‘The us today is like a house where the owners are busy tearing down the walls to throw bricks at the neighbors

Figure 9: Autocorrelation functions for the three regions

Method: Tsal.fit. Takes the rank-size city distribution: q and κ are now uncorrelated. This is good.

To fit a power law, Spss needs the cumulative distribution. Easiest way to do this in the Chandler file, sorted by period and size, is to create a second column to the right of the size column, and make it a cumulative sum for all cities. Then go to each largest city turn this row yellow, and pull over the largest city size. Then Paste the series for one period into a column of Spss. ChandleCumPop.sav has CumPol by rank. Create a variable called rank from 1 to n. If you want only the top cities make n=10 (TopTen). To compute the Pareto 1/Beta coefficient in Spss/Analyze/Regression/Curve Estimation [x] Power law. This gives Beta=alpha+1 where alpha=1 is Zipf. To do the whole series with [x] Display Anova table, select “Top10” as the independent variable and enter all the regionYears (e.g., “c1200”) as dependent variables (run them all at once. Then when you enter the results in a spreadsheet do a transformation on 1/Beta to calculate Beta as it’s reciprocal, e.g. .5 becomes 2.0, the Zipfian Beta. For a single period select “Rank” as the dependent variable and cYear (e.g., “c1200”) as the Independent Variable and you get Beta directly.

CHINA and MID-ASIA: indirect effect of conquest?

opposite to predicted: China q Up SPI down! As predicted: China

Does SPI(max) at LOW beta and then predict beta recovery? (possibly)

1 Our special thanks to Constantino Tsallis, the inspiration for this study, who patiently taught Doug and Nataša the fundamentals of q-exponential concepts and methods and then answered questions as they proceeded through the substantive analysis, and to Peter Turchin for generously providing data and suggestions for analysis. Any errors however remain our own. Thanks also to Robert McC. Adams, George Modelski and William Thompson for critical commentary and suggestions. Support from EU project ISCOM, “Information Society as a Complex System,” headed by David Lane, Sander van der Leeuw, Geoff West, and Denise Pumain, is acknowledged for the city sizes project. We thank the leaders and member of the project for critical commentary. The larger project, Civilizations as Dynamic Networks, forms part of a Santa Fe Institute Working group for which SFI support is acknowledged. An early draft of the paper was presented to the Seminar on “Globalization as Evolutionary Process: Modeling, Simulating, and Forecasting Global Change,” sponsored by the Calouste Gulbenkian Foundation, meeting at the International Institute for Applied Systems Analysis (IIASA), Laxenburg, Austria, April 6-8, 2006. Sponsored by the Calouste Gulbenkian Foundation, meeting at the International Institute for Applied Systems Analysis (IIASA), Laxenburg, Austria, April 6-8, 2006. The reliability of the final analysis would not have been possible without the help of Cosma Shalizi, who derived and then programmed an MLE solution to the problems of estimating the parameters of historical city-size distributions, many of which had relatively small lists of largest city sizes once we got to the regional level.

2 Institute for Mathematical Behavioral Sciences, University of California, Irvine.

3 External Faculty, Santa Fe Institute.

4 Faculty of Social Sciences, University of Ljubljana, Slovenia.

5 Noting from the shared database that the top echelon of cities in a single region may be swept away in a short period by interregional competition, Batty refers to our work on instabilities at the level of city systems.

6 For example, excluding two primate city outliers, the next largest 16 cities for 1998 in the U.S. (over 11 million) show a steep log-log slope, those ranking down to .5 million show a shallower slope, those to 1 million a much shallower slope, and then the power-law disappears altogether (Malacarne et al. 2001:2).

7 Commentary on our White et al. (2004) paper on the present topic elicited the suggestion from Cosma Shalizi that we should consider MLE estimates for fitting q-exponentials to city size data. Given our small sample sizes from the Chandler data as we moved from a world sample (75 data points) to regional samples, we called on Shalizi in late 2006 to derive the MLE equation for us, which he did,. He then found earlier derivations such as those of Arnold (1983) and others. We are greatly indebted to Shalizi for writing functions in R to make MLE (unbiased) estimates of the q-exponential parameters and bootstrap estimates of the standard errors of these estimates.

8 There are other advantages that we do not exploit here:

1. 
   Y_q estimates an expected “largest city size” M consistent with the body of the size distribution. This requires simultaneous estimation of M and Y(0) to solve Y(0) P_Ө,σ (X ≥ M)=M.
   
2. 
   The total urban population can be estimated from Y_q without having data on all smaller cities, although this feature is not utilized here.
   
3. 
   Equation (1) and Y_q may be fitted without the largest city so as to derive an expected size for the largest city given our model.
   
4. 
   This gives our model a ratio measure of the largest city size to its expected size from Y_q.
   
5. 
   Y_q has a known derivative Y_q′(x)=Y0/κ[1-(1-q)x/κ]^(q/(1-q)) giving the slope of the curve Y_q(x) for any city size x.
   
6. 
   Solving for Y_q(M)=M for the estimated largest city size M consistent with Y_q gives Y_q′(M) as the slope of the Y_q at M and converges with β=1/(q-1) for the Pareto power-law slope.
   

9 In some of our earlier analyses there were tenuous indicators of 400-year periods and possibly 200-year periods of city size oscillation that seemed to correspond more closely with Turchin’s secular cycles. With more accurate MLE measures, however, we see longer periods of stability, and we attribute the appearance of shorter oscillations in the earlier analysis to biased estimations that, in introducing intermittent error, tended to break longer periods of stability into what appeared to be shorter ones.

10 Technically, the mathematics assigned to chaos is a deterministic departure from randomness in which a dynamic trajectory never settles down into equilibrium, and small differences in initial conditions lead to divergent trajectories. The link between empirical history and “edge of chaos” is typically done by simulation.