Doug White Irvine 30 minutes, 30 slides

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Eurasian city system dynamics in the last millennium

Complex Dynamics of Distributional Size and Spatial Change, mediated by networks

Doug WhiteIrvine

30 minutes, 30 slides

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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 q-exponential 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

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Zipf’s law? Population at rank r

M the largest city and α=1 Chandler Rank-Size City Data (semilog) for

Eurasia (Europe, China, Mid-Asia)


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m1970m1950m1925m1914m1900m1875m1850m1825m1800m1750m1700m1650m1600m1575m1550m1500m1450m1400m1350m1300m1250m1200m1150m1100m1000

m900c1970c1950c1925c1914c1900c1875c1850c1825c1800c1750c1700c1650c1600c1575c1550c1500c1450c1400c1350c1300c1250c1200c1150c1100c1000c900

e1970e1950e1925e1914e1900e1875e1860e1825e1800e1750e1700e1650e1600e1575e1550e1500e1450e1400e1350e1300e1250e1200e1150e1100e1000e900ZipfCum

Transforms: natural logcity rank

RTr =

r

i

Mi1

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Zipfian=top curve

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Zipf’s law? Population at rank r

M the largest city and α=1

city systems in the last millennium city rank

RTr =

r

i

Mi1

Take a closer look:

Lots of variation from a Zipfian norm

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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:

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.

“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.”


6city systems in the last millennium

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

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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

Tr=

r

iiS

1

RTr =

r

i

Mi1

Rank size power law M~S1

Empirical cumulative city-population distribution

P(X≥x)


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)

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Probability distribution q shapes for a person being in a city with at least population x (fitted by MLE estimation) Pareto Type II

city systems in the last millennium Shalizi (2007) right graphs=variant fits

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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)

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Variations in q and the power-law slope β for 900-1970 in 50 year intervals


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date

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MinQ_BetaBeta10MLEqExtrap

China Europe Mid-Asia

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MinQ_BetaBeta10MLEqExtrap

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Random walk or Historical Periods? Runs Test Results


Runs Tests at medians across all three regions

MLE-q Beta10 Min(q/1.5,

Beta/2) Test Value(a) 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) 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

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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


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N Minimum Maximum Mean Std. Deviation Std.Dev/Mean MLEqChinaExtrap 25 .56 1.81 1.5120 .25475 .16849 MLEqEuropeExtrap 23 1.02 1.89 1.4637 .19358 .13225 MLEqMidAsIndia 25 1.00 1.72 1.4300 .16763 .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


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



etc.


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Lag Number

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mle_MidAsia with mle_China

Lower Confidence Limit

Upper Confidence Limit

Coefficient

Time-lagged cross-correlation effects of Mid-Asian q on China

(1=50 year lagged effect)


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Lag Number

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mle_China with mle_Europe



Coefficient

Time-lagged cross-correlation effects of China q on Europe

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China with Europe4



Coefficient

(100 year lagged effect)

(non-MLE result for q)


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logSilkRoad with EurBeta10



Coefficient

Time-lagged cross-correlation effects of the Silk Road trade on Europe

(50 year lagged effect)

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mle_MidAsia with mle_Europe



Coefficient

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Fmle_Europe with ParisPercent



Coefficient


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.

21Chinese phase diagram

Turchin 2005:

Dynamical Feedbacks in

Structural Demography

Key:

Innovation

http://eclectic.ss.uci.edu/~drwhite/Civ/Spuf/Tur/ChinaPopWarPhaseSpace.jpg

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Turchin 2005 validates statistically the interactive prediction versus the inertial prediction for England, Han China (200 BCE -300 CE), Tang China (600 CE - 1000)

http://eclectic.ss.uci.edu/~drwhite/Civ/Spuf/Tur/ChinaEnglandPopWarPredictions.jpg


J. S. Lee measure of SPI for China region (internecine wars )


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


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.

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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.


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

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 scale-free 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), log-normal, becoming power-law with Pi(t) > Pmin(t), i.e., “losers eliminated”

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– 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/cond-mat/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.

References


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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).


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TAILS AND BODIES OF CITY-SIZE DISTRIBUTIONS VARY INDEPENDENTLY

Partial independence of q and β

Doug White Irvine 30 minutes, 30 slides

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