The pace of life in the city:
urban population size dependence of the dynamics of
disease, crime, wealth and innovation
Luís M. A. Bettencourt
Theoretical Division
Los Alamos National Laboratory
ASU - February 4, 2006
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Collaboration & Support:
José Lobo :
Geoffrey West:
Global Institute of Sustainability, ASU
Santa Fe Institute
Dirk Helbing & Christian Kuhnert, T.U. Dresden
Support from ISCOM: European Network of Excellence
Special thanks to Sander van der Leeuw:
School of Human Evolution and Social Change, ASU
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Scaling in Biological Organization
R=R0 Mb
b=3/4
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Power law solves: R(a N)=ab R(N)
Scale Invariance
Cells in organisms are constrained by
resource distribution networks:
-Hierarchical Branching
-[3d] Space filling
-Energy Efficient
-Terminate at invariant area units
R  BM ,
b
3
d
b 
4 d 1
Total metabolic rate

r
R
M
 BM

metabolic rate/mass
Larger organisms are slower!!
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1
4
The city as a ‘natural organism’
Until philosophers rule as kings or those who are now called kings
and leading men genuinely and adequately philosophize, that is,
until political power and philosophy entirely coincide, […]
cities will have no rest from evils,...
nor, I think, will the human race.
Plato: [Republic 473c-d]
Raphael's School of Athens (1509-1511)
[…] it is evident that the state [polis] is a creation of nature,
and that man is by nature a political animal.
The proof that the state is a creation of nature and prior to
the individual is that the individual, when isolated, is not
self-sufficing; and therefore he is like a part in relation to the whole.
Aristotle: Politics [Book I]
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Is there are analogue between
biological and social scaling?
•
Metabolic Rates ~ Nd/(d+1)
Energy/resource consumption
•
•
Rates decrease ~N-1/(d+1)
Times increase ~N1/(d+1)
Is 3> d ~2 ?
We set forth to search for data and estimate power laws:
Y(N)=Y0 Nb
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Energy consumption vs. city
size
Germany: year 2002
Data source:
German Electricity
Association [VDEW]
Courtesy of
Christian Kuehnert
super-linear
growth
economy
of scale
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Structural Infrastructure
optimized global design for economies of scale
observation
s
Country/
year
Y
b
95% CI
adj.- R2
Gasoline
Stations
0.77
[0.74,0.81]
0.93
318
USA/2001
Gasoline
Sales
0.79
[0.73,0.80]
0.94
318
USA/2002
Length of
electrical cables
0.88
[0.82,0.94]
0.82
387
Germany/2001
Note that although there are economies of scale in cables the
network is still delivering energy at a superlinear rate:
Social rates drive energy consumption rates, not the opposite
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Basic Individual needs
proportionality to population
adj.- R2
observations
Country/yea
r
Y
b
95% CI
Total
establishments
0.98
[0.95,1.02]
0.95
331
USA/2001
Total
employment
1.01
[0.99,1.02]
0.98
331
USA/2001
Total Household
electrical consumption
1.00
[0.94,1.06]
0.70
387
Germany/2001
Total Household
electrical consumption
1.05
[0.89,1.22]
0.91
295
China/2002
Total Household water
consumption
1.01
[0.89,1.11]
0.96
295
China/2002
Also true for the scaling of number of housing units
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The urban economic miracle
across time, space, level of development or economic
system
observation
s
Country/
year
Y
b
95% CI
Total
Wages/yr
1.12
[1.09,1.13]
0.96
361
USA/2002
GDP/yr
1.15
[1.06,1.23]
0.96
295
China/2002
GDP/yr
1.13
[1.03,1.23]
0.94
37
Germany/200
3
GDP/yr
1.26
[1.03,1.46]
0.64
196
EU/2003
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adj.-
R2
Innovation as the engine
Y
b
95% CI
adj.- R2
observations
Country/yea
r
New
Patents/yr
1.27
[1.25,1.29]
0.72
331
USA/2001
Inventors/yr
1.25
[1.22,1.27]
0.76
331
USA/2001
Private R&D
employment
1.34
[1.29,1.39]
0.92
266
USA/2002
“Supercreative”
Professionals
1.15
[1.11,1.18]
0.89
287
USA/2003
R&D
employment
1.67
[1.54,1.80]
0.64
354
France/1999*
R&D
employment
1.26
[1.18,1.43]
0.93
295
China/2002
* France/1999 data courtesy Denise Pumain, Fabien Paulus
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Innovation measured by Patents
Source data:
U.S. patent
office
Includes all patents
between 1980-2001
From “Innovation in the city: Increasing returns to scale in urban patenting”
Bettencourt, Lobo and Strumsky Data courtesy of Lee Fleming, Deborah Strums
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Employment patterns
b=1.15 ( 95% C.I.=[1.11,1.18] )
adjusted R2= 0.89
Data courtesy of Richard
Florida and Kevin
Stolarick.
Plot by Jose Lobo
Supercreative professionals [Florida 2002, pag. 327-329] are “Computer and
Mathematical, Architecture and Engineering, Life Physical and Social Sciences Occupations,
Education training and Library, Arts, Design, Entertainment, Sports and Media Occupations”.
Derived
Statistics from Standard Occupation Classification System of the U.S. Bureau of Labor
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Social Side Effects
Y
b
95% CI
adj.- R2
observations
Country/yea
r
Total elect.
consumption
1.09
[1.03,1.15]
0.72
387
Germany/2001
Cost of
housing
0.09
[0.07,1.27]
0.21
240
USA/2003
New AIDS cases
1.23
[1.18,1.29]
0.76
93
USA/2002
Serious
Crime
1.16
[1.11,1.18]
0.89
287
USA/2003
Walking Speed
0.09
[0.07,0.11]
0.79
21
Several/1979
Disease transmission is a social contact process:
dT
 c SI
Standard Incidence
dt
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The Pace of Life
walking speed vs. population
Borstein & Bornstein
Nature 1976
Bornstein, IJP 1979
b
CI [0.071,0.115]
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But cities to exist at all must also satisfy:
Basic individual needs (house, job, basic necessities)
Require city-wide infrastructure:
- larger population
- higher density

