Urban and ecosystem dynamics:
past, present, future
Douglas White
1-23-07
Workshop on aspects of Social and Socio-Environmental
Dynamics
School of Human Evolution and Social Change
and
Center for Social Dynamics and Complexity
1
Thanks to
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•
•
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Laurent Tambayong, UC Irvine
Nataša Kejžar, U Ljubljana
Constantino Tsallis, Ernesto Borges, Centro Brasileiro de Pesquisas Fısicas, Rio de
Janeiro
Peter Turchin, U Conn
Céline Rozenblat, U Zurich
Numerous ISCOM project and members, including Denise Pumain, Sander v.d.
Leeuw, Luis Bettencourt
Commentators Michael Batty, William Thompson, George Modelski
2
Outline
• Measure for city size deviations from Zipfian constructed and fitted to three
Eurasian world regions.
• Does the shape parameter q of these distributions oscillate historically in
longer periods than expected at random?
• Does fall in q away from Zipfian correlate with other measures of instability,
e.g., internecine warfare or sociopolitical violence?
• Do variations in shape parameter q represent alternating periods of stability
and instability?  Are city distributions historically unstable, as argued by
Michael Batty, Nature 2006, (citing White et al. 2005)
• Does shape parameter q for China affect Europe q with a time lag (diffusion
of innovation, Silk Route trade)?  Do city size instabilities affect worldsystem centers?
3
Michael Batty (Nature, Dec 2006:592), using some
of the same data as do we for historical cities
(Chandler 1987), states the case made here:
“It is now clear that the evident macro-stability in such
distributions” as urban rank-size hierarchies at different times
“can mask a volatile and often turbulent micro-dynamics, in
which objects can change their position or rank-order rapidly
while their aggregate distribution appears quite stable….”
Further, “Our results destroy any notion that rank-size scaling
is universal… [they] show cities and civilizations rising and
falling in size at many times and on many scales.”
Batty shows legions of cities in the top
echelons of city rank being swept away
as they are replaced by competitors,
largely from other regions.
4
City Size Distributions for
Measuring Departures from Zipf
construct and measure the shapes of cumulative city size
distributions for the n largest cities from 1st rank size S1 to the
smallest of size Sn as a total population distribution Tr for all
people in cities of size Sr or greater, where r=1,n is city rank
Cumulative citypopulation distribution
r
Tr=  Si
i 1
Rank size power law M~S1
r

Mi
RTr = 
i 1
5
City Size Distributions for
Measuring Departures from Zipf
This typical form of the city-size distribution tends toward a
power-law in the tail, with a crossover C where smaller city
sizes tend more toward an exponential distribution.
Cumulative citypopulation distribution
r
Tr=  Si
i 1
Rank size power law
r

Mi
RTr = 
i 1
6
City Size Distributions for
Measuring Departures from Zipf
This typical form of the city-size distribution tends toward a
power-law in the tail, with a crossover C where smaller city
sizes tend more toward an exponential distribution. This closely
fits the q-exponential, Yq(S ≥ x) = Y0 (1-(1-q)x/κ)1/(1-q)
Cumulative citypopulation distribution
r
Tr=  Si
i 1
Rank size power law
r

Mi
RTr = 
i 1
7
City Size Distributions for
Measuring Departures from Zipf
The q-exponential, Yq(S ≥ x) = Y0 (1-(1-q)x/κ)1/(1-q)
asymptotes toward a power law in the tail when q>1, and
levels at smaller city sizes toward a finite urban population Y0
as governed by a crossover parameter κ (kappa).
Cumulative citypopulation distribution
r
Tr=  Si
i 1
Rank size power law
r

Mi
RTr = 
i 1
8
City Size Distributions for
Measuring Departures from Zipf
The q-exponential, Yq(S ≥ x) = Y0 (1-(1-q)x/κ)1/(1-q)
asymptotes toward a power law in the tail when q>1, and
levels at smaller city sizes toward a finite urban population Y0
as governed by a crossover parameter κ (kappa).
Cumulative citypopulation distribution
r
Lower q
steeper α
in the tail
Tr=  Si
i 1
Rank size power law
r

Mi
RTr = 
i 1
9
City Size Distributions for
Measuring Departures from Zipf
The q-exponential, Yq(S ≥ x) = Y0 (1-(1-q)x/κ)1/(1-q)
asymptotes toward a power law in the tail when q>1, and
levels at smaller city sizes toward a finite urban population Y0
as governed by a crossover parameter κ (kappa).
Cumulative citypopulation distribution
r
Higher q
flatter α
in the tail
Tr=  Si
i 1
Rank size power law
r

