopped.

Fitting the simulated result with the real data
First, we investigated fitting the simulated result with two points of municipalities’ real data (1980
and 2006), because the number of municipalities has decreased greatly in this period. For 1980 data,
we set β = 1.16, baverage = 0.99 and number of sites = 3257, respectively, and searched for faveragecorresponding to these constants. We obtained faverage as 67.62, sum of MSE as 930000.1, and MSE as
1.49. For 2006 data, we set β = 1.31, baverage = 0.99, and number of sites = 1822, respectively, andobtained faverage as 215.70, sum of MSE as 174322.2, and MSE as 1.15.
Similarly, we fitted simulated result with the average of prefectures real data from 1980 to 2006.
We set β = 2.49, baverage = 0.99 and number of sites = 47, respectively, and obtained faverage as 34763.9,
sum of MSE as 2.44, and MSE as 0.17.
In Figure 6, not only the approximation line of simulated result and real data, but also the whole
distributions, overlap almost completely. These results show the simulated results generated by the
RMP and the real data are in very good agreement.

(b)

104

10

3

10

2

10

1

10

0

102

1980 realsim

104

2006 real
sim

N(x)

N(x)

(a)

103

Prefecture real
sim

101

102
101
100
102

2

10

103

3

10

104

105

10

106

4

107

105

106

107

100

6

7

10

10

x

x

Fig. 6. Log-log plots of the rank of the sites N(x) vs. the population x from 1980 to 2006; (a) municipalities and (b)prefectures. The solid lines and dotted lines represent the approximation lines of real data and simulated result,respectively. Symbol (○) and (☓) stand for real data and simulated result, respectively.

Trade-off relationship between variables
In the previous subsection, we searched for faverage in the setting of β, baverage and number of sites.
However essentially, even if one variable is fixed or changed, specified distribution can be generated by
the RMP. baverage is on trade-off relations with faverage. Here, we research the trade-off relation between

baverage and faverage when distribution of municipalities in 2006 is generated. Figure 7 shows therelationship between baverage and faverage. It is obvious that baverage is proportional to faverage. Because bi(t)is a non dimensional variable (growth rate) and fi(t) is a dimensional variable (number of people), thosescales are different. When the average of the growth rate (baverage) is small, large faverage is necessary to
reproduce the population distribution of Japan. On the contrary, faverage may be small, when the
average of the growth rate (baverage) is large.

This implies that there are many combinations of noises

which can reproduce real population distribution. One of these combinations might correspond to thenoise of the real world. It will be a further study to estimate what kinds of combinations are realistic.
6000
5000

f average

4000
3000
2000
1000
0
0.94

0.95

0.96

0.97
0.98
b average

0.99

Fig. 7. The relationship between baverage and faverage.

1

6.

Concluding Remarks
In summary, we have investigated the cumulative population distribution of all sites(municipalities and prefectures) in Japan from 1980 to 2006, and they seem not to have changed somuch in these 26 years. Especially, the exponent in tail part (β ) has hardly changed. This appears
similar to the double Pareto distribution, which is closely matched to the head of a lognormal
distribution, and the tail of a power law distribution. We reproduced these distributions by the RMP
which combined two kinds of noises recursively. In the RMP, we indicated that the entire distributionof population can be reproduced by using the exponent in tail part. It is very interesting to be able toreproduce the population distribution by such a simple mechanism. Additionally, we clarified thefeatures of the distribution generated by the RMP under noise control. Multiplicative noise baverage ison trade-off relations with additive noise faverage. The controllability by changing noises can be applied
to a power law distribution in various other fields.
For future work, we should investigate the social meaning of noises. Specifying the meaning ofthese noises and knowing methods to control them may be very useful for planning various policies.

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