# The Lévy Flight of Cities: Analyzing Social-Economical Trajectories with Auto-Embedding

Authors: Linfang Tian; Kai Zhao; Jiaming Yin; Huy Vo; Weixiong Rao
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

Lévy flights are useful in stochastic measurement, such as biology, human mobility, and financial mathematics, where trajectories contain many short flights and some long flights, and the step-lengths follow a power-law distribution. For cities inhabited by human beings, this paper proposes to use latent factors to study social-economical development trajectories and observes that the trajectories of four Asian cities exhibit Lévy flight characteristics. We collect the social and economical data such as GDP, goods producing industries, population, and general tertiary education, and map these data into social or economical factor through a deep-learning embedding method auto-encoder. We find that the step-lengths of these urban social-economical trajectories can be fitted approximately as a power-law distribution. We use the stochastic multiplicative processes (SMPs) to explain such pattern, wherein the presence of a boundary constraint, the SMP, leads to a power-law distribution. It means that these urban social-economical trajectories follow a Lévy flight pattern, where some years have large changes and many years have little changes.


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## Body

## 1. Introduction

We cannot see the financial flows that fund the economical development, although we see the economical prosperity; we cannot see the cultural atmosphere that nourished the social progress, although we see the social civilization. In a large-dimensional factor model [1, 2], statistical (latent) factors estimated from the data are particularly appealing due to the underlying factor structure is usually unknown. Latent factors have been successfully used in economics and finance, such as analyzing, predicting and forecasting U.S. macroeconomic indicators [3], and asset pricing [4, 5]. In this paper, we use latent factors, namely, social factor and economical factor, to construct the urban social-economical development trajectory.Many researchers have explored computational social or economical science models. Reference [6] explores the control of quantitative economic management variables under the numerical method based on stochastic ordinary differential equations; reference [7] analyzes the influence of the industrial economy on the social economy based on multiple linear regression equation; reference [8] explores the term structure of economic management rate under the parameter analysis of the estimation model based on common differential equation; [9] explores the Ivancevic option pricing model in the economy; reference [10] analyzes the relationship between industrial agglomeration and regional economic growth based on the multiobjective optimization model; and reference [11] explores multiattribute decision method based on normal random variable in economic management risk control. Previous researches have shown that trajectories in human society exhibit the statistically similar Lévy flight characteristics, such as the traces of bank note [12], human mobility [13], and COVID-19 confirmed cases [14]. Many studies have shown that the rank-order of cities in the United States [15], India [16], and Spain [17] shows Lévy flights characteristics, following Pareto law [15, 18] or Zipf’s law [19, 20]. Researchers have also employed power-law adjusted metropolitan indicators [21] to assess urban land use efficiency in Chinese cities. In this paper, we explore the Lévy flight characteristics of urban social-economic development trajectory.We study the data sets of four Asian cities: two in China including Hong Kong and Shanghai, the third is Singapore, and the fourth is Tokyo, Japan (see Table1). In Asia, Tokyo emerged as a major financial center in the 1980s as the Japanese economy became one of the largest economies in the world [22]. Hong Kong and Singapore developed soon after leveraging their links with London and Britain [23]. In the 21st century, Shanghai has emerged as a hub for international trade and finance.Table 1
Data sets of Hong Kong, Shanghai, Singapore, and Tokyo. AFMQ represents agriculture, fishing, mining, and quarrying. GPI represents goods producing industries. SPI represents services producing industries. Per capita GDP represents per capita GDP at current market prices.
ClassDescriptionHKSHASGTYOEconomyGDP at current market prices1981–20191978–20191960–20191968–2019Agriculture, fishing, mining and quarrying1981–20191978–2019NANAProportion of AFMQ1981–20191978–2019NANAGoods producing industries1981–20191978–20191960–20191968–2019Proportion of GPI1981–20191978–20191960–20191968–2019Services producing industries1981–20191978–20191960–2019NAProportion of SPI1981–20191978–20191960–2019NAPer capita GDP1981–20191978–2019NANAPer capita GDP (dollar)NA1978–2019NANAGovernment revenue1981–20191978–2019NANAGovernment expenditure1981–20191978–20191960–2019NAPersonal incomeNANANA1968–2019Original insurance incomeNA1978–2019NANAOriginal insurance pays outNA1978–2019NANAGross fixed capital formationNA1978–20191960–2019NAGross agricultural productionNA1978–2019NANAGross industrial productionNA1978–20191960–20191968–2019SocietyPopulation1981–20191978–20191960–20191968–2019LaborNANANA1968–2019General tertiary education1981–20191978–20191960–20191968–2019Ordinary secondary school1981–20191978–20191960–20191968–2019Ordinary primary school1981–20191978–20191960–20191968–2019Book publicationsNA1978–2019NANAJournals publicationsNA1978–2019NANANewspaper publicationsNA1978–2019NANAWe assume that cities evolve in two directions [24]: one is economical growth, and the other is the development of social civilization. Economical factors include GDP at current market prices, goods producing industries, and service producing industries; and social factors contain population, education, and publication. This paper covers the most commonly used data types for measuring urban development [17]. Firstly, it is clear that the research object is the step-lengths of urban development trajectory [25], with economical development as horizontal axis and social development as vertical axis. Here, we perform min-max normalization on all the data. Then, we apply the recently popular artificial neural network embedding technique, namely auto-encoder [26], on all economical factors to extract a latent vector, and the same is done on social factors. Secondly, we use a designed collection of distribution models to fit the step-lengths distribution of urban trajectory, and determine the optimal one according to Akaike information criterion [27]. The results show that the step-lengths distribution of Hong Kong’s development trajectory is more in line with a truncated power-law distribution [28]; Shanghai and Tokyo are more power-law; Singapore is more in the pattern of exponential distribution. To the best of our knowledge, this is the first work that explores urban social-economical development trajectory using computational social science models and explains the general model behind it.The contribution of this paper is three-fold. In Section 1, we extract the urban social-economical development trajectory. In Section 2, we demonstrate that log-normal distribution [29–31] in the presence of a lower-boundary constraint, approximately yields a power-law distribution. This result is a step towards explaining the emergence of Lévy flight patterns in urban development. In Section 3, we consider city as an evolving organism [32] and use the stochastic multiplicative processes to explain the power-law patterns in development trajectory step-lengths.


