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Human Survival and Complex Network Dynamics at Continental Scales

Complex Dynamics of Distributional Size, Networks, and Spatial Scales

Douglas R. WhiteUC Irvine

50 minutes+ discussion, 54 slidesUC Campus Complexity Videoconference April 20, 2007

This pdf available athttp://eclectic.ss.uci.edu/~drwhite/center/cac.html#White4

http://tinyurl.com/279ejp

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Abstract. Two projects on macrosocial systems are examined: food collecting societies varying through time and space in harshness of environmental conditions; and regional city systems networked by trade and war with coordination problems and interregional competition. Both involve complex network dynamics that support human survival. In each case, severe external conditions, competition, and instability often cause create system crashes. A first question in each case concerns the instability of complex systems and the role of networks in resilient outcomes and recovery from system crashes. A second question concerns system dynamics of rise and fall, and problems of coordination. My focus is on how, through network and population dynamics, different but related forms of resilience develop to solve (and hopefully solve in the future) problems of human survival. These problems are particularly acute since we now face potentially catastrophic consequences of global warming; population collapse in many parts of the world; and a climate of fear and preemptive panic in the grab for resources that are more and more limited relative to urban system demands and pressures to solve resource pressures through warfare rather than adaptive innovation.

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Problems addressed:

Two projects on macrosocial systems are examined. Both involve complex network dynamics that support human survival:

(1) food collecting societies varying through time and space in harshness of environmental conditions;

(2) regional city systems networked by trade and war with coordination problems and interregional competition.

In each case, severe external conditions, competition, and instability create system crashes.

The first question in each case concerns the instability of complex systems and the role of networks in resilient outcomes and recovery from system crashes.

The second question concerns system dynamics of rise and fall, and problems of coordination.

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My focus is on how, through network and population dynamics, different but related forms of resilience develop to solve (and hopefully solve in the future) problems of human survival.

These problems are particularly acute since we now face: Potentially catastrophic consequences of global warming;

Population collapse in many parts of the world;

A climate of fear and preemptive panic in the grab for resources that are more and more limited relative to urban system demands and pressures to solve resource pressures through warfare rather than adaptive innovation.

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forager systems

Foragers radiate to fill habitats and operate relatively autonomously at low densities and relatively small language groups, e.g., with populations <500 in each language group.

Problem: how as “isolated small populations” they can survive stochastic demographic variations, consequences of genetic drift and inbreeding depression?

city systems

Cities begin as networks of productive complementarity networked by trade and competition needed for their survival.

They extend trade networks and colonize new regions, forming (1) greater %ages of the worlds population at (2) accelerating growth rates leading to (3) system crashes and (4) problems of resilience.

(to elucidate aspects of complex network dynamics)Different but parallel problems

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Data - Generally accepted views that are contradictory

• Birdsell, from ethnographic data, estimates that the populations of Indigenous Australian language groups were consistently small, averaging perhaps 500 people each.

• Marriage rules are taken by ethnographers to imply endogamous marriage as both a norm and a logical requirement.

• Social dynamics, by these paradoxical assumptions, would seem to be the opposite of urban systems.

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• Dousset (2005:91) "rapid diffusion [of section terms] was most probably linked to the strong networks that local groups in the Western Desert maintained among themselves, with frequent ceremonial exchanges and intermariages"

• yet fairly consistent ethnographer reports of strict endogamy

The Australian Paradox 1

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• Paleodemographers argue that small reproductively closed human populations are doomed due to stochastic variations in birth rates and sex ratios, and inbreeding depression.

• If both the population estimates and the models

are right, how did these small closed societies avoid extinction and indeed persist in Australia for 40,000 years and more?

The Australian Paradox 2

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In the 19th century, anecdotal discussions of intergroup marriages may have over-reported them because of a naïve assumption that “group marriage” was ubiquitous and intergroup marriage was seen as its logical extension.

But the 20th century, in reaction, saw dismissive assumptions that intergroup marriages were byproducts of colonization and concomitant detribalization rather than of longstanding traditions. Hence authentic cases of intergroup marriage probably were under-reported.

Much potentially valuable data on intergroup marriages is thus defective or missing, biased by different assumptions at different times, with the end result being a paucity of modern understanding of intergroup marriages among Indigenous Australians. — D. White and W.W.Denham 2007. Dousset (2005) upsets this whole applecart.

