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# History & Mathematics. Trends and Cycles

Leonid Grinin, Andrey Korotayev

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Leonid Grinin, Andrey Korotayev (Dir.). History & Mathematics. Trends and Cycles. Uchitel, pp.328, 2014, Leonid Grinin, 978-5-7057-4223-3. hprints-01862988

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RUSSIAN ACADEMY OF SCIENCES
Keldysh Institute of Applied Mathematics

INSTITUTE OF ORIENTAL STUDIES
VOLGOGRAD CENTER FOR SOCIAL RESEARCH

HISTORY & MATHEMATICS

Trends and Cycles

Edited by
Leonid E. Grinin,
and Andrey V. Korotayev

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'Uchitel'
Publishing House
Volgograd
‘History & Mathematics’ Yearbook

*Editorial Council:* Herbert Barry III (Pittsburgh University), Leonid Borodkin (Moscow State University; Cliometric Society), Robert Carneiro (American Museum of Natural History), Christopher Chase-Dunn (University of California, Riverside), Dmitry Chernavsky (Russian Academy of Sciences), Thessaleno Devezas (University of Beira Interior), Leonid Grinin (National Research University Higher School of Economics), Antony Harper (New Trier College), Peter Herrmann (University College of Cork, Ireland), Andrey Korotayev (National Research University Higher School of Economics), Alexander Logunov (Russian State University for the Humanities), Gregory Malinetsky (Russian Academy of Sciences), Sergey Malkov (Russian Academy of Sciences), Charles Spencer (American Museum of Natural History), Rein Taagapera (University of California, Irvine), Arno Tausch (Innsbruck University), William Thompson (University of Indiana), Peter Turchin (University of Connecticut), Douglas White (University of California, Irvine), Yasuhide Yamanouchi (University of Tokyo).

**History & Mathematics: Trends and Cycles. Yearbook** / Edited by Leonid E. Grinin and Andrey V. Korotayev. – Volgograd: ‘Uchitel’ Publishing House, 2014. – 328 pp.

The present yearbook (which is the fourth in the series) is subtitled *Trends & Cycles*. It is devoted to cyclical and trend dynamics in society and nature; special attention is paid to economic and demographic aspects, in particular to the mathematical modeling of the Malthusian and post-Malthusian traps' dynamics.

An increasingly important role is played by new directions in historical research that study long-term dynamic processes and quantitative changes. This kind of history can hardly develop without the application of mathematical methods. There is a tendency to study history as a system of various processes, within which one can detect waves and cycles of different lengths – from a few years to several centuries, or even millennia. The contributions to this yearbook present a qualitative and quantitative analysis of global historical, political, economic and demographic processes, as well as their mathematical models.

This issue of the yearbook consists of three main sections: (I) *Long-Term Trends in Nature and Society*; (II) *Cyclical Processes in Pre-industrial Societies*; (III) *Contemporary History and Processes*.

We hope that this issue of the yearbook will be interesting and useful both for historians and mathematicians, as well as for all those dealing with various social and natural sciences.

*The present research has been carried out in the framework of the project of the National Research University Higher School of Economics.*

‘Uchitel’ Publishing House
143 Kirova St.,
400079 Volgograd,
Russia

ISBN 978-5-7057-4223-3
Volgograd 2014

© ‘Uchitel’ Publishing House, 2014
III. Contemporary History and Processes

<table><tr><td>Andrey V. Korotayev,<br>Sergey Yu. Malkov, and<br>Leonid E. Grinin</td><td>A Trap at the Escape from the Trap? Some Demographic<br>Structural Factors of Political Instability in Modernizing<br>Social Systems</td><td>201</td></tr><tr><td>Arno Tausch and<br>Almas Heshmati</td><td>Labour Migration and ‘Smart Public Health’</td><td>268</td></tr><tr><td>Anthony Howell</td><td>Is Geography ‘Dead’ or ‘Destiny’ in a Globalizing World?<br>A Network Analysis and Latent Space Modeling Approach<br>of the World Trade Network</td><td>281</td></tr><tr><td>Kent R. Crawford and<br>Nicholas W. Mitiukov</td><td>The British-Italian Performance in the Mediterranean<br>from the Artillery Perspective</td><td>300</td></tr><tr><td>Alisa R. Shishkina,<br>Leonid M. Issaev,<br>Konstantin M. Truevtsev,<br>and Andrey V. Korotayev</td><td>The Shield of Islam? Islamic Factor of HIV Prevalence in<br>Africa</td><td>314</td></tr><tr><td>Contributors</td><td></td><td>322</td></tr><tr><td>Guidelines for Contributors</td><td></td><td>328</td></tr></table>
**Introduction**