## 2. Methods

### 2.1. Data Sets

We collect the official data set of Hong Kong (see Table2), Shanghai (see Table 3), Singapore (see Table 4), and Tokyo (see Table 5) in our work. The data sets of the four Asian cities are collated and matched (see Table 1).Table 2
The URL of HK dataset.
DescriptionURLGDPhttps://www.censtatd.gov.hk/sc/web_table.html?id=31\#Per capita GDPhttps://www.censtatd.gov.hk/sc/web_table.html?id=31\#Primary industryhttps://www.censtatd.gov.hk/sc/web_table.html?id=35\#Secondary industryhttps://www.censtatd.gov.hk/sc/web_table.html?id=35\#Tertiary industryhttps://www.censtatd.gov.hk/sc/web_table.html?id=35\#Proportion of primary industryhttps://www.censtatd.gov.hk/sc/web_table.html?id=36\#Proportion of secondary industryhttps://www.censtatd.gov.hk/sc/web_table.html?id=36\#Proportion of tertiary industryhttps://www.censtatd.gov.hk/sc/web_table.html?id=36\#Government revenuehttps://www.censtatd.gov.hk/sc/web_table.html?id=193\#Government expenditurehttps://www.censtatd.gov.hk/sc/web_table.html?id=194\#Populationhttps://www.censtatd.gov.hk/sc/web_table.html?id=1A\#Labour forcehttps://www.censtatd.gov.hk/sc/web_table.html?id=6\#Primary schoolhttps://www.censtatd.gov.hk/sc/scode370.html\#section6Secondary schoolhttps://www.censtatd.gov.hk/sc/scode370.html\#section6Universityhttps://www.censtatd.gov.hk/sc/scode370.html\#section6Table 3
The URL of SHA dataset.
DescriptionURLGDPhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C0401.htmPrimary industryhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C0401.htmSecondary industryhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C0401.htmTertiary industryhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C0401.htmIndustryhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C0401.htmGeneral public budget revenuehttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C0501.htmGeneral public budget expenditurehttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C0501.htmProportion of primary industryhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C0404.htmProportion of secondary industryhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C0404.htmProportion of tertiary industryhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C0404.htmProportion of industryhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C0404.htmTotal fixed asset investmenthttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C0701.htmGeneral public budget revenuehttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C0501.htmGeneral public budget expenditurehttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C0501.htmGross agricultural productionhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C1201.htmGross industrial productionhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C1301.htmOriginal insurance incomehttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C1801.htmOriginal insurance pays outhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C1801.htmResident population at year-endhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C0201.htmRegistered population at year-endhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C0201.htmGeneral higher educationhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C2103.htmOrdinary secondary schoolhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C2103.htmOrdinary primary schoolhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C2103.htmBook print runhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C2316.htmJournal print runhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C2317.htmNewspaper print runhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C2318.htmTable 4
The URL of SG dataset.
DescriptionURLGDPhttps://tablebuilder.singstat.gov.sg/table/TS/M015241Goods producing industrieshttps://tablebuilder.singstat.gov.sg/table/TS/M015241Services producing industrieshttps://tablebuilder.singstat.gov.sg/table/TS/M015241Goods proportionhttps://tablebuilder.singstat.gov.sg/table/TS/M015241Services proportionhttps://tablebuilder.singstat.gov.sg/table/TS/M015241Government consumptionhttps://tablebuilder.singstat.gov.sg/table/TS/M015241Gross fixed capital formationhttps://tablebuilder.singstat.gov.sg/table/TS/M015051Total populationhttps://tablebuilder.singstat.gov.sg/table/TS/M810001\#!Government expenditure on eduhttps://tablebuilder.singstat.gov.sg/table/TS/M850011Primary schoolshttps://tablebuilder.singstat.gov.sg/table/TS/M850011Secondary schoolshttps://tablebuilder.singstat.gov.sg/table/TS/M850011Tertiaryhttps://tablebuilder.singstat.gov.sg/table/TS/M850011Literacy ratehttps://tablebuilder.singstat.gov.sg/table/TS/M850001Table 5
The URL of TYO dataset.
DescriptionURLLoanshttps://www.toukei.metro.tokyo.lg.jp/tnenkan/2019/tn19q3i015.htmManufactured goodshttps://www.toukei.metro.tokyo.lg.jp/tnenkan/2019/tn19q3i016.htmGDPhttps://www.toukei.metro.tokyo.lg.jp/tnenkan/2019/tn19q3i016.htmPrefectural incomehttps://www.toukei.metro.tokyo.lg.jp/tnenkan/2019/tn19q3i016.htmPopulationhttps://www.toukei.metro.tokyo.lg.jp/tnenkan/2019/tn19q3i002.htmLaborhttps://www.toukei.metro.tokyo.lg.jp/tnenkan/2019/tn19q3i002.htmChildren and studentshttps://www.toukei.metro.tokyo.lg.jp/tnenkan/2019/tn19q3i017.htmElementary schoolshttps://www.toukei.metro.tokyo.lg.jp/tnenkan/2019/tn19q3i017.htmJunior secondaryhttps://www.toukei.metro.tokyo.lg.jp/tnenkan/2019/tn19q3i017.htmSenior secondaryhttps://www.toukei.metro.tokyo.lg.jp/tnenkan/2019/tn19q3i017.htmUniversitieshttps://www.toukei.metro.tokyo.lg.jp/tnenkan/2019/tn19q3i017.htmHong Kong data set (see Table2) is a set of official data with 14 factors, such as GDP at current market prices, agriculture, fishing, mining and quarrying, population, and general tertiary education, collected over 39 years (from 1981 to 2019) on the official website of Census and Statistics Department of Hong Kong, China. This data set contains 546 numerical values.Shanghai data set (see Table3) is a set of official data with 23 factors, such as GDP at current market prices, goods producing industries, population, and general tertiary education, collected over 42 years (from 1978 to 2019) on the official website of Municipal Statistics Bureau of Shanghai, China. This data set contains 966 numerical values.Singapore data set (see Table4) is a set of official data with 12 factors, such as GDP at current market prices, goods producing industries, population, and general tertiary education, collected over 60 years (from 1960 to 2019) on the official website of Singapore Department of Statistics. This data set contains 720 numerical values.Tokyo data set (see Table5) is a set of official data with 10 factors, such as GDP at current market prices, population, labor, and general tertiary education, collected over 52 years (from 1968 to 2019) on the official website of statistics division, Bureau of General Affairs, Tokyo, Japan. This data set contains 520 numerical values.