Mi
RTr = 
i 1
10
City Size Distributions for
Measuring Departures from Zipf
In shifting to a semilog rather than a log-log plot of Tr in
which the Zipfian is expressed as a straight line, we see that
many of the empirical distributions in semilog are relatively
Zipfian but some bow concavely from a straight line when
α>1.
Cumulative citypopulation distribution
c900
12
c1400
c1450
r
c1500
c1550
10
c1700
c1750
c1800
c1825
8
c1850
c1875
c1900
c1914
c1925
6
c1950
Straightline in
semilog
for Zipfian
Tr=  Si
i 1
Rank size power law
bin
r

Mi
RTr = 
4
2
40
50
63
80
100
126
159
202
252
317
400
508
635
800 1008 1280 1600 2016 2540 3225 4032 5080 6400 8127 9999
bins
i 1
11
City Size Distributions for
Measuring Departures from Zipf
In shifting to a semilog rather than a log-log plot of Tr in
which the Zipfian is expressed as a straight line, we see that
many other empirical distributions in semilog bow concavely
from a straight line either when α>1 for a rank-size power-law,
or when the q-exponential has a higher crossover.
Cumulative citypopulation distribution
c1000
12
c1100
c1150
r
c1200
c1250
10
c1300
c1350
c1575
c1600
8
c1650
c1970
bin
6
4
2
40
50
63
80
100
126
159
202
252
317
400
508
635
800 1008 1280 1600 2016 2540 3225 4032 5080 6400 8127 9999
bins
Transforms: natural log
Bowedline in
semilog
nonZipfian
Tr=  Si
i 1
Rank size power law
r

