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“Oscillatory dynamics of city-size distributions

in world historical systems” 1


Douglas R. White,2,3 Nataša Kejžar,4 and Laurent Tambayong2

for:


G. Modelski, T. Devezas and W. Thompson, eds. Globalization as Evolutionary Process: Modeling, Simulating, and Forecasting Global Change

9:03PM 2/11/2007

Vienna2\WhiteKejzarTambaVienna_K_April.doc G has deletions
INTRODUCTION p. 1

PART I: CITY SYSTEM INSTABILITIES p. 3

PART II: DATA p. 4

PART III: THE Q/β SCALING AND HYPOTHESES p. 6

PART IV: CROSS-CORRELATION OF THE SCALING MEASURES p. 15

PART V: HISTORICAL NETWORK AND INTERACTION PROCESSES p. 20

PART VI: CONCLUSIONS p. 24

Abstract. Oscillatory patterns of expansion/contraction have long characterized the dynamics of demographic, economic, and political processes of human societies, including those of exchange economies and globalization. Major perturbations in city-size distributions are shown to exist for major regions in Eurasia in the last millennium and to exhibit some of the characteristics of cyclical oscillations on the scale of 100s of years as well as longer fluctuations, up to 800 years, between periods of collapse. Variations in timing, irregularities in amplitudes, and ups and downs in our measure appear to correlate with some of the peaks and troughs in urban population growth and show long-cycle correlations with J.S. Lee’s (1931) sociopolitical instability (SPI) data on the durations of internecine wars for China.

We hypothesize an embedding of dynamical processes that runs from trading zone network sizes and city-size distributions that cycle roughly 200, 400, or 800 years, partly dependent on the severity of the decline. Our interpretation of city-size distribution oscillations is that they follow, with a generational time lag, rises and falls in the expansion/contraction of multi-connected trade network macro zones. More Zipfian city-size hierarchies tend to rise with trade network expansions and fall with contractions. These might be longer oscillations than Turchin’s population (“secular”) SPI cycles that average about 220 years. Turchin cycles seem to embed two leading polity cycles (Modelski and Thompson 1996) that average about 110 years. These are averages, and actual timings vary, but we give explanations for why such these average cycle-lengths might tend to diminish by half as each embedded process tends to operate at successively smaller spatial scales. We do find evidence that rise and fall in Silk Road connectivities between China and Europe had time lagged effects on the growth of power law tails in European urban hierarchies.

Time-lagged synchronies in the dating of phases for city distributions in different regions that are connected by multiple routes of trade, as noted tentatively by Chase-Dunn and Manning (2002:21), at least in the rising and more Zipfian phase, lend credibility to the existence of city-system rise and fall cycling. We focus here on central civilization, including China and Europe, and the Mid-Asian region between, and the world cities database. These data are likely to reflect changes in the macro regions connected by trade networks, where we would expect synchronization. Our three larger regions show dominant time-lagged effects of city-system rise and fall for Mid-Asia affecting China in this millennium, and China affecting Europe, at different time lags.
INTRODUCTION

Globalization, world-system, and historical dynamic theory offer complementary perspectives for the study of city systems as the politicoeconomic engine of interstate networks. Here we combine these perspectives to examine a dynamical perspective on systems of cities. Globalization theory applied to Eurasia in the last millennium (e.g., Modelski and Thompson 1996) focuses on centers of economic innovation and political power and their successive periods of rise and fall in dominance. Units of larger scale, as for example polities, are shown to operate at successively longer time scales in rise and fall than the economic innovation centers within those polities. World-system theory for similar regions and processes (e.g., Chase-Dunn and Hall 1997) differs in focusing as well on innovation at the peripheries of states and empires, that is, on the marcher or boundary polities that resist the encroachment of expanding empires. Marcher states that amalgamate to defeat the spread of empire often defeat polities formally organized on a much larger scale through superior cohesive or decentralized organization able to forge a superior combative skill or technology. World-systems theory often limits itself to the more prominent types of relations, such as trade in bulk goods and interstate conflict, that form distinct macroregional networks. The structural demographic approach to the political economics of agrarian empires (e.g., Turchin 2003, 2006) is capable of yielding a more dynamical historical account of how central polities rise and fall as their internal cohesion disintegrates with population growth into factional conflict and of how once dominant polities and economies contend with marcher states that coalesce into formidable opponents on their frontiers.

