Allometric Scaling of Countriesswarmagents.cn.13442.m8849.cn/jake/slides/allometric... · 2011. 3. 7. · GDP GovernmentConsumption Eco-Eco 0.955073 0.954618 GDP GovernmentDebt Eco-Eco

Allometric Scaling of Countries国家的异速生长标度律研究

Jiang Zhang

Department of Systems Science

School of Management

Beijing Normal University

Outline

• Introduction and Background

– Metabolic theory of ecology

– （代谢生态学）

– Allometric Scaling of Human Cities

• Allometric Scaling of Countries

– Static allometric scaling

patterns

– Population and GDP

– Dynamic patterns

• Further studies

F~M3/4

West, G.B. and Brown, J.H., Review : the origin of allometric scaling laws in biology from genomes to ecosystems: towards a quantitative unifying theory

of biological structure and organization, The Journal of Experimental Biology, 2005,208: 1575-1592

鼠

培养的动物细胞

线粒体（动物的肌细胞）

Metabolic Theory of Ecology

• An empirical unified formula:

• Other allometric scaling with

body size

• T~M1/4

• f~M-1/4

• Some important constants

• The number of heart beat per

life time ~1.5*109

kTEeMF /4/3~

Brown, J.H., Toward a metabolic theory of ecology,

Ecology, 2004,85,7: 1771~1789

Allometric Scaling of Human Cities

Allometric Scaling of Human Cities

1~/

~

b

b

PPY

PY

If b>1, scaling efficient

If b<1, scaling inefficient

If b=1, scaling invariant

Macroeconomic regularities

• Some allometric scaling

relations:

• Population and export

• X~P3/4

• Grouping countries by GDP

Per Capita

Bertrand B., Macroeconomic regularities in the growth of nations: an empirical inquiry,

International journal of systems science. 1984, 15: 917-936

Allometric Scaling of Countries

Motivations

• Is the allometric scaling a

universal pattern for complex

systems

• Are there some patterns behind

the macro-economic variables?

• Are there regularities in growth

of nations?

Data Source

• A Mathematica’s command:

CountryData can get the

information of countries

• The data are from:

• Encyclopaedia Britannica. Britannica Book

of the Year. Encyclopaedia Britannica, 2006

• United States National Geospatial-

Intelligence Agency. "NGA GEOnet Names

Server (GNS)." 2007

• University of California at Berkeley. "GADM:

Global Administrative Areas." 2008

• University of Pennsylvania. "Penn World

Tables." 2006. »

• van der Heyden, J. "GeoHive: Global

Statistics." 2008

• World Health Organization. "WHO Mortality

Database: Tables." 2007

• …

• Totally 237 countries as the

sample sets

Mathematica 6.0

Properties

• Totally 128 numeric properties of

each country are considered

• We classified these properties

into 8 categories

Category Properties Abb

Geographical properties Area, Boundary Length, Water Area, etc. Geo

Properties related to naturalresources and features

Arable Land Area, Irrigated Land Area, etc. Nat

Demographic properties Adult Population, Female Adult Population,Population, Total Fertility Rate etc.

Dem

Economic-related properties GDP, GDP At Parity, GDPPerCapita, GovernmentDebt, etc.

Eco

Trade-related properties Export Value, Import Value, External Debt,Economic Aid, etc.

Tde

Energy-related propertie Electricity Consumption, Natural Gas Consumption,Oil Consumption, etc.

Ene

Communications-relatedproperties

Phone Lines,Cellular Phones, Internet Users, etc. Com

Transportation-relatedproperties

Airports,Road Length, Railway Length, Pipelines,etc.