### 2.2. Obtaining Each Factor’s Step-Lengths

For each factor, we use this year’s amount minus last year’s amount to get year change amount as raw step-length. We find that change amount of the GDP ranges from dozens to thousands, while that of proportion of goods producing industries ranges between−0.2 and 0.5. These raw step-lengths of different factors differ greatly. We suffer from an information overload of these factors, even though more factors provide the opportunity to understand social or economical phenomena better. Thus, we have to use dimension reduction to organize these data.


### 2.3. Obtaining Social-Economical Trajectory by Embedding

In this paper, we try to get embedding layer through training auto-encoder (AE), which is a type of artificial neural network. Firstly, we classify the data into two classes: regarding GDP, per capita GDP, goods producing industries, etc., as economical variables; regarding population, general tertiary education, etc., as social variables. Secondly, we use min-max scaling to make sure variables that are measured at different scales contribute equally to the model fitting. Thirdly, in the AE, the input feature, the data set of economical variables or social variables, is transformed into one latent space with the encoder and then reconstructed from latent space with decoder. The encoder is used as a dimension reducer. To train this AE, an Adam algorithm is applied as an optimizer and mean square error (MSE) as a loss function.We use two-layer fully connected networks as the encoder and decoder, and the loss function is(1)ℒx,x′=x−x′2=x−fhx2,where x∈ℝn is the input feature of one year and n is the number of variables. The data from m years construct m training samples. x′ is the output of the decoder. The computation of the encoder and decoder is defined as(2)hx=σW2σW1x+b1+b2,fy=σ′W2′σ′W1′y+b1′+b2′,where W1,W2,W1′,W2′,b1,b2,b1′,b2′ are learnable parameters of the network and σ,σ′ are the activation functions. We train the AE by minimizing the MSE of the input feature and the output of the decoder. And the output of the encoder hx is the embedding of the original input. Using the embedding technique to reduce the dimensions of the data set into two dimensions: economy and society. Then, we draw the urban development trajectory (see Figure 1) with economical growth as horizontal coordinate and social growth as the vertical coordinate. We extract the step lengths, as the information from the graph.Figure 1
The socialeconomical trajectories of urban development. We extract economical and social coordinates by Auto-Encoder, respectively, and draw (a) HK Trajectory, (b) SHA Trajectory, (c) SG Trajectory, and (d) TYO Trajectory with economy as the horizontal axis and society as the vertical axis.
(a)(b)(c)(d)


### 2.4. Identifying the Scaling Exponent

To fit a heavy-tailed distribution such as a power-law distribution, we need to determine what portion of the data to fitxmin and the scaling exponent α. We use the methods from [37] to determine xmin and α. We create a power-law fit starting from each value in the data set. Then, we select the one that results in the minimal Kolmogorov–Smirnov distance between the real data and the fitted one as the optimal value of xmin. After that, the scaling exponent α in the power-law distribution is given by(3)α=1+n∑i=1nlnxixmin−1,where xi is the observed value of xi>xmin and n is the number of samples.


### 2.5. Akaike Weights

In this paper, we use a designed collection of distribution models to explore different working hypotheses and select the best model in that Table6 collection.Table 6
The designed collection of distribution models.
DistributionProbability density function (pdf)Exponentialλe−λxPower-lawCx−αLog-normal1/xσ2πexp−lnx−μ2/2σ2Truncated power-lawCx−αe−γxWe use Akaike weights [38] to choose the best-fitted distribution. An Akaike weight is a normalized distribution selection criterion. Its value is between 0 and 1. The larger the value is, the better the distribution is fitted.Akaike’s information criterion (AIC) is used in combination with maximum likelihood estimation (MLE). MLE finds an estimator ofθ^ that maximizes the likelihood function Lθ^|data of one distribution. AIC is used to describe the best fitting one among all fitted distributions,(4)AIC=−2logLθ^|data+2K.Here,K is the number of estimable parameters in the approximating model.After determining the AIC value of each fitted distribution, we normalize these values as follows. First of all, we extract the difference between different AIC values calledΔi,(5)Δi=AICi−AICmin.Then, Akaike weightsWi are calculated as follows:(6)Wi=exp−Δi/2∑r=1Rexp−Δi/2.