Several of the problems in extending these kinds of complementary approaches to globalization, world-systems, and historical dynamics relate to how networks – social, political, and economic – fit into the processes of change and dynamical patterns that are observed historically. One aspect of major problems that might engage network research involves how changing network fluctuations of long distance trade influence inter- and intra-regional dynamics. The Silk Road trade so important in the connections through the marcher states and later empires of Mongol Central Asia between China and the Middle East, for example, also facilitated the diffusion of economic inventions from East to West that were crucial in the rise of the European city system. These included paper money, institutions of credit, and vast new knowledge, weaponry, and technologies. Spufford (2002), for example, shows how important were the transmissions of innovations from China from a European perspective, while Temple (1987) summarizes the work of Needham (1954-2004) to show the debt of the West to China. A related problem, among many other open network problems in historical research is how regional and long-distance trade networks are coupled, along with conflicts and wars, to the rise and fall of cities and city systems. That is the problem we take up here.

We approach the problems of the rise and fall of commercial trade networks, regional city systems, regional conflicts, and the historical dynamics of globalization and world-system interactions in Eurasia, during the last millennium, with a concern for valid comparative measurement of large scale phenomena. Tertius Chandler (1987) and other students of historical city sizes (Pasciuti 2006, Modelski 2003, Bairoch 1958, Braudel 1992, and others) have made it possible to compare the shapes of city-size distribution curves. These are the data we examine here in order to gauge and compare the dynamics of city system rise and fall, both within distinct regions and as changes in one region (such as China) affect changes in others (such as Europe, to give but one example).

Our approach here is to divide up Chandler’s (1987) Eurasian largest city-sizes data into three large regions – China, Europe, and the Mid-Asian region in between – and measure variations over time that depart from the Zipfian rank-size distribution. Zipfian rank-size is the tendency for cities ranked 1 to n in size to approximate a size of M/r where r is a city’s rank compared to the largest city and M is a maximum city size M that gives the best fit for the entire distribution (this formulation allows the rank 1 largest city size S1 to differ from its expected value under a Zipfian fitted to an extensive set of the larger cities). The Zipfian distribution has been taken to be a recurrent and possibly universal pattern for city sizes as well as many other complex system phenomena. What we find for Eurasia and regions within Eurasia is that there are systematic historical periods that show significant deviations from the Zipfian. Some of these deviations show the characteristics of a regional collapse of city systems from which there is eventual recovery (unlike cataclysmic collapse exemplified by the Mayan cities system).

The periods of rise and fall of city systems for each Eurasian region, however, are different. This allows us to test the hypothesis that the rise and fall measure for China anticipates with a time lag the rise and fall measure for Europe, which is a prediction for the period starting in 900 CE consistent with Modelski and Thompson (1996), Temple (1987) and Needham (1954-2004). Finally, for the region of China we have sufficient time-series data to test the predictions from the historical dynamics model of Turchin (2005). This allows some limited results on whether some of the same processes are operative for the rise and fall of city systems as for the historical dynamics of state and empire rise and fall.

