Eurasian city system dynamics in the last millennium Complex Dynamics of Distributional Size and Spatial Change, mediated by networks

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1 Eurasian city system dynamics in the last millennium Complex Dynamics of Distributional Size and Spatial Change, mediated by networks Doug White Irvine 3 minutes, 3 slides

2 Abstract (as in handout) A period historical scaling of city sizes in regions of Eurasia ( CE-) 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

3 Zipf s law? Population at rank r M the largest city and α= Chandler Rank-Size City Data (semilog) for Eurasia (Europe, China, Mid-Asia) Zipfian=top curve city rank top Transforms: natural log 8 ZipfCum e e e e e e e3 e3 e4 e4 e e e e6 e6 e e e8 e8 e86 e8 e e4 e e e c c c c c c c3 c3 c4 c4 c c c c6 c6 c c c8 c8 c8 c8 c c4 c c c m = m m m m m m3 m3 m4 m4 m m m m6 m6 m m m8 m8 m8 m8 m m4 m m m r i= Mi α city systems in the last millennium 3

4 Zipf s law? Population at rank r M the largest city and α= r = i= Mi α Take a closer look: Lots of variation from a Zipfian norm city rank city systems in the last millennium 4

5 Michael Batty (Nature, Dec 6:), using some of the same data as do we for historical cities (Chandler 8), states (and cites) the case made here (in a previous article, ) 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

6 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 CE. Gradual reduction in shifts until the Industrial Revolution. US shift lower, largest 83-. UK similar, with a - 6 suburbanization shift UK World city systems in the last millennium 6

7 City Size Distributions for Measuring Departures from Zipf construct and measure the shapes of cumulative city size distributions for the n largest cities from st rank size S to the smallest of size S n as a total population distribution T r for all people in cities of size S r or greater, where r=,n is city rank P(X x) Empirical cumulative citypopulation distribution r T r= i= r R Tr = i= S i Rank size power law M~S Mi α

8 For the top cities, fit the standard Pareto Distribution slope parameter, is beta (β) in P β (X x) = (x/xmin) -β (Xmin/x) β () 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=+/ө. P Ө,σ (X x) = ( + x/σ) -Ө () city systems in the last millennium 8

9 Probability distribution q shapes for a person being in a city with at least population x (fitted by MLE estimation) Pareto Type II Shalizi () right graphs=variant fits city systems in the last millennium

10 Examples of fitted curves for the cumulative distribution Curved fits measured by q shape, log-log tail slopes by β Low q fittings for years in High q fittings for years in cumulative number of people [ 3 ] e+3 3 e+3 4 e+4 e+ 6 e cumulative number of people [ 3 ] e+3 3 e+3 4 e+4 e+ 6 e+6 P β (X x) = (x/x min ) -β (top ten cities) city size [ ] city size [ ] city systems in the last millennium

11 Variations in q and the power-law slope β for - in year intervals city systems in the last millennium date China Europe Mid-Asia MinQ_Beta Beta MLEqExtrap

12 Random walk or Historical Periods? Runs Test Results Runs Tests at medians across all three regions MLE-q Beta Min(q/., Beta/) Test Value(a)...88 Cases < Test Value Cases >= Test Value Total Cases 3 3 Number of Runs Z Asymp. Sig. (-tailed)..3.3 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 8 Number of Runs 4 Z Asymp. Sig. (-tailed).8.4. a Median city systems in the last millennium

13 Fitted q parameters for Europe, Mid-Asia, China, -CE, year lags. Vertical lines show approximate breaks between Turchin s secular cycles for China and Europe Downward arrow: Crises of the 4th, th, and th Centuries city systems in the last millennium 3

14 N Minimum Maximum Mean Std. Deviation Std.Dev/Mean MLEqChinaExtrap MLEqEuropeExtrap MLEqMidAsIndia BetaTopChina BetaTopEur BetaTopMidAsia

15 Are there inter-region synchronies? Cross-correlations give lag = perfect synchrony lag = state of region A predicts that of B years later lag = state of region A predicts that of B years later lag 3 = state of region A predicts that of B years later etc. city systems in the last millennium

