Eurasian city system dynamics in
the last millennium
Complex Dynamics of Distributional Size and Spatial Change, mediated by networks
Doug White
Irvine
30 minutes, 30 slides
1
Abstract
(as in handout)
•
A 25 period historical scaling of city sizes in regions of Eurasia (900 CE-1970) shows both rises and falls
of what are unstable city systems, and the effects of urban rise in the Middle East on China and of urban
rise on China on Europe. These are indicative of some of the effects of trade networks on the robustness
of regional economies. Elements of a general theory of complex network dynamics connect to these
oscillatory "structural demographic" instabilities.
•
The measurements of instability use maximum likelihood estimates (MLE) of Pareto II curvature for city
size distributions and of Pareto power-laws for the larger cities. Collapse in the q-exponential curve is
observed in periods of urban system crisis. Pareto II is equivalent under reparameterization to the qexponential distribution. Further interpretation of the meaning of changes in q-exponential shape and
scale parameters has been explored in a generative network model of feedback processes that mimics,
in the degree distributions of inter-city trading links, the shapes of city size distributions observed
empirically.
•
The MLE parameter estimates of size distributions are unbiased even for estimates from relatively few
cities in a given period, They are sufficiently robust to support further research on historical urban system
changes, such as on the dynamical linkage between trading networks and regional city-size distributions.
The q-exponential results also allow the reconstruction of total urban population at different city sizes in
successive historical periods.
city systems in the last millennium
2
Zipf’s law? Population at rank r
r
RTr =
M the largest city and α=1
12
 Mi

i 1
Chandler Rank-Size City Data (semilog) for
Eurasia (Europe, China, Mid-Asia)
11
10
12
Zipfian=top curve
11
10
9
8
7
6
5
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
top20
city rank
18
19
20
ZipfCum
c900
m1000
e900
c1000
m1100
e1000
c1100 9
m1150
e1100
c1150
m1200
e1150
c1200
m1250
e1200
c1250
m1300
e1250
c1300 8
m1350
e1300
c1350
m1400
e1350
c1400
m1450
e1400
c1450
m1500
e1450
c1500 7
m1550
e1500
c1550
m1575
e1550
c1575
m1600
e1575
c1600
m1650
e1600
c1650 6
m1700
e1650
c1700
m1750
e1700
c1750
m1800
e1750
c1800
m1825
e1800
c1825 5
m1850
e1825
c1850
m1875
e1860
c1875
m1900
e1875
c1900
m1914
e1900
c1914
m1925
e1914
c1925
m1950
e1925
c1950
m1970
e1950
c1970
e1970
m900
1
2
3
4
5
6
7
8
9
11
top20
Tr
Transforms: natural log
city systems in the last millennium
10
3
Zipf’s law? Population at rank r
RTr =
M the largest city and α=1
r
 Mi

i 1
Take a
closer
look:
Lots of
variation
from a
Zipfian
norm
city rank
city systems in the last millennium
4
Michael Batty (Nature, Dec 2006:592), using some of
the same data as do we for historical cities (Chandler
1987), states (and cites) the case made here (in a
previous article, 2005) for city system instability:
“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.
city systems in the last millennium
5
Color key: Red to Blue:
Early to late city entries
The world system &
Eurasia are the most
volatile. Big shifts in
the classical era until
around 1000 CE.
Gradual reduction in
shifts until the Industrial
Revolution. US shift
lower, largest 1830-90.
UK similar, with a 195060 suburbanization shift
UK
World
city systems in the last millennium
6
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
Empirical cumulative citypopulation distribution
P(X≥x)
r
Tr=  Si
i 1
Rank size power law M~S1
r

Mi
RTr = 
i 1
7
For the top 10 cities, fit the standard Pareto
Distribution slope parameter, is beta (β) in
Pβ(X ≥ x) = (x/xmin)-β ≡ (Xmin/x)β
(1)
To capture the curvature of the entire city
population, fit the Pareto II Distribution
shape and scale parameters theta and
sigma (Ө,σ) to the curve shape q=1+1/Ө.