In Part I we pose the problem of instabilities in city sizes and systems drawing on Chandler’s data for 26 historical periods from 900 CE to 1970. Part II examines ways of measuring departure from Zipfian distributions of city sizes and introduces the data used for city sizes and possible correlates of city system change. Part III gives the results of the scaling of city sizes for different regions so as to measure city system changes. In Part IV we examine the time-lagged interregional cross-correlations for these measures, and summarize results for cross-region synchrony. Part V examines correlations and time-lags between our three Eurasian regions and for other variables related to known historical oscillations where we have adequate data for hypothesis tests. The variables tested include such variables as trade connectivity, internecine warfare within China and development of credit and currency systems that facilitate international exchange as well as innovative national markets. VI concludes with a summary and implications of the findings.
PART I: CITY SYSTEM INSTABILITIES p. 3

Jen (2005:8-9) defines stability in terms of dynamical recoveries from small perturbations that return to an original state. Seriousness of the question of city system instability and of major departures from the Zipfian derive from the assumption that city economies are organized as networks that involve trade and war, and depend on innovation to join the leading economic or political sectors of more global networks. The two main factors that make for instability are competition and population growth.

Economic competition, aided by power politics, tends to make for oscillations that may return to what might be called structural stability. That is, they make for economic and political limit cycles rather than conservative stationarity. Populations of polities, empires, regions, and global world systems, also exhibit limit cycles if we average out trends of population growth, e.g., over the last several millennia. Jen defines structural stability as the ability to return from instability through other dynamics than the original (e.g., by varying external parameters) that are qualitatively similar to the original dynamic, as for example the Lotka-Volterra type of oscillatory limit cycle. While economic and political systems are not stable in the strict sense they may have the resilience to return to structural stabilities as they pass through oscillatory limit cycles with differing but qualitatively similar dynamics. Major population growth trends, however, as they interact with dynamical oscillations or limit cycles, may lead to structural instability, an inability to return to stability even through other dynamics than the original but that recover qualitative similarity to the original. The imperative of incessant competitive innovation for successful cities and city systems forms part of what leads to overgrowth of population relative to resources and to subsequent system crashes. Historically, these instabilities lead eventually to industrial revolutions that, rather than conserve materials and energies, may push extravagant degradation of resources into dynamically irreversible crises such as global warming and problems of structural instabilities. Unless innovation turns toward conservation the problems created will not be solved in the next century or possibly not in next millennium. The issues here are ones of scale, expansions of scale (size of cities, size of polities and empires, size of economies), the dynamic interactions that operate at different scales, and how these couple spatially and temporally (as described, for example, in Modelski and Thompson 1996).

The first questions of this study, then, are whether city systems as central economic actors and sites for multitudes of agents are stable or unstable, and if unstable, what kinds of models are appropriate for consistency with their dynamics. The thesis here is that it is not just individual cities that grow and decline but entire regional (and global) city systems. Here, drawing on our earlier work (White, Kejžar, Tsallis and Rozenblat 2005), Michael Batty (2006:592) states our case for us. “It is now clear that the evident macro-stability in such distributions” as urban rank-size or Zipfian 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.” What Batty shows, using the same data as do we for historical cities (Chandler 1987), is legions of cities in the top echelons of city rank being swept away as they are replaced by competitors, largely from other regions.5


PART II: DATA p. 4

City Size Data for Historical Eurasia
Chandler’s (1987) database on historical city sizes is complemented by overlapping UN population data from 1950 to the present (in the interest of brevity we do not present these results here). Chandler reconstructed urban populations from many data sources. These included areas within city walls times number per unit area (see Appendix A), connected house-to-house suburbs lying outside the municipal area, data from city histories provided by city librarians, estimates from numbers of houses times numbers per house, and the cross-checking of different estimates (see Pasciuti and Chase-Dunn 2002). From 900 CE to 1970 his size estimates cover over 26 historical periods, usually spaced at 50 year intervals, always comprise a set of largest cities suitable for scaling in a single period. These data include 80 Chinese, 91 European, and in between a much larger number of Mid-Asian cities.

Figure 1 shows numbers of cities in the dataset for in each period when they fall below 21. As numbers decline from 1200-1650 for China, for example, China becomes less hegemonic as many of the top 75 world cities appear in other regions of the world. Resurgence of China begins to occur after the end of the last (Qing) dynasty.