16 Time-lagged cross-correlation effects of Mid-Asian q on China mle_midasia with mle_china (= year lagged effect)..6 Coefficient Upper Confidence Limit Lower Confidence Limit.3 CCF Lag Number city systems in the last millennium 6

17 Time-lagged cross-correlation effects of China q on Europe mle_china with mle_europe ( year lagged effect)..6 Coefficient Upper Confidence Limit Lower Confidence Limit China with Europe4 CCF Coe Upp Limi Low Limi CCF Lag Number Lag Number (non-mle result for q) city systems in the last millennium

18 Time-lagged cross-correlation effects of the Silk Road trade on Europe logsilkroad with EurBeta ( year lagged effect)..6 Coefficient Upper Confidence Limit Lower Confidence Limit mle_midasia with mle_europe CCF Coefficient Upper Confidence Limit Lower Confidence Limit Lag Number Lag Number city systems in the last millennium 8

19 mle_europe with ParisPercent..6 Coefficient Upper Confidence Limit Lower Confidence Limit.3 CCF Lag Number 3 4 6

20 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 () and then relate to the city system shape dynamics. city systems in the last millennium

21 Turchin : Dynamical Feedbacks in Structural Demography Key: Innovation Chinese phase diagram

22 Turchin validates statistically the interactive prediction versus the inertial prediction for England, Han China ( BCE -3 CE), Tang China (6 CE - )

23 J. S. Lee measure of SPI for China region (internecine wars ) city systems in the last millennium 3

24 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 years. Shows the effect of internecine wars in China on city size distributions city systems in the last millennium 4

25 Interpretation: SPI conflict correlates with a higher ratio of q to (low) β, but as β recovers slightly in the next 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

26 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: () change in the log-log slope (β) of the power-law tail of city sizes () 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. 6

27 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 CE. Gradual reduction in shifts until the Industrial Revolution. US shift lower, largest 83-. UK similar, with a - 6 suburbanization shift Gibrat law: proportionate random growth? P i (t) = [Г+ε i (t)] P i- (t), lognormal, becoming power-law with P i (t) > P min (t), i.e., losers eliminated Back to Batty: UK 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. 3; 6; Theil ; Tsallis 88). World city systems in the last millennium

28 References Batty, Michael. 6. Rank Clocks. Nature (Letters) 444:-6. Batty 6 Entropy in Spatial Aggregation Geographical. Analysis 8:- Chandler, Tertius. 8. Four Thousand Years of Urban Growth: An Historical Census. Lewiston, N.Y.: Edwin Mellon Press. Goldstone, Jack. 3. 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. 3. The periodic recurrence of internecine wars in China. The China Journal (March- April) -63. Shalizi, Cosma.. Maximum Likelihood Estimation for q-exponential (Tsallis) Distributions, math.st/84 Theil, Henri.. Statistical Decomposition Analysis. Amsterdam: North Holland. Tsallis, Constantino. 88. Possible generalization of Boltzmann-Gibbs statistics, J.Stat.Phys., 4. (q-exponential) Turchin, Peter. 3. Historical Dynamics. Cambridge U Press. Turchin, Peter.. Dynamical Feedbacks between Population Growth and Sociopolitical Instability in Agrarian States. Structure and Dynamics ():Art. White, Douglas R. Natasa Kejzar, Constantino Tsallis, Doyne Farmer, and Scott White.. A generative model for feedback networks. Physical Review E 3, 6:-8 White, Douglas R., Natasa Keyzar, Constantino Tsallis and Celine Rozenblat.. Ms. Generative Historical Model of City Size Hierarchies: 43 BCE. Santa Fe Institute working paper. city systems in the last millennium 8

29 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 4) Chris Chase-Dunn, UC Riverside (final dataset, Chandler 8) 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

30 Partial independence of q and β TAILS AND BODIES OF CITY-SIZE DISTRIBUTIONS VARY INDEPENDENTLY 3