PӨ,σ (X ≥ x) = (1 + x/σ)-Ө
(2)
city systems in the last millennium
8
Probability distribution q shapes for
a person being in a city with at least
population x (fitted by MLE estimation)
Pareto Type II
Shalizi (2007) right graphs=variant fits
city systems in the last millennium
9
Examples of fitted curves for the cumulative distribution
Curved fits measured by q shape, log-log tail slopes by β
Pβ (X ≥ x) = (x/xmin)-β
(top ten cities)
city systems in the last millennium10
Variations in q and the power-law slope
β for 900-1970 in 50 year intervals
3.0
China
Mid-Asia
Europe
MLEqExtrap
2.5
Beta10
MinQ_Beta
2.0
1.5
1.0
0.5
0.0
911111111111111111111111191111111111111111111111119111111111111111111111111
001122334455566778888999900112233445556677888899990011223344555667788889999
0 0 0 5 0 5 0 590 1
50
05
02
01
0 5101510 1
50
05
25
70
50
5 7101510 1
50
25
1 5171015 1
10121517 1
151710 1
1 5101517 9
101510 1
1 7101215 1
1 0101510 1
15101510 1
1 2951710 1
171
000000000005000005050500 000000000005000005050500 000000000005000005050500
1111111111111111111
001122334455566778888999900112233445556677888899990011223344555667788889999
000505050505705050257025700050505050570505025702570005050505057050502570257
0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 5 0 5 0 5date
00 000000000005000005050500 000000000005000005050500
date
city systems in the last millennium11
Random walk or Historical Periods?
Runs Test Results
Runs Tests at medians across all three regions
1.51
1.79
Min(q/1.5,
Beta/2)
.88
Cases < Test Value
35
36
35
Cases >= Test Value
36
37
38
Total Cases
71
73
73
Number of Runs
20
22
22
-3.944
.0001
-3.653
.0003
-3.645
.0003
MLE-q
Test Value(a)
Z
Asymp. Sig. (2-tailed)
Beta10
Runs Test for temporal variations of q in the three regions
mle_Europe
mle_MidAsia
mle_China
Test Value(a)
1.43
1.45
1.59
Cases < Test Value
9
11
10
Cases >= Test Value
9
11
12
Total Cases
18
22
22
Number of Runs
4
7
7
Z
-2.673
-1.966
-1.943
Asymp. Sig. (2-tailed)
.008
.049
.052
a Median
city systems in the last millennium12
Fitted q parameters for Europe, Mid-Asia, China, 900-1970CE, 50 year lags. Vertical lines show
approximate breaks between Turchin’s secular cycles for China and Europe
Downward arrow: Crises of the 14th, 17th, and 20th Centuries
city systems in the last millennium13
N
MLEqChinaExtrap
25
Minimum
.56
Maximum
1.81
Mean
1.5120
Std. Deviation
.25475
Std.Dev/Mean
.16849
MLEqEuropeExtrap
MLEqMidAsIndia
23
25
1.02
1.00
1.89
1.72
1.4637
1.4300
.19358
.16763
.13225
.11722
BetaTop10China
23
1.23
2.59
1.9744
.35334
.17896
BetaTop10Eur
23
1.33
2.33
1.6971
.27679
.16310
BetaTop10MidAsia
25
1.09
2.86
1.7022
.35392
.20792
14
Are there inter-region synchronies? Cross-correlations give
lag 0 = perfect synchrony
lag 1 = state of region A predicts that of B 50 years later
lag 2 = state of region A predicts that of B 100 years later
lag 3 = state of region A predicts that of B 150 years later
etc.