BIAS IN PERSPECTIVES ON INTERGROUP MARRIAGES

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Australian forager systems

Coast, desert, rivers 12000 BCE-2000 CE

Paradox of extinction risk in closed systems (Bocquet-Appel and Masset 1982). Crises of underpopulation.

Question: was there a dynamic open system interacting with stressors?

Eurasian city systems

Mid-asia, China, Europe as regions; 900-2000 CE

Paradox of power-law growth singularity (Kremer 1993). Crises of overpopulation v. resources

Problem: open system competition interacts with over-growth & collapse.

Contrast of city system with forager dynamics

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If you are live streaming this talk this slide will show again in 1 minute: note down one of these urls and you can download this pdf for cleaner images

http://eclectic.ss.uci.edu/~drwhite/center/cac.html#White4

http://tinyurl.com/279ejp

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power-law city growth and world population 'response'

Start Year t k C Up to (following period) Length

-5000 or earlier n.a. exponential Classical Antiquity n.a.

-200 0.26 36000 Medieval Renaissance c.7000

1250 0.175 19000 Industrial Revolution c.1450

1750-1860 0.09 11000 Consumer Economy c.610

Post-1962 ? (log time length linear decr.) c.100?

Kremer data; Fitted Coefficients of Equation Nt = C/(t0

– t)k

1250

Village City

%urbanUnsustainable singularities at t0

Billions of people

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Start Year k C Up to (following period) Length

-5000 or earlier n.a. exponential Classical Antiquity n.a.

-200 0.26 36000 Medieval Renaissance c.7000

1250 0.175 19000 Industrial Revolution c.1450

1750-1860 0.09 11000 Consumer Economy c.610

Post-1962 ? (log time length linear decr.) c.100?

Kremer data; Fitted Coefficients of Equation Nt = C/(t0

– t)k

Artificial power-law pop growth curves

to match actual world

population 10,000 BCE —

2020 CE0

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If live streaming: note down one of these urls and you can download this

pdf for cleaner images

http://eclectic.ss.uci.edu/~drwhite/center/cac.html#White4

http://tinyurl.com/279ejp

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Australian forager systems

Is each language group really a closed system? As derived from observations that within these societies everyone is considered a relative, all marriages are considered as between relatives, and that it is only kinship terminology that is extended beyond a group’s boundaries? (for Kariera see Romney and Epling 1958:68, “Kariera is a closed system” ??)

Eurasian city systems

In each regional population, and that of the world, city networks as open systems develop innovations, trade and extract resources that attract and maintain more people at accelerating growth rates that eventually outstrip resources to support them: (will come back to this)

Crisis of growth (Kremer 1993) Singularity paradox of power-law growth

Supposed contrast: Open vs. Closed?

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The Australian ethnographic confusion is about marriage in terms of

(1).. section categories which are not descent groups with common ancestors but distinctions embodied in kin terms involving extended statuses (adjacent generations and even/odd implicit marriage moieties).

(2).. rules that are more specific as to what classificatory or proper kin one should marry within a given section.

(3).. negotiations as to which specific choices are made among classificatory or proper kin in local groups.

But ethnographers and modelers have incorrectly assumed that the logical closure of (1) entails local closure of (3) to marriage with relatives of the same language group!!! (the abstract logic of (1) entails no such claim!)

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In the 19th century, anecdotal discussions of intergroup marriages may have over-reported them because of a naïve assumption that “group marriage” was ubiquitous and intergroup marriage was its logical extension.

But the 20th century, in reaction, saw dismissive assumptions that intergroup marriages were byproducts of colonization and concomitant detribalization rather than of longstanding traditions. Hence authentic cases of intergroup marriage probably were under-reported.

Much potentially valuable data on intergroup marriages is thus defective or missing, biased by different assumptions at different times, with the end result being a paucity of modern understanding of intergroup marriages among Indigenous Australians. —D. White & W.W.Denham 2007

BIAS IN PERSPECTIVES ON INTERGROUP MARRIAGES

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One of the few actual network studies (Denham and White 2005; data from Denham 1971) showed

intergroup marriages firmly integrated into consistent marriage patterns, 98% of 114 local marriages consistent

with section memberships

ARANDA ARANDA-ALYAWARRA ALYAWARRA

slanted marriage tie arrows ≡ age-skewed generations, a possible indicator of reproductive stress on availability of mates

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Our Hypothesis: Reproductive Stress intergroup formation of marriage networks

• Forager paleodemographics may be steady state at the continental level, but show temporal variations in response to changing reproductive stress levels locally, pressing outward to find mates in adjacent societies.