(Above 20)


Figure 1: Number of Cities in Top 75 World Cities in each region when they fall below 21
Trade Routes (Eurasia). The total length of the Eurasian long-distance trade routes between 3500 BCE and 1500 CE at 50 year intervals have been calculated by Ciolek (2005) from trade-route maps drawn by Sherratt (2003). World-system pulsations in expansion and contraction of trade routes are shown by Turchin (2007). Turchin calculated a connectivity index for the Silk Routes between China and England. His data also show how instances of epidemics are concentrated near the high points of trade before periods of collapse.

Sociopolitical Instability: Internecine Wars (China only)
Turchin (2003:164) transcribed J. S. Lee’s (1931) coded 5-year interval data on internecine wars in China from 221 BCE (unification by the first Ch’in Dynasty Emperor) to 1929 to create a ten-year sequential intensity index to 1710. D.R. White converted Turchin’s codes into a 25-year index running from CE to 1700 and coded 1725-1925 directly from Lee (1931). Lee coded the period to the end of the Ming Dynasty from a remarkably systematic inventory of conflicts by Chih Shao-nan from the Tih Wang Nien Piao and checked for accuracy by Tung-Kien (Lee 1931:114). The Tabula Annua (Seikainenkan), cross-checked and supplemented by the Tai Ping Tien Kuo Chan Ssi proved a reliable record of Ch’ing Dynasty wars, with those following from Lee’s memory. The systematic pattern discussed by Lee for his graphs is one of two 800-year periods (200 BCE-600CE, then to 1385 CE) in which many more of the intra-territorial China conflicts occur in the last 400-500 than in the early 400-300 years, and a partial repetition up to 1927 of that same pattern. The other evident pattern is for conflicts to become much more likely at the transitions between dynastic periods.

Total Population and Population Change Data (Eurasia)
The early data on Chinese population from 900 to 1300 are controversial. White had data from Chao and Hsieh (1988), provided by Turchin (2003:164-165), he consulted Ho (1956, 1959), Steurmer (1980), Mi Hong (1992), Durand (1960), Heilig (1997, 1999, 2002), and the radical revision proposed by Heijdra (1995) and Mote (1999) that was critiqued by Marks (2002), and others. Given the uncertainty in absolute figures, we coded a binary variable for each 25-year period where 1 is given for a date at which there is a population peak before collapse, with 0 otherwise. The different total population estimates available to us for China over our full time frame agreed very closely as to where these population peaks occurred. In some cases two adjacent peaks were indicated.

Turchin (2006, 2007) provides population, carrying capacity, detrended population, and a misery index (inverse wages) that is useful for England into comparison to our European city data.



Monetary Liquidity (China only)
We devised a robust index of monetization (liquidity) for China from Temple (1986:117-119,168), coding data on credit, paper money, and banking for 900-1700 into a scale from 0-10, treating inflation as lower liquidity.

PART III: THE Q/β SCALING AND HYPOTHESES p. 6



Measuring Departures from Zipf's Law for City Size Distributions

We begin by examining the city size distributions of macroregions in the Eurasian continent, over roughly the last millennium, which is the millennium of modern globalization. We examine the extent of instability of city systems in ways that are visually evident by inspection of changes of the shapes of Eurasian city size distributions, and measured by a shape index. Figure 2 shows a semilog graph of the cumulative rank-size distribution for divisions of most of Eurasia (excluding Japan/Korea) into three regions: China (c900-c1970), Europe (e900-1970) and the Mid-Asian (m900-c1970) remainder. The curve to which power-law distributions should correspond is shown by the top (ZipfCum) power-law curve. Cumulative population size is logged on the y axis and the x axis is city size rank. The Zipfian curve forms a straight line when rank is also logged, with a Pareto log-log slope of 2. As can be seen visually, there is some departure from perfect parallelism in the empirical curves: some lines are more curved or less curved for the top cities than the Zipfian, most lines are flatter that the Zipfian for the smaller cities and many of the curves bend at different city ranks.