city systems in the last millennium15
Time-lagged cross-correlation effects of Mid-Asian q on China
(1=50 year lagged effect)
mle_MidAsia with mle_China
Coefficient
Upper Confidence
Limit
0.9
Lower Confidence
Limit
0.6
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
Lag Number
city systems in the last millennium16
Time-lagged cross-correlation effects of China q on Europe
(100 year lagged effect)
mle_China with mle_Europe
Coefficient
Upper Confidence
Limit
0.9
Lower Confidence
Limit
0.6
China with Europe4
0.3
Coe
Upp
Limi
Low
Limi
0.6
0.0
0.3
-0.3
CCF
CCF
0.9
0.0
-0.3
-0.6
-0.6
-0.9
-0.9
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
Lag Number
-7
-6
-5
-4
-3
-2
-1
0
1
Lag Number
2
3
4
5
6
7
(non-MLE result for q)
city systems in the last millennium17
7
Time-lagged cross-correlation effects of the Silk Road trade on Europe
(50 year lagged effect)
logSilkRoad with EurBeta10
Coefficient
Upper Confidence
Limit
0.9
Lower Confidence
Limit
0.6
mle_MidAsia with mle_Europe
0.3
Coefficient
Upper Confidence
Limit
Lower Confidence
Limit
0.0
0.6
0.3
-0.3
CCF
CCF
0.9
0.0
-0.3
-0.6
-0.6
-0.9
-0.9
-7
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
Lag Number
Lag Number
city systems in the last millennium18
mle_Europe with ParisPercent
Coefficient
Upper Confidence
Limit
0.9
Lower Confidence
Limit
0.6
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
Lag Number
19
Are there synchronies with Turchin et al Historical Dynamics? i.e.,
Goldstone’s Structural Demography?
E.g., Where population growth relative to resources result in
sociopolitical instabilities (SPI) and intrasocietal conflicts;
precipitating fall in population and settling of conflicts, then followed
by a new period of growth.
(secular cycles = operating at scale of centuries)
Illustrate with an example from Turchin (2005) and then relate to the
city system shape dynamics.
city systems in the last millennium20
Turchin 2005:
Dynamical
Feedbacks in
Structural
Demography
Key:
Innovation
Chinese phase diagram
21
Turchin 2005 validates statistically the interactive prediction versus the inertial
prediction for England, Han China (200 BCE -300 CE), Tang China (600 CE - 1000)
22
J. S. Lee measure of SPI for China region
(internecine wars )
city systems in the last millennium23
Interpretation: SPI conflict correlates
with low β, a thin tail of elite cities
(people migrate out of cities to
smaller cities or zones of safety),
with a more normal β restored in 150
years. Shows the effect of
internecine wars in China on city size
distributions
city systems in the last millennium
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Interpretation: SPI conflict correlates
with a higher ratio of q to (low) β, but
as β recovers slightly in the next 50
years instead of the q/β ratio going
down q goes up relative to β as
migrations re-shape the smaller city
sizes, perhaps seeking refuge in
mid-size cities.
city systems in the last millennium
25
Conclusions: city systems in the last millennium
City systems unstable; have historical periods of rise and fall over
hundreds of years; exhibit collapse.
Better to reject the “Zipf’s law” of cities in favor of studying deviations from
the Zipfian distribution, which can nevertheless be retained as a norm for
comparison. It represents an equipartition of population over city sizes that
differ by constant orders of magnitude, BUT ONLY FOR LARGER CITIES.
Deviations from Zipf occur in two ways: (1) change in the log-log slope (β)
of the power-law tail of city sizes (2) change in the curve away from a
constant slope (q), also in where this change occurs (not presented here).
TAILS AND BODIES OF CITY-SIZE DISTRIBUTIONS VARY INDEPENDENTLY.
Both deviations are dynamically related to the structural demographic
historical dynamics (SDHD). The SPI conflicts (e.g., internecine wars or
periods of social unrest and violence) that interact in SDHD processes with
population pressure on resources are major predictors of city-shape (β,q)
changes that are indicators of city-system crisis or decline.
City system growth periods in one region, which are periods of innovation,
have time-lagged effects on less developed regions if there are active
trade routes between them. NETWORKS AFFECT DEVELOPMENT.
26
UK
Color key: Red to Blue:
Early to late city entries
The world system &
Eurasia are the most
volatile. Big shifts in
the classical era until
around 1000 CE.
Gradual reduction in
shifts until the Industrial
Revolution. US shift
lower, largest 1830-90.
UK similar, with a 195060 suburbanization shift
Back to Batty:
“Gibrat’s model provides
universal scaling
behavior for city size
distributions” but the
rank clocks reveal very
different micro-dynamics.