Tra

Pre-processing

• Select one of the important

properties in each group as the

x-variable, and any property as

the y-variable

• Regression under the log-log

coordinate

• Only the regressions with

R^2>0.75 are selected to be

shown

Area ~ X

Property 1 Property 2 Category -Category Υ R2

Area PopulationGeo-Dem

0.674938 0.843486

Area BoundaryLengthGeo-Geo

0.501577 0.937516

Area NaturalResources Geo-Nat 0.195118 0.788516

Area RoadLengthGeo-Tra

0.708611 0.885626

Area AirportsGeo-Tra

0.522199 0.856899

Area ArableLandArea Geo-Nat 0.947424 0.89975

AX ~

Area ~ X

\gamma=0.6749, R2=0.8435, Observations=212

A fractal nature of population distribution

100

101

102

103

104

105

106

107

108

103

104

105

106

107

108

109

1010

China

UnitedStates

Japan

France

Zambia

Area

Popula

tion

Population ~ X

PX ~

Property 1 Property 2 Category -Category Υ R2

Population ArableLandArea Dem-Nat 1.16303 0.90824

Population GDP Dem-Eco 0.791498 0.765477

Population GDPAtParity Dem-Eco 0.888756 0.863696

Population MilitaryAgePopulation Dem-Mil 0.997812 0.995439

Population PhoneLines Dem-Com 0.803596 0.765464

Population InternetUsers Dem-Com 0.87745 0.783515

Population RoadLength Dem-Tra 0.90181 0.90508

All properties may have allometric scaling with population with R2>0.75 are listed

Thus, population is not a good variable as the size of the nation

GDP~ X

GX ~

GDP is a good property standing for the size of nations

GDP Population Eco-Dem 0.740312 0.765477GDP NationalIncome Eco-Eco 0.993397 0.998384GDP FixedInvestment Eco-Eco 0.970636 0.986625GDP TotalConsumption Eco-Eco 0.961682 0.992202GDP GovernmentConsumption Eco-Eco 0.955073 0.954618GDP GovernmentDebt Eco-Eco 1.00138 0.920243GDP MiscellaneousValueAdded Eco-Eco 1.027779 0.981446GDP TradeValueAdded Eco-Eco 0.979636 0.984025GDP TransportationValueAdded Eco-Eco 0.972213 0.979671GDP HouseholdConsumption Eco-Eco 0.970239 0.988671GDP IndustrialValueAdded Eco-Eco 1.151521 0.981725GDP ManufacturingValueAdded Eco-Eco 1.187693 0.977217GDP ImportValue Eco-Tde 0.897532 0.972931GDP ExportValue Eco-Tde 1.065288 0.940181GDP OilConsumption Eco-Ene 0.900448 0.943824GDP ElectricityConsumption Eco-Ene 1.094962 0.894063GDP ElectricityProduction Eco-Ene 1.054752 0.922121GDP PhoneLines Eco-Com 0.918304 0.937089GDP InternetUsers Eco-Com 0.971305 0.92314

GDP~ X

9.0~ GI

\gamma=0.897532 R2=0.972931, Observations=227

106

108

1010

1012

1014

105

106

107

108

109

1010

1011

1012

1013

GDP

Import

Valu

e

G

I

E

GDP~ X

EX ~

Property 1 Property 2 Category -Category

Υ R2

OilConsumption GDP Ene-Eco 0.989289 0.943824

OilConsumption IndustrialValueAdded Ene-Eco 1.119745 0.931072

OilConsumption ManufacturingValueAdded Ene-Eco 1.150913 0.924392

OilConsumption ExportValue Ene-Tde 1.026125 0.900231

OilConsumption ElectricityConsumption Ene-Ene 1.112515 0.95569

OilConsumption ElectricityProduction Ene-Ene 1.103107 0.941575

Energy Consumption

100

102

104

106

108

106

107

108

109

1010

1011

1012

1013

OilConsumption

Ele

ctr

icityC

onsum

ption

\gamma=1.1125, R2=0.9557, Observation=211

Population and GDP

\gamma=0.7915, R2=0.7655, Observations=212

103

104

105

106

107

108

109

1010

107

108

109

1010

1011

1012

1013

1014

China

UnitedStates

Japan

France

Zambia

Population

GD

P

Grouping Countries

• Low income(<=$935)

• Argentina, Brazil, Bulgaria,

Chile, Costa Rica, Mexico,

• Lower middle income

($935~3705)

• Benin, Burkina Faso,

DemocraticRepublicCongo

• Upper middle income ($3706~

$11,455)

• Albania, China, Colombia,

Egypt,…

• High income: nonOECD(>=$

11,456)

• Andorra, Antigua and Barbuda,

Aruba, Bahamas, Bahrain,…

• High income: OECD (>=$ 11,456)