Figure 2: The Chandler Rank-Size City Data (semilog) for Eurasia (Europe, China, Mid-Asia)
Figure 3 shows the same data (all three regions) in a log-log plot where power-law city distributions would all be straight lines and Zipfian distributions would all have the same slope. The lines are neither parallel nor of the same slope, nor do they have curvatures in the same places. Our measurements of the properties of these distributions will be aimed at the hypothesis that these variations provide indicators useful to showing how city system fluctuations fit into economic and historical dynamics.


Figure 3: The Chandler Rank-Size City Data (log-log) for Eurasia (Europe, China, Mid-Asia)
We use two measures of how these curves vary. Each measurement model is based on a standard continuous function with parameter fitting based on recasting the data as a cumulative probability in its complementary form. P(X ≥ x) is the probability that an urbanite will reside in a city of at least size x and each model, as a cumulative complementary distribution function (CCDF), is aimed at capturing the shape of the empirical P(X ≥ x). It is especially important to use MLE or maximal likelihood estimates in fitting the parameters of distributional models from small samples. MLE estimates are unbiased, meaning that with n independent samples of the same data, the expected values of the estimated parameters for each sample would converge to the true parameter values.

The first measure is the standard (type I) Pareto distribution with a single parameter β where the Zipfian is the special case of β =2:



Pβ(X ≥ x) = x-β (1)

Fit to this distribution captures the extent to which the lines are straight in Figure 3.

Fit to a second function, a type II generalized Pareto distribution with a cut-off σ (Arnold 1983), captures the extent to which a given log-log distribution is curved. This function allows recognition that Zipf’s and power laws for city sizes almost always have one or more cut-offs of a lower size below which the power-law changes or ceases to apply.6 This function fits to the urban distribution a shape parameter Ө, analogous to β, and a scale parameter σ.

PӨ,σ (X ≥ x) = (1 + x/σ)-Ө (2)

We were fortunate to have Cosma Shalizi (2007) write an R program for our use of maximal likelihood estimation (MLE) methods (Arnold 1983) for estimating parameters Ө,σ in equation (2).7 He also provided us with an R program for standard MLE with the Pareto distribution. The type II Pareto has an extra σ parameter, but given unbiased ML estimation of the Ө,σ parameters, there are advantages of the extra cut-off parameter σ that specifies where the curve breaks for a power-law or Zipfian tail:8



  1. It can fit an exponential function or a collapsed tail in cases where a power-law or Zipfian tail is inapplicable.

  2. Equation (1) and its parameters Ө,σ are transformations in 1-to-1 functional correspondence with the q-exponential Yq(S x) = Y0 (1-(1-q)x/κ)1/(1-q) model (see Tsallis 1988) and its shape (q) and scale (κ, or kappa) parameters. Here, σ = κ/(q-1) and Ө=1/(q-1). The Yq model has been previously used (Malacarne, Mendes, and Lenzi 2001) for city-size distribution scaling.

  3. With MLE the size of the sample of largest cities does not bias the parameter estimates.

  4. Bootstrap estimates of the standard error and confidence limits of the q,κ parameters are provided by Shalizi’s (2007) R program for MLE. The standard errors are typically very small for q and the true values of the parameters are likely to be within these limits.

  5. MLE runs can be batched for multiple datasets, and the MLE commands are easy to copy and paste into R, e.g.,

china.900 <- c(500,150,90,81,75,75,70,65,60,58,49,47,40,40)

china.900.tsal.fit <- tsal.fit(china.900,xmin=40) # Assigns the results of the fit to the object

china.900.tsal.fit # Displays the estimated parameters and information about the fit

(these estimates run in seconds)

china.900.tsal.errors <- tsal.bootstrap.errors(china.900.tsal.fit,reps=100)

china.900.tsal.errors # Displays the bootstrapped error estimates

(the latter estimates run in minutes)
HYPOTHESES

Several linked hypotheses build on one another, each supposing the previous hypotheses to be supported, and each adding greater specificity in relation to the parameters of the two models, β for the Pareto, and q, κ, and Y(0) for the q exponential or Type II Pareto:

H1. The q-exponential model provides appropriate measures to fit city size distributions: q for slope in the tail, σ for crossover to a more exponential body, and the total urban population asymptote, Y(0).