## 2.1. Data Sets

We collect the official data set of Hong Kong (see Table2), Shanghai (see Table 3), Singapore (see Table 4), and Tokyo (see Table 5) in our work. The data sets of the four Asian cities are collated and matched (see Table 1).Table 2
The URL of HK dataset.
DescriptionURLGDPhttps://www.censtatd.gov.hk/sc/web_table.html?id=31\#Per capita GDPhttps://www.censtatd.gov.hk/sc/web_table.html?id=31\#Primary industryhttps://www.censtatd.gov.hk/sc/web_table.html?id=35\#Secondary industryhttps://www.censtatd.gov.hk/sc/web_table.html?id=35\#Tertiary industryhttps://www.censtatd.gov.hk/sc/web_table.html?id=35\#Proportion of primary industryhttps://www.censtatd.gov.hk/sc/web_table.html?id=36\#Proportion of secondary industryhttps://www.censtatd.gov.hk/sc/web_table.html?id=36\#Proportion of tertiary industryhttps://www.censtatd.gov.hk/sc/web_table.html?id=36\#Government revenuehttps://www.censtatd.gov.hk/sc/web_table.html?id=193\#Government expenditurehttps://www.censtatd.gov.hk/sc/web_table.html?id=194\#Populationhttps://www.censtatd.gov.hk/sc/web_table.html?id=1A\#Labour forcehttps://www.censtatd.gov.hk/sc/web_table.html?id=6\#Primary schoolhttps://www.censtatd.gov.hk/sc/scode370.html\#section6Secondary schoolhttps://www.censtatd.gov.hk/sc/scode370.html\#section6Universityhttps://www.censtatd.gov.hk/sc/scode370.html\#section6Table 3
The URL of SHA dataset.
DescriptionURLGDPhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C0401.htmPrimary industryhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C0401.htmSecondary industryhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C0401.htmTertiary industryhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C0401.htmIndustryhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C0401.htmGeneral public budget revenuehttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C0501.htmGeneral public budget expenditurehttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C0501.htmProportion of primary industryhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C0404.htmProportion of secondary industryhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C0404.htmProportion of tertiary industryhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C0404.htmProportion of industryhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C0404.htmTotal fixed asset investmenthttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C0701.htmGeneral public budget revenuehttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C0501.htmGeneral public budget expenditurehttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C0501.htmGross agricultural productionhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C1201.htmGross industrial productionhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C1301.htmOriginal insurance incomehttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C1801.htmOriginal insurance pays outhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C1801.htmResident population at year-endhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C0201.htmRegistered population at year-endhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C0201.htmGeneral higher educationhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C2103.htmOrdinary secondary schoolhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C2103.htmOrdinary primary schoolhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C2103.htmBook print runhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C2316.htmJournal print runhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C2317.htmNewspaper print runhttp://tjj.sh.gov.cn/tjnj/nj20.htm?d1=2020tjnj/C2318.htmTable 4
The URL of SG dataset.
DescriptionURLGDPhttps://tablebuilder.singstat.gov.sg/table/TS/M015241Goods producing industrieshttps://tablebuilder.singstat.gov.sg/table/TS/M015241Services producing industrieshttps://tablebuilder.singstat.gov.sg/table/TS/M015241Goods proportionhttps://tablebuilder.singstat.gov.sg/table/TS/M015241Services proportionhttps://tablebuilder.singstat.gov.sg/table/TS/M015241Government consumptionhttps://tablebuilder.singstat.gov.sg/table/TS/M015241Gross fixed capital formationhttps://tablebuilder.singstat.gov.sg/table/TS/M015051Total populationhttps://tablebuilder.singstat.gov.sg/table/TS/M810001\#!Government expenditure on eduhttps://tablebuilder.singstat.gov.sg/table/TS/M850011Primary schoolshttps://tablebuilder.singstat.gov.sg/table/TS/M850011Secondary schoolshttps://tablebuilder.singstat.gov.sg/table/TS/M850011Tertiaryhttps://tablebuilder.singstat.gov.sg/table/TS/M850011Literacy ratehttps://tablebuilder.singstat.gov.sg/table/TS/M850001Table 5
The URL of TYO dataset.
DescriptionURLLoanshttps://www.toukei.metro.tokyo.lg.jp/tnenkan/2019/tn19q3i015.htmManufactured goodshttps://www.toukei.metro.tokyo.lg.jp/tnenkan/2019/tn19q3i016.htmGDPhttps://www.toukei.metro.tokyo.lg.jp/tnenkan/2019/tn19q3i016.htmPrefectural incomehttps://www.toukei.metro.tokyo.lg.jp/tnenkan/2019/tn19q3i016.htmPopulationhttps://www.toukei.metro.tokyo.lg.jp/tnenkan/2019/tn19q3i002.htmLaborhttps://www.toukei.metro.tokyo.lg.jp/tnenkan/2019/tn19q3i002.htmChildren and studentshttps://www.toukei.metro.tokyo.lg.jp/tnenkan/2019/tn19q3i017.htmElementary schoolshttps://www.toukei.metro.tokyo.lg.jp/tnenkan/2019/tn19q3i017.htmJunior secondaryhttps://www.toukei.metro.tokyo.lg.jp/tnenkan/2019/tn19q3i017.htmSenior secondaryhttps://www.toukei.metro.tokyo.lg.jp/tnenkan/2019/tn19q3i017.htmUniversitieshttps://www.toukei.metro.tokyo.lg.jp/tnenkan/2019/tn19q3i017.htmHong Kong data set (see Table2) is a set of official data with 14 factors, such as GDP at current market prices, agriculture, fishing, mining and quarrying, population, and general tertiary education, collected over 39 years (from 1981 to 2019) on the official website of Census and Statistics Department of Hong Kong, China. This data set contains 546 numerical values.Shanghai data set (see Table3) is a set of official data with 23 factors, such as GDP at current market prices, goods producing industries, population, and general tertiary education, collected over 42 years (from 1978 to 2019) on the official website of Municipal Statistics Bureau of Shanghai, China. This data set contains 966 numerical values.Singapore data set (see Table4) is a set of official data with 12 factors, such as GDP at current market prices, goods producing industries, population, and general tertiary education, collected over 60 years (from 1960 to 2019) on the official website of Singapore Department of Statistics. This data set contains 720 numerical values.Tokyo data set (see Table5) is a set of official data with 10 factors, such as GDP at current market prices, population, labor, and general tertiary education, collected over 52 years (from 1968 to 2019) on the official website of statistics division, Bureau of General Affairs, Tokyo, Japan. This data set contains 520 numerical values.