Mi
RTr = 
i 1
12
City Size Distributions for
Measuring Departures from Zipf
Either way. Log-log or semilog, we carry out curve fitting to
the q-exponential, Yq(S ≥ x) = Y0 (1-(1-q)x/κ)1/(1-q)
and do so with Chandler’s largest historical cities, 900-1970,
for China, Europe, Middle Asia in between, and the Mideast.
China with Europe4
Coefficient
3.5
Upper Confidence
Limit
0.9
Lower Confidence
Limit
3.0
0.6
China
Europe
0.3
CCF
2.5
0.0
2.0
-0.3
1.5
-0.6
1.0
-0.9
-7
0.5
0.0
900
-0.5
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
Lag Number
1000
1100 1200 1300
1400 1500
1600 1700 1800
1900 2000
q is usually < 3, and >0 and
China q leads Europe q by
50 years (diffusion time) 13
City Size Distributions for
Measuring Departures from Zipf
Middle Asia, caught between China and Europe as connected
by the Silk Roads, has a different profile and interaction.
3.0
Mideast India Afghanistan
2.5
2.0
1.5
1.0
0.5
0.0
900
1000
1100
1200
1300
Coefficient
Upper Confidence
0.9
Limit
Upper Confidence
Limit
Lower Confidence
Limit
Lower Confidence
Limit
0.6
0.6
0.3
0.3
0.0
0.0
-0.3
-0.3
-0.6
-0.6
-0.9
-0.9
-5
-4
-3
-2
-1
0
1
Lag Number
2
3
4
5
6
1600
Coefficient
CCF
CCF
0.9
-6
1500
1700
1800
1900
2000
Europe4 with MidEastIndiaAfghan
China with MidEastIndiaAfghan
-7
1400
7
q is usually < 3, and >0 and
both China q and Europe q
depress the Middle Asia q
within 50 years
(competition?)
14
-7
-6
-5
-4
-3
-2
-1
0
1
Lag Number
2
3
4
5
6
7
China boosted by Middle Asia q
15
Europe not boosted by Middle Asia
q
16
City Size Distributions as Measured
by q Departures from Zipf are
historically unstable: Middle Asia
3.0
Mideast India Afghanistan
2.5
2.0
1.5
1.0
0.5
0.0
900
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000
q is usually < 3, and >0 and
both China q and Europe q
depress the Middle Asia q
within 50 years
(competition?)
17
City Size Distributions as Measured
by q Departures from Zipf are
historically unstable: Europe
3.0
2.5
Europe
2.0
1.5
1.0
0.5
0.0
900
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000
q is usually < 3, and >½ and
China q leads Europe q by
50 years (diffusion time)
18
City Size Distributions as Measured
by q Departures from Zipf are
historically unstable: China
3.5
3.0
China
2.5
2.0
1.5
1.0
0.5
0.0
900
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000
q is usually < 3, and >0 and
China q leads Europe q by
50 years (diffusion time)
19
City Size Distributions as Measured
by q Departures from Zipf are
correlated with instability: China
SPIm with q
Coefficient
0.9
Upper Confidence
Limit
0.6
Lower Confidence
Limit
CCF
0.3
0.0
-0.3
-0.6
-0.9
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
Chinese SPIm=Sociopolitical
Instability (moving average)
as measured by Internecine
wars (Lee 1931), 25 year
periods interpolated for q
Lag Number
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Conclusion
• Measure for deviations from Zipfian (q, kappa) constructed and fitted to
three Eurasian world regions.
• Shape parameter q of these distributions oscillates historically in longer
periods than expected at random (detrended kappa also).
• Fall in q away from Zipfian explored for China, found to be strongly
correlated with internecine warfare, more generally SP Instability.
• Variations in shape parameter q represent periods of stability, instability 
city distributions are historically unstable.
• Shape parameter q for China affects Europe q with 50 year lag (diffusion of
innovation, Silk Route trade). China and Europe q affect Middle Asia q
negatively (competition)  city size instabilities affect world-system
centers.
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Implications
• Connect these results with those of Geoff West, Luis Bettencourt and Jose
Lobo (Chapter 1, ISCOM book), showing the energetic inefficiency of
larger cities.
• That, along with city system instabilities, has implications for more severe
consequences as urban population systems grow in size.
• Energy inefficiencies that are cumulative, growing since the industrial
revolution  severe global warming with no end in sight.
• This includes a 240-300 foot rise in oceans by 22nd C., and flooding of
huge number of coastal cities, displacing 10% or more of world population.
• Need to consider new design principles for redesigning cities that are
energetically efficient in self sustaining local and global systems.
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Policy Research on Urban
Redesign, Energy, and Ecosystem
• With a 240-300 foot rise in oceans by 22nd C., and flooding of
huge number of coastal cities, displacing 10% or more of
world population and greater infrastructural efficiency energy
inefficiencye of larger cities (and conversely for smaller
cities), need to study redesign of
• new cities inland in the smaller range that are energetically
efficient in self-sustaining local and global systems.
• Existing cities inland in the larger range that are energetically
efficient and sustainable in global systems.
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Cohesive Info.Redesign, Minimum
Energy, and Ecocoupling
• If the network hubs found in cities attract population, then membership in
cohesive netgroups per capita might be lower in cities because of
centralization, road design, and now, developer design of suburbs.
• In the era prior to developer design of segmentary suburbs (tree-like
intaburb streets, aparteid in sociopolitical effects), ecological psychology
found greater productive role density and satisfaction in smaller
settlements. This could become a renewed design principle.
• Similarly, in large cities, cohesive designs could be tested against
segmentary aparteid principles and used in design principles for energetic
and ecological efficiencies and sustainabilities.
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Scaling Issues
• Good deal of time devoted to finding reliable and unbiased estimates of the
q-exponential parameters.
• Excel solver can be used, solving a whole series of distributions at once.
• Spss /Analyze/Regression/Nonlinear can be used, one distribution at a time.
• We are testing a candidate model for unbiased MLE of the q-exponential.
• Current findings replicated by different fitting methods.
• Crucial problems:
– Accuracy when there are relatively few cases
– Accuracy and unbiased estimation when the lower-sized cities are missing
– Consistency of results when there are fewer or greater top ranked cities.
• Examination of possible biases in historical distributions.
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Replication and Consistency
• Data for other continents beside China will be run against indices of
sociopolitical instability. Some such data are available from Peter Turchin.
• Regional variability in q studied for China in relation to Turchin’s historical
dynamic models. Questions of accuracy of total population data for China,
possibly other regions.
• Tests of population peaks for China show predicted dynamic lags to
changes in SPI indices.
• Consistency tests work for Y0 estimates < Total population, and give
estimates of percentage urbanization that for China appear to improve on
Chinese census estimates.
• The kappa crossover parameter plays a role in the dynamics.
• The Yq distribution is differentiable. Use of the derivative allows direct
mapping into size-specific processes of urban demographic change.
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