• These reproductive stresses, and the mechanisms for responding to them, constitute the motive force that “powers” our model, entailing as a consequence that:

• Each language group will form a local cluster within a connected “small world” in a large space of continent-wide connections.

• If so, then the universality of section systems in Australia would be explained by selective pressure on institutional mechanisms – shared inter-cultural understandings – that would allow local societies to integrate, not unlike city networks!

(common factors in complex network dynamics)

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Much Diffusional Evidence for Reproductively Open Systems

Dousset (2005): Western Desert diffusion of section names implies: exogamous alignments

Trade + travel networks, dreaming tracks, ritual interactions at boundaries

Ubiquitous generation and descent moieties underlie region-wide relational system: source, direction, onset, rate, process of alignment, outcomes

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The answer to the paradox is thus: what appeared to ethnographers as “closed” section systems with local marriage rules are a means of broader institutional integration creating a continental “small world” of intermarriages and exchange

Diffusion of sections (and intermarriage)

The new data here is on the Western Desert, from Dousset 2005:87

Trade Routes as they relate to

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Dreaming Tracks as they relate to Diffusion of sections (and intermarriage)

… the new data from Dousset 2005:88, etcetera

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A Counter-Intuitive Hypothesis as to why sections facilitate integration• Widespread restrictions on marriages may

reduce choices locally, but facilitate integration of populations globally by forcing people to marry outside their own local group. Local restrictions encourage the dispersion of marriages. When mates are scarce this may help to promote links outside the language group.

• And … sections as a logic of extension of recognized social ties, are abstract (for cities, much like universal civil rights). If you marry an outsider, they and their relatives become classificatory kin.

(This deals with factors of cohesion in complex network dynamics)

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Australian forager systems

Virtually all Australian marriage models assumed these societies were endogamous (H. White, Bush, Kemeny, Hammel), except perhaps for Levi-Strauss (1971:219-220).

Our model and evidence, along with Dousset (2005), shows open marriage networks.

Key to resilience: actual open system, dynamic interaction, with stressors.

Eurasian city systems

Gibrat: Can city growth be stable independent of size?

That would contradict the actual demographic growth curves in each region and the world. It might be true on average but what about dynamics?

Key to resilience? Actual open system, dynamic interaction, with stressors?

(we begin now to see parallels in complex network dynamics)

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World cities and thus world population tend to grow proportional to their size over long periods.

This abnormal growth is sustained by migration into cities and, eventually, increases in longevity, adding those urban populations that are rapidly expanding before demographic transitions occur.

Obviously populations cannot and do not continue to grow proportional to their size, which is power-law growth.

Does expected population growth overall follow Gibrat, while the micro-dynamics of growth is volatile and unstable?

Eurasia, Cities, and World Population

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Michael Batty (Nature, Dec 2006:592), using some of the same data as do we for historical cities (Chandler 1987), cites the case made here and in our 2005 article 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.”

city systems in the last millennium

28city systems in the last millennium

Color key: Red to Blue in time of the rank clock are early to late city entries: Rank 1 is at the center. Low rank Blues, latest in time, rise quick to the top, displacing older yellow cities and ancient red.

The world system & Eurasia are the most volatile. Big shifts occur 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

RANK CLOCK Slide from Michael Batty, Nature 2006

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To see how city system rise and fall plays out regionally we fit log-log slopes for the cumulative city size distribution tail

measured by β and a curve-shape q=1+1/Ө to the body

Maximum likelihood estimation (MLE) procedures provided by Cosma Shalizi

Pβ (X ≥ x) = (x/xmin)-β

(top ten cities)

PӨ,σ (X ≥ x) = (1 + x/σ)-Ө

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

city systems in the last millennium

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Random walk or Historical Periods? Runs Tests at medians for q and β for Eurasia

Runs Test Results

MLE-q Beta10 Min(q/1.5,

Beta/2) Test Value (Median) 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_q Europe mle_q MidAsia mle_q China

Test Value (Median) 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

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Mean q~1.5>1~(No) Gibrat stationarity

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

β tail

q body

Mean β<2~Zipfian tails

33city systems in the last millennium

Are there synchronies with Turchin et al Historical Dynamics? i.e., Goldstone’s Structural Demography?

I.E., 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)

An example from Turchin (2005) will illustrate so as to relate to the city system shape dynamics.