## 2.2. Obtaining Each Factor’s Step-Lengths

For each factor, we use this year’s amount minus last year’s amount to get year change amount as raw step-length. We find that change amount of the GDP ranges from dozens to thousands, while that of proportion of goods producing industries ranges between−0.2 and 0.5. These raw step-lengths of different factors differ greatly. We suffer from an information overload of these factors, even though more factors provide the opportunity to understand social or economical phenomena better. Thus, we have to use dimension reduction to organize these data.


## 2.3. Obtaining Social-Economical Trajectory by Embedding

In this paper, we try to get embedding layer through training auto-encoder (AE), which is a type of artificial neural network. Firstly, we classify the data into two classes: regarding GDP, per capita GDP, goods producing industries, etc., as economical variables; regarding population, general tertiary education, etc., as social variables. Secondly, we use min-max scaling to make sure variables that are measured at different scales contribute equally to the model fitting. Thirdly, in the AE, the input feature, the data set of economical variables or social variables, is transformed into one latent space with the encoder and then reconstructed from latent space with decoder. The encoder is used as a dimension reducer. To train this AE, an Adam algorithm is applied as an optimizer and mean square error (MSE) as a loss function.We use two-layer fully connected networks as the encoder and decoder, and the loss function is(1)ℒx,x′=x−x′2=x−fhx2,where x∈ℝn is the input feature of one year and n is the number of variables. The data from m years construct m training samples. x′ is the output of the decoder. The computation of the encoder and decoder is defined as(2)hx=σW2σW1x+b1+b2,fy=σ′W2′σ′W1′y+b1′+b2′,where W1,W2,W1′,W2′,b1,b2,b1′,b2′ are learnable parameters of the network and σ,σ′ are the activation functions. We train the AE by minimizing the MSE of the input feature and the output of the decoder. And the output of the encoder hx is the embedding of the original input. Using the embedding technique to reduce the dimensions of the data set into two dimensions: economy and society. Then, we draw the urban development trajectory (see Figure 1) with economical growth as horizontal coordinate and social growth as the vertical coordinate. We extract the step lengths, as the information from the graph.Figure 1
The socialeconomical trajectories of urban development. We extract economical and social coordinates by Auto-Encoder, respectively, and draw (a) HK Trajectory, (b) SHA Trajectory, (c) SG Trajectory, and (d) TYO Trajectory with economy as the horizontal axis and society as the vertical axis.
(a)(b)(c)(d)


## 2.4. Identifying the Scaling Exponent

To fit a heavy-tailed distribution such as a power-law distribution, we need to determine what portion of the data to fitxmin and the scaling exponent α. We use the methods from [37] to determine xmin and α. We create a power-law fit starting from each value in the data set. Then, we select the one that results in the minimal Kolmogorov–Smirnov distance between the real data and the fitted one as the optimal value of xmin. After that, the scaling exponent α in the power-law distribution is given by(3)α=1+n∑i=1nlnxixmin−1,where xi is the observed value of xi>xmin and n is the number of samples.


## 2.5. Akaike Weights

In this paper, we use a designed collection of distribution models to explore different working hypotheses and select the best model in that Table6 collection.Table 6
The designed collection of distribution models.
DistributionProbability density function (pdf)Exponentialλe−λxPower-lawCx−αLog-normal1/xσ2πexp−lnx−μ2/2σ2Truncated power-lawCx−αe−γxWe use Akaike weights [38] to choose the best-fitted distribution. An Akaike weight is a normalized distribution selection criterion. Its value is between 0 and 1. The larger the value is, the better the distribution is fitted.Akaike’s information criterion (AIC) is used in combination with maximum likelihood estimation (MLE). MLE finds an estimator ofθ^ that maximizes the likelihood function Lθ^|data of one distribution. AIC is used to describe the best fitting one among all fitted distributions,(4)AIC=−2logLθ^|data+2K.Here,K is the number of estimable parameters in the approximating model.After determining the AIC value of each fitted distribution, we normalize these values as follows. First of all, we extract the difference between different AIC values calledΔi,(5)Δi=AICi−AICmin.Then, Akaike weightsWi are calculated as follows:(6)Wi=exp−Δi/2∑r=1Rexp−Δi/2.