Historical dynamics of
proportionate random
growth generating scalefree effects (in tails) can
be informed from rank
clocks, as for networks.
Expected growth rate can
be composed into overall
growth and change at
different spatial scales
(information distance),
enabling different
systems (and types) to be
integrated through a
hierarchy of information
hierarchies (Batty p. 593;
1976; Theil 1972; Tsallis
1988).
Gibrat’ law: proportionate random
growth? Pi(t) = [Г+εi(t)] Pi-1(t), lognormal, becoming power-law with
Pi(t) > Pmin(t), i.e., “losers eliminated”
World
city systems in the last millennium27
References
–
Batty, Michael. 2006. Rank Clocks. Nature (Letters) 444:592-596. Batty 1976 Entropy in Spatial
Aggregation Geographical. Analysis 8:1-21
–
Chandler, Tertius. 1987. Four Thousand Years of Urban Growth: An Historical Census. Lewiston, N.Y.:
Edwin Mellon Press.
–
Goldstone, Jack. 2003. The English Revolution: A Structural-Demographic Approach. In, Jack A.
Goldstone, ed., Revolutions - Theoretical, Comparative, and Historical Studies. Berkeley: University of
California Press.
–
Lee, J.S. 1931. The periodic recurrence of internecine wars in China. The China Journal (MarchApril) 111-163.
–
Shalizi, Cosma. 2007. Maximum Likelihood Estimation for q-Exponential (Tsallis) Distributions,
math.ST/0701854 http://arxiv.org/abs/math.ST/0701854
–
Theil, Henri. 1972. Statistical Decomposition Analysis. Amsterdam: North Holland.
–
Tsallis, Constantino. 1988. Possible generalization of Boltzmann-Gibbs statistics, J.Stat.Phys. 52,
479. (q-exponential)
–
Turchin, Peter. 2003. Historical Dynamics. Cambridge U Press.
–
Turchin, Peter. 2005. Dynamical Feedbacks between Population Growth and Sociopolitical
Instability in Agrarian States. Structure and Dynamics 1(1):Art2.
http://repositories.cdlib.org/imbs/socdyn/sdeas/
–
White, Douglas R. Natasa Kejzar, Constantino Tsallis, Doyne Farmer, and Scott White. 2005. A
generative model for feedback networks. Physical Review E 73, 016119:1-8 http://arxiv.org/abs/condmat/0508028
–
White, Douglas R., Natasa Keyzar, Constantino Tsallis and Celine Rozenblat. 2005. Ms. Generative
Historical Model of City Size Hierarchies: 430 BCE – 2005. Santa Fe Institute working paper.
city systems in the last millennium
28
Thanks to those who contributed to this
project
•
•
•
•
•
•
•
•
•
•
•
Laurent Tambayong, UC Irvine (co-author on the paper, statistical fits)
Nataša Kejžar, U Ljubljana (co-author on the paper, initial modeling, statistics)
Constantino Tsallis, Ernesto Borges, Centro Brasileiro de Pesquisas Fısicas, Rio de
Janeiro (q-exponential models)
Cosma Shalizi (the MLE statistical estimation programs in R: Pareto, Pareto II, and a
new MLE procedure for fitting q-exponential models)
Peter Turchin, U Conn (contributed data and suggestions)
Céline Rozenblat, U Zurich (initial dataset, Chandler and Fox 1974)
Chris Chase-Dunn, UC Riverside (final dataset, Chandler 1987)
Numerous ISCOM project and members, including Denise Pumain, Sander v.d.
Leeuw, Luis Bettencourt (EU Project, Information Society as a Complex System)
Commentators Michael Batty, William Thompson, George Modelski (suggestions and
critiques)
European Complex System Conference organizers (invitation to give the initial
version of these findings as a plenary address in Paris; numerous suggestions)
Santa Fe Institute (invitation to work with Nataša Kejžar and Laurent Tambayong at
SFI, opportunities to collaborate with Tsallis and Borges, invitation to give a later
version of these findings at the annual Science Board meeting).
city systems in the last millennium
29
Partial independence of q and β
TAILS AND BODIES OF CITY-SIZE DISTRIBUTIONS VARY INDEPENDENTLY
30