Complex Dynamics of Distributional Size and Spatial Change, mediated by networks
30 minutes, 30 slides
1

(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

Population at rank r
R
Tr
=
M the largest city and α=1 i r 

1
12
Mi
 
12
11
10
9
8
7
6
5
11
10
1 2
Zipfian=top curve
3 4 5 14 15 16 17 18 19 20 6 7 8 9 10 11 top20
12 13 city rank
Transforms: natural log c900 c1000 c1100 9 c1150 c1200 c1250 c1300 8 c1350 c1400 c1450 c1500 7 c1550 c1575 c1600 c1650 6 c1700 c1750 c1800 c1825 5 c1850 c1875 c1900 c1914 c1925 c1950 c1970 m900
ZipfCum e900 e1000 e1100 e1150 e1200 e1250 e1300 e1350 e1400 e1450 e1500 e1550 e1575 e1600 e1650 e1700 e1750 e1800 e1825 e1860 e1875 e1900 e1914 e1925 e1950 e1970 m1000 m1100 m1150 m1200 m1250 m1300 m1350 m1400 m1450 m1500 m1550 m1575 m1600 m1650 m1700 m1750 m1800 m1825 m1850 m1875 m1900 m1914 m1925 m1950 m1970
1 2 3 4 5 6 7 8 city systems in the last millennium 3
9 10 11 12 top20
13 14 15 16 17 18 19 20
Transforms: natural log e1550 e1575 e1600 e1650 e1700 e1750 e1800 e1825 e1860 e1875 e1900 e1914 e1925 e1950 e1970
ZipfCum e900 e1000 e1100 e1150 e1200 e1250 e1300 e1350 e1400 e1450 e1500 c1575 c1600 c1650 c1700 c1750 c1800 c1825 c1850 c1875 c1900 c1914 c1925 c1950 c1970 m900 c900 c1000 c1100 c1150 c1200 c1250 c1300 c1350 c1400 c1450 c1500 c1550 m1000 m1100 m1150 m1200 m1250 m1300 m1350 m1400 m1450 m1500 m1550 m1575 m1600 m1650 m1700 m1750 m1800 m1825 m1850 m1875 m1900 m1914 m1925 m1950 m1970

7
6
5
9
8
12
11
10
Population at rank r
R
Tr
=
M the largest city and α=1 i r 

1
Mi
 
Take a closer look:
Lots of variation from a
Zipfian norm city rank city systems in the last millennium 4
1 2 3 4 5 6 7 8 9 10 11 12 top20
13 14 15 16 17 18 19 20
e1350 e1400 e1450 e1500 e1550 e1575 e1600 e1650 e1700 e1750 e1800 e1825 e1860 e1875 e1900 e1914 e1925 e1950 e1970
ZipfCum e900 e1000 e1100 e1150 e1200 e1250 e1300 c1400 c1450 c1500 c1550 c1575 c1600 c1650 c1700 c1750 c1800 c1825 c1850 c1875 c1900 c1914 c1925 c1950 c1970 m900 c900 c1000 c1100 c1150 c1200 c1250 c1300 c1350 m1450 m1500 m1550 m1575 m1600 m1650 m1700 m1750 m1800 m1825 m1850 m1875 m1900 m1914 m1925 m1950 m1970 m1000 m1100 m1150 m1200 m1250 m1300 m1350 m1400

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 1950-
60 suburbanization shift
UK
World city systems in the last millennium 6

n
S
1
people
S n
T r
S r
r
n
P(X ≥x)
Empirical cumulative citypopulation distribution r
T r= i


1
S i
Rank size power law M~S
1
R
Tr
= i r 

1
Mi
 
7

P
x
x/x min
)
min
x
P
(X ≥ x) = (1 + x/σ)
city systems in the last millennium 8

Pareto Type II
Shalizi (2007) right graphs=variant fits city systems in the last millennium 9

P
β
(X ≥ x) = (x/x min
) β
(top ten cities) city systems in the last millennium 10

3.0
3.0
2.5
2.5
2.0
2.0
1.5
1.5
1.0
1.0
China Europe Mid-Asia
MLEqExtrap
Beta10
MinQ_Beta
0.5
0.5
0.0
0.0
9
0
1
0
0 0
0
1
1
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0 0
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0 0
0
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1
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1 7
1
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1 0
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1 0
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1 5
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1 5
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1 7 1
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0 0
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5 0 5 7 0 5
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0
1 0
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1 0
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0
1 2
3
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1 0 1 0
5
0
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0
1 5
5
0
1
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1
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1 5
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0
1 0
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0
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0 5 0 2 5
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7 0 2 5
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0
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7 0 0 0
1
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1 7 1
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0
0
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5
1 1
3 3
0 5
1 1
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0
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1 1
5 5
0 5
1 1 1
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0 5
1 1
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0
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1 1
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0 2
1 1 1 1
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7 0
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2
1 1
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0 0 0 5 0 date
0 0 0 0 5 0 5 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 5 0 5 0 5 0 0 city systems in the last millennium 11
MLEqExtrap
Beta10
MinQ_Beta