• Australia, France, Japan, United

Kingdom, United States

104

105

106

107

108

109

1010

107

108

109

1010

1011

1012

1013

Population

GD

PPopulation and GDP

Low

income

Lower middle

income

Upper middle

income

High income: OECD

High income: nonOECD

Population and GDP

γ=0.8481, R2=0.9404, #=21

104

105

106

107

108

109

108

109

1010

1011

1012

1013

Population

GD

P

105

106

107

108

109

107

108

109

1010

1011

1012

Population

GD

P

104

105

106

107

108

109

1010

107

108

109

1010

1011

1012

1013

Population

GD

P

105

106

107

108

109

1010

1011

1012

1013

Population

GD

P

104

105

106

107

109

1010

1011

1012

Population

GD

P

γ=0.8505, R2=0.9404, #=21γ= 1.0136, R2= 0.9918, #=36

γ= 1.0253, R2= 0.9273, #=41γ=1.0236,, R2= 0.9749, #=46

Low

income

Lower middle

income

Upper middle

income High income: OECDHigh income: nonOECD

Dynamic patterns

• CountryData command may also

provide some properties of

countries in each year through

1971~2006

• Population, GDP, ImportValue are

mainly studied.

• We mainly observe the allometric

scaling exponent and R^2

change with time and their

distributions

Area and Population

Low

income

1970 1975 1980 1985 1990 1995 2000 2005 20100.65

0.655

0.66

0.665

0.67

0.675

0.68

0.685

Year

1970 1975 1980 1985 1990 1995 2000 2005 20100.82

0.825

0.83

0.835

0.84

0.845

0.85

0.855

Year

R2

0.65 0.655 0.66 0.665 0.67 0.675 0.68 0.6850

1

2

3

4

5

6

7

Fre

quency

0.82 0.825 0.83 0.835 0.84 0.845 0.85 0.8550

2

4

6

8

10

R2

Fre

quency

GDP and ImportValue

Low

income

1970 1975 1980 1985 1990 1995 2000 2005 20100.82

0.84

0.86

0.88

0.9

0.92

Year

1970 1975 1980 1985 1990 1995 2000 2005 20100.955

0.96

0.965

0.97

0.975

Year

R2

0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.9 0.91 0.920

1

2

3

4

5

6

7

Fre

quency

0.958 0.96 0.962 0.964 0.966 0.968 0.97 0.972 0.974 0.9760

1

2

3

4

5

6

7

R2

Fre

quency

Population and GDP

Low

income

1970 1975 1980 1985 1990 1995 2000 2005 20100.75

0.8

0.85

0.9

0.95

Year

1970 1975 1980 1985 1990 1995 2000 2005 20100.74

0.76

0.78

0.8

0.82

0.84

0.86

0.88

0.9

Year

R2

0.76 0.78 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.940

1

2

3

4

5

6

7

8

Fre

quency

0.76 0.78 0.8 0.82 0.84 0.86 0.880

2

4

6

8

10

R2

Fre

quency

Population and GDP by Grouping

Low

incomeLower middle

income

Upper middle

income

High income: OECDHigh income: nonOECD

Pattern of Growth

Growth of GDP and Population

836415624 968369811.1898 1121141289.4695 129801422610

10

1011

1012

1013

Population

GD

P

China

γt=6.5857

212124541 238853866.0754 268951291.8695 30284122510

12

1013

1014

Population

GD

P

United State

γt=6.5888

52338813 55723102.2291 59326223.5815 6316232710

11

1012

1013

Population

GD

P

France

γt=12.3480

0 50 100 150 20010

-1

100

101

102

Countries(Rank)

t

0 50 100 150 2000

0.5

1

Countries(Rank)

R2

-40 -20 0 20 400

20

40

60

t

Fre

quen

cy

0 0.5 10

10

20

30

40

R2

Fre

quen

cy

Distributions of Exponents

1.409 0.9682

Conclusions

• Allometric Scaling is a universal

relationship between macro-

variables although the exponents

may be variant.

• Area, Population and GDP are

important variables as geometric,

demographic, and economic size

of nations.

• GDP exhibits scale-inefficient as

the per-capita income increases

• The growth of GDP and

Population show multiple power

law regularities in time.

A Theoretical Problem

0 2 4 6 8 10 12 14

x 108

0

2

4

6

8

10

12

14x 10

12

China

UnitedStates

Japan

France

Zambia

Population

GD

P

106

108

1010

1012

1014

10-3

10-2

10-1

100

China

UnitedStates

Japan

France

Zambia

Pr(

X

x)

x:GDP

103

105

107

109

10-3

10-2

10-1

100

China

UnitedStates

Japan

France

Zambia

Pr(

X

x)

x:Population

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