Optimization of system level
distribution networks
Result: 3 categories:
Social
- interpersonal interactions - grow with # effective relations
Individual - no interactions
- proportional to population
Structural - global urban optimization - economies of scale
 b  1
Social

b
Scaling Law: Y  Y0 N
Individual
b  1
b  1
Structural


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Scale, Pace and Growth
Consider the energy balance equation:
dN
R  NRc  E 0
dt
available
resources
costs

growth
dN Ra b Rc

N 
N
dt E 0
E0
General Solution:
1
1b
R  1b


R
R
N(t)   a  N (0)  a exp[ c (1 b)t]
Rc 
E0
Rc 

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b<1 implies limited carrying capacity
biological population dynamics
1
1b
Ra 
N   
Rc 

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b>1 : Finite time Boom and Collapse
1
1b
R  1b


R
R
N(t)   a  N (0)  a exp[ c (1 b)t]
Rc 
E0
Rc 


tcrit 
E0
T
N1b (0)  50 b1
( 1)Ra
n
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years.
Escaping the singularity with b>1:
cycles of successive growth & innovation
tcrit 
E0
T
N1b (0)  50 b1 years.
( 1)Ra
n
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tcrit shortens with N
Consequences for epidemiology

 
Epidemic dynamics
over quasi-static background
Consider: SIR : N  S,I
S
S  BN   I  S
N


 S

I    (  )I
 N


b
SF  BN b , IF 0
Endemic fixed point:

 is a small parameter
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Disease free fixed point:

1

SE 
N, IE  0BN b1 1N
0


dynamics at disease free fixed point

I  (  ) 0 BN
b1
1I,  0 


 0 BN b1
b>1
Unstable if:
 0 BN
b1
1
1
Even if initially stable b>1 eventually
leads to endemic state.
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b=1
b<1
N
Dynamics at endemic steady
state
Infected as a fraction of the population:
IE 
  0 BN b1 1
N 
b>1
IE
N
b=1

b<1
N
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Oscillations and decay
at endemic state
Eigenvalues:


b1


1 
BN

1
0
  0 BN b1  (0 BN b1 ) 2  4

2 
0




 (N )t
cos(N)t
Solution: e

N
N
b>1

b>1
b=1
b<1
b<1
N
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b=1
N
General picture
Scali ng
Exponen t
Driving Force
Organ ization
Growth
b<1
Optimi zation, Effi ciency
Stable equilibrium
Biological
Sigmoid al
b>1
Creation of Information, Wealt h
and Resour ces
Social
nonequilibrium, constant adaptation
b=1
Single Individua l Maintenanc e
Long term f inite attractor
Boom / Coll apse
Finite time singularity
Increasing acceleration / discontinuities
Trivial, Free
Exponen tial
Infinite time divergence
Human social organization is a compromise over many social activities
Epidemiological dynamics is affected by large-scale
human organization and behavior
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An argument for the smallness of the
superlinearity in social scaling exponents
It is expected that a social process scales with the number of contacts.
Naively for a homogeneous population N:
nc  N(N 1)/2
Or per capita ncpc=(N-1)/2
Clearly in a large population, N>1000, not all contacts can be realized.

This naïve estimate is the wrong result, an unattainable upper bound.
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Now, still assume that the number of effective contacts increases with
N but is constrained (time, cognition, energy) to be much smaller:
n cpc (N)
1
2
A
~ 10 10
n cpc (N 0 )
between largest
and smallest city
Now equate the change in productivity per capita R= R1N1with this
increase in effective contacts:


 N  1
log( A)
1 2
A      1
~
 0.14  0.29
log( N /N 0 )
7
N 0 
Note that ncpc(N) may itself scale, but with a very small exponent
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