## 3. Results

### 3.1. Power-Law Fit for Step-Lengths Distribution of Urban Development Trajectory

First, we extract the social-economical trajectories of cities (see Figure1) to get an overview. Second, we get the step-length histograms (see Figure 2) and find that all of them show heavy tails. Third, we use the designed collection of distribution models, including truncated power-law [39], log-normal, power-law [28, 40–42], and exponential distributions [43] (see method section), to fit the step-lengths of the Hong Kong city, the Shanghai city, the Singapore, and the Tokyo city (see Figure 3). Table 7 illustrates these urban trajectory step-lengths best fitted distributions according to Akaike weights. Table 8 shows the corresponding fitted parameters. Hong Kong’s development step-lengths are relatively rich, with more long-distance flights. The fitted truncated power-law distribution has a thicker tail with α=1.3547. The development step size of Shanghai has more short-distance flights and less long-distance flights. It is still in a stable development period. The scaling exponent of the fitted power-law distribution is α=2.2829. Singapore is better fitted by exponential distribution with λ=1.6075. With the exponential transformation, the exponential distribution can deducted to power-law distribution with α=1+λ=2.6075. Tokyo has been developing earlier than Hong Kong, the fitted power-law distribution has a larger tail α=2.6016. All of these step-lengths follow probability density function Pl∝l−α with exponent α between 1.3 and 2.7. Our results are consistent with many other researches [44], and the exponent α is close to their results.Figure 2
Step-lengths histograms. (a–d) All of these histograms are positive skewed and show obvious “heavy tail” feature, which follows the Lévy flight characteristics of frequent short-distance flights and occasional long-distance flights. The trajectories of Hong Kong and Shanghai have more long-distance flights, and their histograms show thicker tails: (a) HK histogram; (b) SHA histogram; (c) SG histogram; (d) TYO histogram.
(a)(b)(c)(d)Figure 3
Power-law fit for step-lengths distribution of urban development trajectory. The green points refer to the step-lengths samples obtained, respectively, from the Hong Kong, Shanghai, Singapore, and Tokyo data sets, while the solid lines represent the best fitted distribution according to Akaike weights. (a) Hong Kong started developing later than both Singapore and Tokyo, but earlier than Shanghai. The trajectory of Hong Kong’s development has more long-distance flights. The fitted truncated power-law distribution has scaling exponentα=1.3547. (b) The trajectory of Shanghai’s development has more short-distance flights and less long-distance flights. The scaling exponent of the fitted power-law distribution is α=2.2829. (c) Singapore’s social and economical development is relatively stable, the fitted exponential distribution has λ=1.6075. The exponential distribution can be deducted to a power-law distribution with scaling exponent α=1+λ=2.6075 by exponential transformation. (d) Japan’s social development changes are small, and the economical development is more stable. The fitted power-law distribution has a larger scaling exponent α=2.6016: (a) HK truncated power-law; (b) SHA power-law; (c) SG exponential; (d) TYO power-law.
(a)(b)(c)(d)Table 7
Akaike weights of fitted distributions for the trajectory step-lengths of four Asian cities.
CitiesExponentialPower-lawLog-normalTruncated power-lawHK0.19790.25680.22260.3227SHA0.04010.46030.21630.2814SG0.67170.00140.12830.1985TYO0.15940.41320.19790.2295Table 8
The calculated and fitted parameters of trajectory step-length distribution of four Asian cities. We take change ratesri in the interval (0.48 and 1.48) [45]. The mean is noted as v′ and variance is D′. α^ is calculated scaling exponent and α is the fitted one. Here, the step-lengths of Hong Kong are better fitted by a truncated power-law distribution rather than a power-law distribution.
Citieslminv′D′v′/D′α^αHK∗0.0752−0.19040.1443−1.32002.32001.3547SHA0.0216−0.10100.0751−1.34532.34532.2829SG0.0262−0.13100.1145−1.14452.14452.6075TYO0.0033−0.117160.0875−1.33902.33902.6016We use Akaike weights (see Methods section) to choose the best fitted distribution and show the Akaike weights for all fitted distribution in Table7. The Akaike weights is a value between 0 and 1. The larger it is, the better the distribution is fitted. The trajectory step-lengths of Hong Kong are fitted a truncated power-law distribution best with weight of 0.3227, Shanghai has fitted a power-law distribution best with weight of 0.4603, Singapore has fitted an exponential distribution best with weight of 0.6717, Tokyo has fitted a power-law distribution best with weight of 0.4132.


### 3.2. Exponential Transformation

The urban trajectory step-lengths of Singapore are fitted exponential distribution best with weight of 0.6717, the probability density function (pdf) of exponential distribution can be transformed into a power-law distribution. LetX be an exponential random variable whose pdf is given by λe−λx,λ>0,x>0, then the cumulative distribution function (cdf) is given by(7)PX≤x=∫0xλe−λtdt=1−e−λx,λ>0,x>0.TransformingX into Y=keXk>0, we get(8)PY≤y=PkeX≤y,=PX≤lnyk,=1−e−λlny/k,=1−yk−λ,=1−kλy−λ.By differentiating, we obtain probability density functionλkλy−1+λ, which follows power-law.Which corresponds to the probability density function of the power-law distribution with scaling exponentα=1+λ. Thus, the scaling exponent for Singapore here is α=1+λ.