H2. Variations in q are deep indicators of rise and fall of urban systems, analogous to those of a β Pareto (power-law) coefficient for cities.

H3. Evaluated in the context of population peaks and troughs and SPI fluctuation, an optimal or normal state of q for city distributions will reflect a body of the distribution that is consistent with a Zipfian tail of β =2=1/(q-1) where, solving for q, q =1.5. This should be the average value of q for the historical period.

H4. Variations in q, as a measure of rise and fall of urban systems, are strongly affected by population and sociopolitical instabilities in ways that reflect deviations up or down from an average q for a long historical period.

H5. Variations the length of oscillatory times for processes involving the successively larger spatial scale of local economies, innovative economic regions, and contention for leading polities (Modelski and Thompson 1996), the population/instability cycles of structural demographic secular cycles, and the cycles of empires (Turchin 2003, 2005, 2006) and global wars, and, finally, city systems (as studied here) are also successively longer in duration.

H6. These embedded or “stacked” spatiotemporal scales tend to occur in political economies according to a rough algorithm of doubling, diagrammed in Figure 4. The doubling is hypothesized to result from interaction between an internal endogenous dynamic and an external dynamic of pairing. In the top diagram below shows a possible explanation for a doubling of cycles between population rise and fall (smaller scale, faster cycle) and trade network rise and fall (larger scale, slower cycle. These couple through (a) an internal endogenous dynamic, the structural demography/secular cycle historical dynamic of Turchin (2003, 2005) where population rise and fall interact endogenously through scarcity and plentitude with sociopolitical instability (internal competition) and (b) a competitive dynamic between neighboring units in trading networks, where periods of scarcity in one region might be offset by economically advantaged neighbors with different sets of resources. Here the combination of a disintegrative secular phase and a contracted network regionally constrained by polity conflicts at the borders of region seems to be sufficient to weaken the border-polity conflicts, and clear the way for more globalized trade patterns to reemerge after being long suppressed in the second of the internal cycles. The second diagram repeats this logic for the endogenous dynamic of polities at a smaller scale and competing polities (populations) at a larger scale. Here within the shorter half-cycle in the secular oscillation (a shorter loop than the first diagram) is a time at which the leading polity is weakened in its growth cycle by social conflict, allowing a competing polity to expand and challenge the renewal of the first polity for leadership. A third embedded loop might involve the temporally variable polity expansion cycle in a population growth phase that might couple to leading economic sector growth for a leading polity. The key processes here are those that involve a weakness to emerge in the internal cycle that facilitates strengthening of the external cycle in a series of nested loops that draw on one another fractally by halvings and doublings.


Lag times in 2:1 Phasing for Secular (ca. 220 years) and Trade Cycles (ca. 440 years)
Lag times in 2:1 Phasing for Leadership (ca. 110 years) / Secular Cycles (ca. 220 years)
replacement
Figure 4: Internal/external completive dynamics exemplified at different spatiotemporal scales
SCALING RESULTS

Using visual and statistical evidence for changes in the shape parameter, White, Kejžar, Tsallis and Rozenblat (2005) were able in an earlier study to date six Q-periods in Eurasia over the last millennial period. These changes and periods were seen to be related to the framework for studying globalization developed by Modelski and Thompson (1996). In studying multiple regions we take a more detailed and dynamical view of this relationship.