34Chinese phase diagram

Turchin 2005:

Dynamical Feedbacks in

Structural Demography

Key:

Innovation

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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)

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Fitted q parameters for Europe, Mid-Asia, China, 900-1970 CE, 50 year lags. Vertical lines show approximate breaks between Turchin’s secular cycles for China and Europe;

Arrows: Crises of the 14th, 17th, and 20th Centuries

qqq

37city systems in the last millennium

Are there inter-region synchronies? Cross-correlations at successive time lags:

lag 0 = synchronic correlation

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

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76543210-1-2-3-4-5-6-7

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-Asia q on China

(1=50 year lagged effect)

city systems in the last millennium

3976543210-1-2-3-4-5-6-7

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Lower Confidence Limit

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Coefficient

(100 year lagged effect)

(non-MLE result for q)

Max.Likelihood: China q predicts Europe q with a 100 year time lag

city systems in the last millennium

40city systems in the last millennium

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logSilkRoad with EurBeta10

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Coefficient

Time-lagged cross-correlation effects of the Silk Road trade on Europe (50 year lagged effect)

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mle_Europe with ParisPercent

Lower Confidence Limit

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Coefficient

Europe q (max. likelihood) with % French Population in Paris

42city systems in the last millennium

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

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Fitted q parameters for Europe, Mid-Asia, China, 900-1970 CE, 50 year lags. Vertical lines show approximate breaks between Turchin’s secular cycles for China and Europe;

Arrows: Crises of the 14th, 17th, and 20th Centuries

WHAT DO THESE OSCILLATIONS HAVE TO DO WITH NETWORKS OF TRADE AND WAR?

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Having developed network analytic predictions for growth and decline in city systems and individual cities in Medieval Europe, Laurent Tambayong and I are now looking for patterns in Eurasia, region by region. The following slides represent data for which we are trying to connect, through mathematical models, the changes in inter-city networks & those of power-law tail (β) and q oscillations in the city size distributions.

In consultation with Michael Batty (1976:593; Theil), we can calculate expected growth rate for each region, decomposed into overall growth and change at different spatial scales (information distance), enabling different systems (and types) to be integrated through spatial and information hierarchies.

The project is open to collaboration. One element we need is someone to write a PERL script or robot search to systematically collect, formatted and dated information about the world’s active trade links, commodities and volumes at different historical periods.

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0900 AD

Zipfian q with global hubs CORRELATES with global network links

From first stirrings of globalization to the 21st Century

Europe Central Asia China

Medit. Near East

India

Bagdad & Changan (Xi’an)

Silk routes

Europe MidAsia China

These slides connect the city network & city size distributions to changes in power-law tails and q-scaling of city sizes

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1000 AD

960: Song capital at Kaifeng, invention of national markets, credit mechanisms diffuse

Silk routes

N~3

Europe MidAsia ChinaZipfian q with global hubs CORRELATES with global network links

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1100 AD

Silk Routes

Europe MidAsia ChinaIndic cities fall with Ghorid/Seljuk invasions; China q remains Zipfian, European Zipf with silk routes

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1150 AD

Song China loses Kaifeng; Seljuks move into Turkey to consolidate silk route links

1127: No. Song capital of Kaifeng conquered, Song move to south, capital at Hangchow

Silk Routes diminish

Europe MidAsia China

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1200 AD

Song capital at Hangchow

Golden Horde silk routes

Silk Routes diminish

Europe MidAsia ChinaZipfian q with global hubs CORRELATES with global network links, silk route integration

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1250 AD

Ghengis Khan breaks network links with conquests, severs Europeans trade (q falls)

cutnodes edgecut

Europe MidAsia China

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1300 AD

Mongol administrative takeover of China pushes q higher (Imperial capitals),

1279: Mongols conquer Song

Kublai Khan Mongol trade

Europe MidAsia China

Administered trade flourishes but punishes merchant cities

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1350 AD

Regionalization with Zipfian q (interregional connections tenuous as Yuan focus on China)

Mongols refocus on Yuan administration of China

Silk routes unimportant

Europe MidAsia China

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1400 AD

Regional markets, Zipfian q

1368 Ming retake China

Silk routes unimportant

Europe MidAsia China

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1450 AD

1421 Ming move capital to Peking

Silk routes unimportant

World population growth turns super-exponential

Europe MidAsia ChinaEast-West competition; mercantile cities in Europe flatten the urban hierarchy (lower q)

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1500 AD

Europe MidAsia ChinaIndic, Near East, China resurgent as European cities vie for dominance; 1453 fall of Constantinople

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1550 AD

Regional markets, Zipfian q, Western trade dominance of Muslim Constantinople

Europe MidAsia China

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Conclusions: Forager societies, counter intuitively and contrary to mistaken “closed world” models of 20th century ethnography, utilize network linkages to construct continental scale “small-worlds” that solved problems of reproductive survival, buffering reproductive crashes and extinctions.