### 3.3. Log-Normal in the Stochastic Multiplicative Processes

A city should be considered as an evolving organism instead of a static one. We assume here that the urban organism may grow or shrink [46] according to a random variable Rt at each step t, then get the trajectory step-lengths lt=rt−1lt−1. These stochastic multiplicative processes [47] assume that the stochastic growth of an organism is expressed as a percentage of its current increment and is independent of its current actual size. Then, we find(9)lt=rtrt−1…r1l0,lnlt=∑i=1tlnri+lnl0.Assuming that the random variablelnRi satisfies independent and identical distributions with mean v and variance D, the central limit theorem says that lnLt=∑i=1tlnRi+lnl0. It converges to a normal distribution with mean vt and variance Dt for sufficiently large t, which means Lt converges to a log-normal distribution. Figures 4 and 5 intuitively show that the logarithms of step-length change rates lnri follow a normal distribution, while ri do not. We used Kolmogorov–Smirnov test to verify that the logarithms of step-length change rates lnri of four cities can be reasonably assumed and satisfy normal distributions with K–S value p≥0.9477 for Hong Kong, Shanghai, and Tokyo and p=0.7399 for Singapore (see Table 9).Figure 4
The step-lengths change ratesr. The change rates r are defined as the change ratios of lengths between two consecutive flights. (a–d) We observe that r is random, but do not follow the characteristics of a normal distribution: (a) HK r value; (b) SHA r value; (c) SG r value; (d) TYO r value.
(a)(b)(c)(d)Figure 5
The Logarithms of step-length change rateslnr. lnr are the natural logarithms of change rates r. (a–d) We observe that lnr are random, and follow the characteristics of a Normal distribution, which the data tend to cluster around the mean: (a) HK in r value; (b) SHA in r value; (c) SG in r value; (d) TYO in r value.
(a)(b)(c)(d)Table 9
Thep value of Kolmogorov–Smirnov test for four city datasets.
Citiesp valueHK0.9804SHA0.9477SG0.7399TYO0.9933Note here that,lt is the step-length of the flight between time t−1 and time t. The probability density function of the step-length with the same change variable is log-normal(10)flt=12πDt1ltexp−12Dtlnlt−vt2=12πDtlt−1+vDexp−12Dtln2lt+v2t2.


### 3.4. Mechanism behind the Power-Law Pattern

Previous studies [40, 48] show that if there exists a lower bound lmin, such that lt=maxlmin,rt−1lt−1, then the random variable Lt converges to a power-law distribution, and log-normal easily pushed to a power-law model.Given the log-normal probability density function (see equation (10)),(11)flt=1lt2πDtexp−lnlt−vt22Dt=1lt2πDtexp−lnlt2−2vtlnlt+v2t22Dt=1lt2πDtexp−lnlt2+v2t22DtexpvlnltD=1lt2πDtltv/Dexp−lnlt2+v2t22Dt=12πDtlt−1+v/Dexp−lnlt2+v2t22Dt.This form shows the similarity between a log-normal distribution and a power-law distribution. Ift⟶∞, then lnlt2/2Dt⟶0.(12)flt⟶12πDtexp−v2t2Dlt−1+v/D⟶Clt−α.The pdf of log-normal distribution is indistinguishable from that of power-law distribution(13)flt=Clt−α,1<α≤3.Here, thev and the D are the normalized mean and variance of lnRi.


## 3.1. Power-Law Fit for Step-Lengths Distribution of Urban Development Trajectory

First, we extract the social-economical trajectories of cities (see Figure1) to get an overview. Second, we get the step-length histograms (see Figure 2) and find that all of them show heavy tails. Third, we use the designed collection of distribution models, including truncated power-law [39], log-normal, power-law [28, 40–42], and exponential distributions [43] (see method section), to fit the step-lengths of the Hong Kong city, the Shanghai city, the Singapore, and the Tokyo city (see Figure 3). Table 7 illustrates these urban trajectory step-lengths best fitted distributions according to Akaike weights. Table 8 shows the corresponding fitted parameters. Hong Kong’s development step-lengths are relatively rich, with more long-distance flights. The fitted truncated power-law distribution has a thicker tail with α=1.3547. The development step size of Shanghai has more short-distance flights and less long-distance flights. It is still in a stable development period. The scaling exponent of the fitted power-law distribution is α=2.2829. Singapore is better fitted by exponential distribution with λ=1.6075. With the exponential transformation, the exponential distribution can deducted to power-law distribution with α=1+λ=2.6075. Tokyo has been developing earlier than Hong Kong, the fitted power-law distribution has a larger tail α=2.6016. All of these step-lengths follow probability density function Pl∝l−α with exponent α between 1.3 and 2.7. Our results are consistent with many other researches [44], and the exponent α is close to their results.Figure 2
Step-lengths histograms. (a–d) All of these histograms are positive skewed and show obvious “heavy tail” feature, which follows the Lévy flight characteristics of frequent short-distance flights and occasional long-distance flights. The trajectories of Hong Kong and Shanghai have more long-distance flights, and their histograms show thicker tails: (a) HK histogram; (b) SHA histogram; (c) SG histogram; (d) TYO histogram.
(a)(b)(c)(d)Figure 3
Power-law fit for step-lengths distribution of urban development trajectory. The green points refer to the step-lengths samples obtained, respectively, from the Hong Kong, Shanghai, Singapore, and Tokyo data sets, while the solid lines represent the best fitted distribution according to Akaike weights. (a) Hong Kong started developing later than both Singapore and Tokyo, but earlier than Shanghai. The trajectory of Hong Kong’s development has more long-distance flights. The fitted truncated power-law distribution has scaling exponentα=1.3547. (b) The trajectory of Shanghai’s development has more short-distance flights and less long-distance flights. The scaling exponent of the fitted power-law distribution is α=2.2829. (c) Singapore’s social and economical development is relatively stable, the fitted exponential distribution has λ=1.6075. The exponential distribution can be deducted to a power-law distribution with scaling exponent α=1+λ=2.6075 by exponential transformation. (d) Japan’s social development changes are small, and the economical development is more stable. The fitted power-law distribution has a larger scaling exponent α=2.6016: (a) HK truncated power-law; (b) SHA power-law; (c) SG exponential; (d) TYO power-law.
(a)(b)(c)(d)Table 7
Akaike weights of fitted distributions for the trajectory step-lengths of four Asian cities.
CitiesExponentialPower-lawLog-normalTruncated power-lawHK0.19790.25680.22260.3227SHA0.04010.46030.21630.2814SG0.67170.00140.12830.1985TYO0.15940.41320.19790.2295Table 8
The calculated and fitted parameters of trajectory step-length distribution of four Asian cities. We take change ratesri in the interval (0.48 and 1.48) [45]. The mean is noted as v′ and variance is D′. α^ is calculated scaling exponent and α is the fitted one. Here, the step-lengths of Hong Kong are better fitted by a truncated power-law distribution rather than a power-law distribution.
Citieslminv′D′v′/D′α^αHK∗0.0752−0.19040.1443−1.32002.32001.3547SHA0.0216−0.10100.0751−1.34532.34532.2829SG0.0262−0.13100.1145−1.14452.14452.6075TYO0.0033−0.117160.0875−1.33902.33902.6016We use Akaike weights (see Methods section) to choose the best fitted distribution and show the Akaike weights for all fitted distribution in Table7. The Akaike weights is a value between 0 and 1. The larger it is, the better the distribution is fitted. The trajectory step-lengths of Hong Kong are fitted a truncated power-law distribution best with weight of 0.3227, Shanghai has fitted a power-law distribution best with weight of 0.4603, Singapore has fitted an exponential distribution best with weight of 0.6717, Tokyo has fitted a power-law distribution best with weight of 0.4132.