Figure 6 shows the q and β slope parameters fitted by MLE for the regions of China, Europe and the region between (Mid-Asia, from the Middle East to India),. Here, a Zipfian tail would have q=1.5 and β=2. The horizontal line shows that this slope and shape is approximated more recently in the early modern and modern period. We also show a normalized minimum of q and β in which we divide q by 1.5 and β by 2.0 to normalize for the Zipfian.


China

Europe

Mid-Asia


Figure 5: Values of q, β, and their normalized minimum
Table 1: Descriptive Statistics




N

Minimum

Maximum

Mean

Std. Deviation

Std.Dev/Mean

MLEqChinaExtrap

25

.56

1.81

1.5120

.25475

.16849

MLEEuropeExtrap

23

1.02

1.89

1.4637

.19358

.13225

MLEMidAsIndia

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

MinQ_BetaChina

25

.37

1.16

.9645

.16247

.16845

MinQ_BetaEurope

23

.68

1.26

.9049

.15178

.16773

MinQ_BetaMidAsia

25

.54

1.01

.8252

.13217

.16017

Valid N (listwise)

19










Mean values for q in the three regions vary around q=1.5±.07, consistent with a Zipfian tail, and similarly for variations around β=2 ±.07 for the top 10 cities Pareto slope. Statistical runs tests of whether the variations around the means are random or patterned into larger temporal periods are shown in Table 2 and 3. The runs tests reject the null hypothesis (p<.01 for Europe, p<.05 for Mid-Asia, and p<.06 for China; p<.00003 overall).



Table 2: 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


Table 3: 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
The time periods of successive values above and below the medians represent the rise and fall of q to Zipfian or steeper-than-Zipfian slopes alternating with low-q periods with truncated tails of the distributions. Relatively long city-slump periods occur in the medieval period for all three regions, a second slump occurs in Europe in 1450-1500, another in Mid-Asia in 1800-1850, and one in China in 1925 (not shown) when to q falls to 1.02. As shown by the lower dotted line in Figure 6, values of q below 1.3 might be considered as approach a city system crash or destruction of primate cities. China and Europe experience an abnormal rise in q in 1900 beyond 1.7 (upper dotted line). This results in a thin tailed distribution (extreme primate cities) that might be considered as a different kind of city system crisis. Some crashes have to do with wars, like the Song loss of their capital to the Jin in 1127. Global wars noted by stars on the lines in Figure 6 might have to do with the punctuations of these periods, but we are unable to evaluate that question statistically. Crashes in q usually occur at long intervals (as in bold) , as in Figure 5, with β falling at shorter intervals. The dates below roughly correlate with secular cycles (Turchin 2007) and Modelski-Thompson (1996) globalization processes:

Mid-Asia: 1100, 1450 1825-75, 1914 (major/minor urban crashes)

China: 1150-1250, 1650 1925 (major/minor urban crashes)

Europe: 1250, 1450-1500, 1950? (major/minor crashes)

Cycle 1 2 3 4 5 (Modelski-Thompson 1996: Table 8.3)

Break 950 1150 1430 1640 1850





Figure 6: Fitted q and β 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

PART IV: CROSS-CORRELATION OF THE SCALING MEASURES p. 15

One of the major patterns of variability in city distributions is the primate city effect: the primate and top ranked cities often form a steeper urban hierarchy in periods of economic boom or empire, or when the primate city is a major international trade center. In periods of decline they may form a truncated tail compared to the body of the distribution. Further, the slope of the tail of the size distribution (β) tends to change faster than the shape body of the distributions. This is tested using data from all three regions using the autocorrelation function (AFC) , in Figure _, where values of a variable in one time period are correlated its values at time lags 1-16 (each lag here is 50 years). The AFC of β compared to q shows a much higher short term continuity (1 lag of 50 years), a recovery period at 5-6 lags, and then autocorrelation largely disappears, while q varies more continuously with more stable long term autocorrelations (up to 16 lags or 800 years). The upper and lower confidence limits are at 95% for a two-tailed significance test (p<.05).

q β

Figure _: Autocorrelation functions of q (long, smooth) and β (short), all regions.

