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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. Bettencourt Theoretical Division Los Alamos National Laboratory ASU - February 4, 2006 Collaboration & Support:

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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. Bettencourt

Theoretical Division

Los Alamos National Laboratory

ASU - February 4, 2006

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Collaboration & Support:

José Lobo : Global Institute of Sustainability, ASU

Geoffrey 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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Total metabolic rate metabolic rate/mass

Larger organisms are slower!!

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The city as a ‘natural organism’ distribution networks:

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 distribution networks: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 distribution networks:

Germany: year 2002

Data source:

German Electricity

Association [VDEW]

Courtesy of

Christian Kuehnert




of scale

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Structural Infrastructure distribution networks:optimized global design for economies of scale

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 distribution networks:proportionality to population

Also true for the scaling of number of housing units

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The urban economic miracle distribution networks:across time, space, level of development or economic system

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Innovation as the engine distribution networks:

* France/1999 data courtesy Denise Pumain, Fabien Paulus

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Innovation measured by Patents distribution networks:

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 StrumskyData courtesy of Lee Fleming, Deborah Strumsky

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Employment patterns distribution networks:

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 from Standard Occupation Classification System of the U.S. Bureau of Labor Statistics

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Social Side Effects distribution networks:

Disease transmission is a social contact process:

Standard Incidence

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The Pace of Life distribution networks:walking speed vs. population

Borstein & Bornstein

Nature 1976

Bornstein, IJP 1979


CI [0.071,0.115]

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But cities to exist at all must also satisfy: distribution networks:

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

Scaling Law:

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Scale, Pace and Growth distribution networks:

Consider the energy balance equation:





General Solution:

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b<1 distribution networks: implies limited carrying capacitybiological population dynamics

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b>1 distribution networks: : Finite time Boom and Collapse

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Escaping the singularity with distribution networks:b>1:cycles of successive growth & innovation

tcrit shortens with N

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Consequences for epidemiology distribution networks:

Epidemic dynamics

over quasi-static background


Disease free fixed point:

Endemic fixed point:

m is a small parameter

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b>1 distribution networks:




dynamics at disease free fixed point

Unstable if:


Even if initially stable b>1 eventually

leads to endemic state.

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b=1 distribution networks:



Dynamics at endemic steady state

Infected as a fraction of the population:


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Oscillations and decay distribution networks:at endemic state













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General picture distribution networks:

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:

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:

between largest

and smallest city

Now equate the change in productivity per capita R= R1Nb-1 with this increase in effective contacts:

Note that ncpc(N) may itself scale, but with a very small exponent