## 3.2. Exponential Transformation

The urban trajectory step-lengths of Singapore are fitted exponential distribution best with weight of 0.6717, the probability density function (pdf) of exponential distribution can be transformed into a power-law distribution. LetX be an exponential random variable whose pdf is given by λe−λx,λ>0,x>0, then the cumulative distribution function (cdf) is given by(7)PX≤x=∫0xλe−λtdt=1−e−λx,λ>0,x>0.TransformingX into Y=keXk>0, we get(8)PY≤y=PkeX≤y,=PX≤lnyk,=1−e−λlny/k,=1−yk−λ,=1−kλy−λ.By differentiating, we obtain probability density functionλkλy−1+λ, which follows power-law.Which corresponds to the probability density function of the power-law distribution with scaling exponentα=1+λ. Thus, the scaling exponent for Singapore here is α=1+λ.


## 3.3. Log-Normal in the Stochastic Multiplicative Processes

A city should be considered as an evolving organism instead of a static one. We assume here that the urban organism may grow or shrink [46] according to a random variable Rt at each step t, then get the trajectory step-lengths lt=rt−1lt−1. These stochastic multiplicative processes [47] assume that the stochastic growth of an organism is expressed as a percentage of its current increment and is independent of its current actual size. Then, we find(9)lt=rtrt−1…r1l0,lnlt=∑i=1tlnri+lnl0.Assuming that the random variablelnRi satisfies independent and identical distributions with mean v and variance D, the central limit theorem says that lnLt=∑i=1tlnRi+lnl0. It converges to a normal distribution with mean vt and variance Dt for sufficiently large t, which means Lt converges to a log-normal distribution. Figures 4 and 5 intuitively show that the logarithms of step-length change rates lnri follow a normal distribution, while ri do not. We used Kolmogorov–Smirnov test to verify that the logarithms of step-length change rates lnri of four cities can be reasonably assumed and satisfy normal distributions with K–S value p≥0.9477 for Hong Kong, Shanghai, and Tokyo and p=0.7399 for Singapore (see Table 9).Figure 4
The step-lengths change ratesr. The change rates r are defined as the change ratios of lengths between two consecutive flights. (a–d) We observe that r is random, but do not follow the characteristics of a normal distribution: (a) HK r value; (b) SHA r value; (c) SG r value; (d) TYO r value.
(a)(b)(c)(d)Figure 5
The Logarithms of step-length change rateslnr. lnr are the natural logarithms of change rates r. (a–d) We observe that lnr are random, and follow the characteristics of a Normal distribution, which the data tend to cluster around the mean: (a) HK in r value; (b) SHA in r value; (c) SG in r value; (d) TYO in r value.
(a)(b)(c)(d)Table 9
Thep value of Kolmogorov–Smirnov test for four city datasets.
Citiesp valueHK0.9804SHA0.9477SG0.7399TYO0.9933Note here that,lt is the step-length of the flight between time t−1 and time t. The probability density function of the step-length with the same change variable is log-normal(10)flt=12πDt1ltexp−12Dtlnlt−vt2=12πDtlt−1+vDexp−12Dtln2lt+v2t2.


## 3.4. Mechanism behind the Power-Law Pattern

Previous studies [40, 48] show that if there exists a lower bound lmin, such that lt=maxlmin,rt−1lt−1, then the random variable Lt converges to a power-law distribution, and log-normal easily pushed to a power-law model.Given the log-normal probability density function (see equation (10)),(11)flt=1lt2πDtexp−lnlt−vt22Dt=1lt2πDtexp−lnlt2−2vtlnlt+v2t22Dt=1lt2πDtexp−lnlt2+v2t22DtexpvlnltD=1lt2πDtltv/Dexp−lnlt2+v2t22Dt=12πDtlt−1+v/Dexp−lnlt2+v2t22Dt.This form shows the similarity between a log-normal distribution and a power-law distribution. Ift⟶∞, then lnlt2/2Dt⟶0.(12)flt⟶12πDtexp−v2t2Dlt−1+v/D⟶Clt−α.The pdf of log-normal distribution is indistinguishable from that of power-law distribution(13)flt=Clt−α,1<α≤3.Here, thev and the D are the normalized mean and variance of lnRi.


## 4. Discussion

A variety of power-law characteristics related to cities, both within and across cities, have been documented [49–53]. In particular, city-size distributions are known to be well approximated by power-laws all around the world [15]. However, the rank-size distributions among cities [54] are mostly static, and the dynamic urban power-law distributions focus on the specific indicators over time, while urban systematic change has not been studied. This article is the first to explore urban social-economical development trajectory and shows the Lévy flight characteristics of trajectory step-lengths, and explains the general mechanism behind it.



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