In comparing β and q we know they relate theoretically in that if every distribution fitted q perfectly, then the relation between them at the asymptotic tail should be β ≡ 1/(1-q) , as shown in the diagonal line in Figure 7. This is not the case empirically, however, because the β for the tail varies independent from the q for the body. The plot of their joint distribution shown in Figure 7, where the observations come from all three regions, shows a regression line for actual β and q that is orthogonal to their relation under the assumption that they fit a common curve with an asymptote of the curve follows a power law where β ≡ 1/(1-q). The regression analysis for variable 1/β with q for the historical data gives = 1/(1-.3q), with R2=.31 (the regression line is the same for q>1.22 and q >1, where β ≡ 1/(1-q) does not apply). This orthogonality shows the need for both measures, one for power-law tendencies in the top 10 cities that are quite independent of the q values that are consistent with a different power-law asymptote.



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Collapse of tail Steeper Pareto tail

q<β Primate city slump
Regression line Theoretical line

β=1/(q-1)

Shallow Pareto tail



q>β Primate city effect


Figure 7: Empirical space of estimated q and β values
The dispersion of q and β values within Figure 7, as shown for Figures 8 and 10, is much less for China than in Europe (Figure 8) and Mid-Asia (Figure 10). The urban system of China as a whole, except for the outlier of the crash of 1150, is more stable. The cross-correlations in Figure 9, where each lag represents 50 years, as in the cross-correlation graphs to follow, show for China and Europe that q and β are positively correlated at one point in time, the lag 0 correlation.


Figure 8: MLE-q and Pareto β for China and Europe
The question addressed by Figure 9 is whether, in time-lagged correlation: Does the shape (q) of the body of the city affects the tail (β) in subsequent periods, or the reverse? We have in mind in posing this question that β might shape q if it is long distance trade has an effect on the larger cities engaged in international trade, but q might shape β if it is the waxing and waning of industries in the smaller cities that feed into the export products for the larger cities. Figure 9 shows that as a departure from the initial high correlation at lag 0 for China and Europe (but not replicated in Mid-Asia), high q (e.g., over 1.5) drives Pareto β down over time (often under 2, reducing the slope of the power law tail below that of the Zipfian), which suggests that high q produces an urban system decline in β. This would contradict an hypothesis of long-distance trade as a driver of rise and fall in the larger cities. It would not contradict, however, the possibility that long-distance trade was directly beneficial to the smaller cities, with these effects feeding into the success of the larger cities with a time lag. For China and Europe, where successful long-distance trade was organized on the basis of the diffusion of effective credit mechanisms available to the smaller merchant cities, this seems a plausible explanation for the time-lag findings. The credit mechanisms that made effective long-distance trading engagements possible in these regions, however, were not so easily available in Mid-Asia where Islam operated to regulate interest rates to prevent excessive usury.

If there is a correlation between long-distance trade and the rise and fall of population pressure in the secular cycles of agrarian empires, our data might support Turchin’s (2007) argument, formulated partly in response to our own studies of the role of trade networks in civilizational dynamics, that it is during the high-pressure (stagflation) period that long-distance trade flourishes. If so, the impact of trade should be reflected first in variations in q, which vary more slowly than β.





Figure 9: MLE-q drives Pareto β for China and Europe
Variables q and β do not cross-correlate for Mid-Asia, but detailed examination of the time-lags in Figure 10 shows a weak cyclical dynamic of Hi-q→Lo-β →Lo-q→Hi-β that holds to 1950.


Figure 10: MLE-q interacts dynamically with Pareto β for Mid-Asia, partly reflecting the patterns of Figure 9
The overall pattern for the three regions is shown in the cross-correlations in Figure 11, supporting that of Figure 9. The strong correlation between q and β at lag 1 (p<.000001) synchronically gives way to falling values of β over time in the 50-year lags 1-3 in the figure.



Figure 11: MLE-q drives Pareto β overall for Eurasian



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