Similarly, urban civilizations utilize network linkages to construct continental scale “small-worlds” that expand productive and population growth. These created problems of survival however, that have not yet been solved: population growth surpassing the capacity for distribution of resources, operating within Lotka-Volterra oscillatory dynamics with periodic city system crashes resulting from inter-regional competition and war.

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Thanks to those who contributed to these projects

• Woodrow Denham, Anthropology Department, Alice Lloyd College• 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).

city systems in the last millennium

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END – Supplementary materials continue, e.g., Eurasia from 1550

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1550 AD

Regional markets, Zipfian q, Western trade dominated by Muslim Constantinople

Europe MidAsia China

Portuguese Asian trade expansion

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1600 AD

Regional markets, Zipfian q, Western inland trade blocked by Muslim Constantinople

Europe MidAsia China

Portuguese Asian trade expansion

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1650 AD

Europe MidAsia ChinaRegional markets, Zipfian q, Western inland trade blocked by Muslim Constantinople

Dutch Indonesian trade expansion

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1700 AD

Europe MidAsia ChinaRegional markets, Zipfian q, Western trade routes to the orient

Dutch Indonesian trade expansion

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low lowhigh[Q3- high

For the next series q-periods fit with

2:1 Secular Population cycles (not shown) 3-4:1 Modelski world leadership cycles, circa 8:1 Kondratiev cycles (doublings)

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1750 AD

Bifurcated world

Europe MidAsia China

British Indic trade expansion

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1800 AD

Bifurcated world, Indic subdominant

Circum-European cities start to overtake China in number

Europe MidAsia China

British Chinese trade expansion

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1825 AD

European cities overtake China in number and size

Industrial revolution

Europe MidAsia China

British Chinese trade expansion and Industrial Revolution

Bifurcated world, Indic subdominant

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1850 AD

Europe MidAsia China

British Chinese trade expansion and European Industrial Revolution

Euro-dominant

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1875 AD

Europe MidAsia China

British Chinese trade expansion and European Industrial Revolution

Euro-dominant

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1900 AD

Europe MidAsia ChinaEuro-dominant, Japan, Russia, Calcutta rising

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1925 AD

Trifurcated - rise of Japan, Soviet Union

Europe MidAsia China

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1950 AD

linked by airlines

Europe MidAsia China

73city 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

Slides from Michael Batty, Nature 2006

74city 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

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 spatial and 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”

Slides from Michael Batty Nature 2006

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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)

76city systems in the last millennium

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 (Ө=1/(q-1).

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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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.

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Cities Abstract• 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.

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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.

– White, Douglas R.,and Woodrow W. Denham. 2007. The Indigenous Australian Marriage Paradox: Small-World Dynamics on a Continental Scale

References

city systems in the last millennium

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Note on the Institutional structure of intersocietal marriage and classificatory kinship linkages in Australia

Alternate generation moieties, named or not, are universal in Australia (Dousset p71), either in the patriline or the matriline (Brendt and Brendt 1983:29), which define the generations.

The nonexogamous (alternating) generation, however, extends not only through siblings and half-siblings, but siblings-in-law.

Parents are not necessarily of the same genealogical generation altho they are of the same classificatory generation.

Once these links are made with a single person, classificatory kinship is extended to everyone in that local or regional network.

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Level of stress Low Mod High

Stress responses

Freq of polygyny Low Mod High# marriage classes Low Mod High

Marriage to close kin High Mod LowGenetic load inbreeding High Mod LowExogamy Low Mod HighStratification Flat Shallow slope Steep slope

Symmetric marriage Yes No NoClosure/Aperture Closure Transitional Aperture

H-W age difference Low Mod HighCognitive Model

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Level of stress Low

Moderate High

Closure/Aperture

Closure Transitional Aperture

H-W age difference

Low slopeMedium slope High slope

S I G N A T U R E S

OF

STRESS