Runs Tests at medians across all three regions
Min( q /1.5,
Beta/2)
Test Value(a)
MLEq Beta10
1.51 1.79 .88
Cases < Test Value 35 36 35
Cases >= Test Value 36 37 38
Total Cases 71 73 73
Number of Runs 20 22 22
Z -3.944 -3.653 -3.645
Asymp. Sig. (2-tailed)
.0001 .0003 .0003
Runs Test for temporal variations of q in the three regions mle_Europe mle_MidAsia mle_China
Test Value(a)
Cases < Test Value
Cases >= Test Value
Total Cases
Number of Runs
Z
Asymp. Sig. (2-tailed)
1.43
9
9
18
4
-2.673
.008
1.45
11
11
22
7
-1.966
.049
1.59
10
12
22
7
-1.943
.052 a Median city systems in the last millennium 12

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 millennium 13

MLEqChinaExtrap
MLEqEuropeExtrap
MLEqMidAsIndia
BetaTop10China
BetaTop10Eur
BetaTop10MidAsia
N
25
23
25
23
23
25
Minimum Maximum
.56 1.81
1.02
1.00
1.89
1.72
1.23
1.33
1.09
2.59
2.33
2.86
Mean
1.5120
1.4637
1.4300
1.9744
1.6971
1.7022
Std. Deviation Std.Dev/Mean
.25475 .16849
.19358
.16763
.13225
.11722
.35334
.27679
.35392
.17896
.16310
.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 millennium 15

0.0
-0.3
-0.6
-0.9
Time-lagged cross-correlation effects of Mid-Asian q on China
(1=50 year lagged effect) mle_MidAsia with mle_China
0.9
Coefficient
Upper Confidence
Limit
Lower Confidence
Limit
0.6
0.3
-7 -6 -5 -4 -3 -2 -1 0 1
Lag Number
2 3 4 5 6 7 city systems in the last millennium 16

Time-lagged cross-correlation effects of China q on Europe mle_China with mle_Europe
(100 year lagged effect)
0.9
0.6
0.3
0.0
-0.3
-0.6
-0.9
Coefficient
Upper Confidence
Limit
Lower Confidence
Limit
China with Europe4
0.3
0.0
0.9
0.6
-0.3
-0.6
-0.9
-7 -6 -5 -4 -3 -2 -1 0 1
Lag Number
2 3 4 5
-7 -6 -5 -4 -3 -2 -1 0 1
Lag Number
2 3 4 5 6 7
6 7
(non-MLE result for q) city systems in the last millennium 17
Coefficient
Upper Confidence
Limit
Lower Confidence
Limit

Time-lagged cross-correlation effects of the Silk Road trade on Europe
(50 year lagged effect) logSilkRoad with EurBeta10
0.9
0.6
0.3
0.0
-0.3
-0.6
-0.9
Coefficient
Upper Confidence
Limit
Lower Confidence
Limit mle_MidAsia with mle_Europe
0.0
-0.3
-0.6
-0.9
0.9
0.6
0.3
-7 -6 -5 -4 -3 -2 -1 0 1
Lag Number
2 3 4 5 6 7
-7 -6 -5 -4 -3 -2 -1 0 1
Lag Number
2 3 4 5 6 7 city systems in the last millennium 18
Coefficient
Upper Confidence
Limit
Lower Confidence
Limit

mle_Europe with ParisPercent
0.0
-0.3
-0.6
0.9
0.6
0.3
-0.9
-7 -6 -5 -4 -3 -2 -1 0 1
Lag Number
2 3 4 5 6 7
Coefficient
Upper Confidence
Limit
Lower Confidence
Limit
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 millennium 20

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 millennium 23

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 24

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

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 1950-
60 suburbanization shift
Gibrat’ law: proportionate random growth?
P i
(t) = [ Г+ε i
(t)] P i-1
(t), lognormal, becoming power-law with
P i
(t) > P min
(t), i.e., “losers eliminated”
UK
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).
World city systems in the last millennium 27

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 (March-
April) 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

•
Laurent Tambayong, UC Irvine (co-author on the paper, statistical fits)
•
Nata
ša 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)
• C éline 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 Nata ša 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