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The pace of life in the city:urban population size dependence of the dynamics of disease, crime, wealth and
innovation
Luís M. A. BettencourtTheoretical Division
Los Alamos National Laboratory
ASU - February 4, 2006
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Collaboration & Support:
José Lobo : Global Institute of Sustainability, ASUGeoffrey West: 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 Power law solves: R(a N)=ab R(N) Scale Invariance
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Cells in organisms are constrained by resource distribution networks:
-Hierarchical Branching-[3d] Space filling-Energy Efficient-Terminate at invariant area units
RBM b , b3
4d
d 1 r R
M BM
1
4
Total metabolic rate metabolic rate/mass
Larger organisms are slower!!
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[…] 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]
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]
The city as a ‘natural organism’
Raphael's School of Athens (1509-1511)
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Is there are analogue betweenbiological 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
economy of scale
Germany: year 2002
Data source: German ElectricityAssociation [VDEW]
Courtesy of Christian Kuehnert
super-lineargrowth
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Structural Infrastructureoptimized global design for economies of scale
Y b 95% CI adj.- R2 observations
Country/year
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
cables0.88 [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 needsproportionality to population
Y b 95% CI adj.- R2 observations
Country/year
Total establishmen
ts0.98 [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
consumption1.00 [0.94,1.06] 0.70 387 Germany/
2001Total 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 miracleacross time, space, level of development or
economic system
Y b 95% CI adj.- R2 observations
Country/ year
Total Wages/y
r1.12 [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/2003
GDP/yr 1.26 [1.03,1.46] 0.64 196 EU/2003
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Innovation as the engine
Y b 95% CI adj.- R2 observations
Country/year
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
From “Innovation in the city: Increasing returns to scale in urban patenting”Bettencourt, Lobo and Strumsky Data courtesy of Lee Fleming, Deborah Strumsky
Source data:U.S. patent officeIncludes all patents between 1980-2001
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Employment patterns
b=1.15 ( 95% C.I.=[1.11,1.18] ) adjusted R2= 0.89
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 from Standard Occupation Classification System of the U.S. Bureau of Labor Statistics
Data courtesy of Richard Florida and Kevin Stolarick. Plot by Jose Lobo
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Social Side Effects
Y b 95% CI adj.- R2 observations
Country/year
Total elect. consumptio
n1.09 [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
dtcSI Standard Incidence
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The Pace of Life walking speed vs. population
Borstein & BornsteinNature 1976Bornstein, IJP 1979
bCI [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 Optimization of system level - higher density 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
Y Y0Nb
b 1 Social
b1 Individual
b 1 Structural
Scaling Law:
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Scale, Pace and Growth
RNRc E0
dN
dt
dN
dtRaE0
N b RcE0
N
N(t) RaRc
N1 b (0) RaRc
exp[
RcE0
(1 b)t]
1
1 b
Consider the energy balance equation:
availableresources
costs growth
General Solution:
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b<1 implies limited carrying capacity
biological population dynamics
N RaRc
1
1 b
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N(t) RaRc
N1 b (0) RaRc
exp[
RcE0
(1 b)t]
1
1 b
tcrit E0
( 1)RaN1 b (0) 50
T
nb 1years.
b>1 : Finite time Boom and Collapse
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Escaping the singularity with b>1:cycles of successive growth & innovation
tcrit E0
( 1)RaN1 b (0) 50
T
nb 1years. tcrit shortens with N
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Consequences for epidemiology
Consider:
SIR :N
S,I
Epidemic dynamics over quasi-static background
S
BN b SNI S
I S
N ( )
I
is a small parameter
Disease free fixed point:
SE 1
0
N, IE
0BNb 1 1 N
Endemic fixed point:
SF BN b , IF 0
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dynamics at disease free fixed point
I( ) 0BN
b 1 1 I, 0
Unstable if:
0BNb 1 1 b=1
1
b<1
b>1
0BNb 1
N
Even if initially stable b>1 eventuallyleads to endemic state.
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Dynamics at endemic steady state
IEN
0BNb 1 1
Infected as a fraction of the population:
N
IEN
b<1
b=1
b>1
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Oscillations and decayat endemic state
1
2 0BN
b 1 (0BNb 1)2 4 BN b 10 1
0
Eigenvalues:
Solution:
e (N )t cos (N)t
N
b>1
b=1
b<1
N
b>1
b=1
b<1
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General pictureScaling
ExponentDriving Force Organization Growth
b<1Optimization, Efficiency
Stable equilibriumBiological Sigmoidal
Long term finite attractor
b>1Creation of Information, Wealth
and Resourcesnonequilibrium, constant adaptation
SocialBoom / Collapse
Finite time singularityIncreasing acceleration / discontinuities
b=1 Single Individual Maintenance Trivial, Free ExponentialInfinite time divergence
Human social organization is a compromise over many social activitiesEpidemiological 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
exponentsIt is expected that a social process scales with the number of contacts.Naively for a homogeneous population N:
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.
nc N(N 1) /2
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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:
between largestand smallest city
Now equate the change in productivity per capita R= R1Nwith this increase in effective contacts:
Ancpc (N)
ncpc (N0)~ 101 102
AN
N0
1
1log(A)
log(N /N0)~
1 2
70.14 0.29
Note that ncpc(N) may itself scale, but with a very small exponent
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