**Modeling and Measuring Cycles, Processes, and Trends**

*Leonid E. Grinin and Andrey V. Korotayev*

The present Yearbook (which is the fourth in the series) is subtitled *Trends & Cycles*. Already ancient historians (see, e.g., the second Chapter of Book VI of Polybius' *Histories*) described rather well the cyclical component of historical dynamics, whereas new interesting analyses of such dynamics also appeared in the Medieval and Early Modern periods (see, e.g., Ibn Khaldūn 1958 [1377], or Machiavelli 1996 [1531]<fnref n="1" />). This is not surprising as the cyclical dynamics was dominant in the agrarian social systems. With modernization, the trend dynamics became much more pronounced and these are trends to which the students of modern societies pay more attention. Note that the term *trend* – as regards its contents and application – is tightly connected with a formal mathematical analysis. Trends may be described by various equations – linear, exponential, power-law, etc. On the other hand, the cliodynamic research has demonstrated that the cyclical historical dynamics can be also modeled mathematically in a rather effective way (see, e.g., Usher 1989; Chu and Lee 1994; Turchin 2003, 2005a, 2005b; Turchin and Korotayev 2006; Turchin and Nefedov 2009; Nefedov 2004; Korotayev and Komarova 2004; Korotayev, Malkov, and Khaltourina 2006; Korotayev and Khaltourina 2006; Korotayev 2007; Grinin 2007), whereas the trend and cycle components of historical dynamics turn out to be of equal importance.

It is obvious that the qualitative innovative motion toward new, unknown forms, levels, and volumes, etc. cannot continue endlessly, linearly and smoothly. It always has limitations, accompanied by the emergence of imbalances, increasing resistance to environmental constraints, competition for resources, etc. These endless attempts to overcome the resistance of the environment created conditions for a more or less noticeable advance in societies. However, relatively short periods of rapid growth (which could be expressed as a linear, exponential or hyperbolic trend) tended to be followed by stagnation, different types of crises and setbacks, which created complex patterns of historical dynamics, within which trend and cyclical components were usually interwoven in rather intricate ways (see, e.g., Grinin and Korotayev 2009; Grinin, Korotayev, and Malkov 2010).

<footnote n="1">¹ For interpretations of their theories (in terms of cliodynamics, cyclical dynamics etc.) see, e.g., Turchin 2003; Korotayev and Khaltourina 2006; Grinin 2012a.</footnote>
Hence, in history we had a constant interaction of cyclical and trend dy-
namics, including some very long-term trends that are analyzed in *Section I* of
the present Yearbook which includes contributions by **Leonid E. Grinin,**
**Alexander V. Markov,** and **Andrey V. Korotayev** ('Mathematical Modeling
of Biological and Social Evolutionary Macrotrends'), **Tony Harper**
('The World System Trajectory: The Reality of Constraints and the Potential
for Prediction') and **William R. Thompson and Kentaro Sakuwa** ('Another,
Simpler Look: Was Wealth Really Determined in 8000 BCE, 1000 BCE, 0 CE,
or Even 1500 CE?').

If in a number of societies and for quite a long time we observe regular
repetition of a cycle of the same type ending with grave crises and significant
setbacks, this means that at a given level of development we confront such
rigid and strong systemic and environmental constraints which the given soci-
ety is unable to overcome.

Thus, the notion of cycle is closely related to the concept of the trap. In the
language of nonlinear dynamics the concept of traps will more or less corre-
spond to the term 'attractor'. Continuing the comparison with nonlinear dy-
namics, we should say that a steady escape from the trap will largely corre-
spond to the concept of a phase transition.

In this Yearbook particular attention is paid, of course, to the Malthusian
trap. The escape from the Malthusian trap in historical retrospect was incredi-
bly difficult (see, e.g., Korotayev *et al*. 2011; Grinin 2012b). Periodically, at-
tempts were made to get out of this trap. However, for many millennia no so-
cieties managed to achieve a final steady escape from it, but those attempts in
the long run led to a systematic increase in the level of technological develop-
ment of the World System.

The problems of the mathematical modeling of the Malthusian trap dynam-
ics are analyzed in the article by **Sergey A. Nefedov** ('Modeling Malthusian
Dynamics in Pre-Industrial Societies: Mathematical Modeling') in *Section II* of
the present issue of the Yearbook. This section also includes the article by **Ser--
gey Gavrilets, David G. Anderson,** and **Peter Turchin** ('Cycling in the Com-
plexity of Early Societies'), as well as the one by **David C. Baker** ('Demo-
graphic-Structural Theory and the Roman Dominate'). These articles deal with
various cycles in the historical dynamics of pre-Modern social systems that
are rather tightly connected with demographic macroprocesses. The first arti-
cle of the next section also deals with the problems of the escape from the
Malthusian trap.

*Section III* deals with Modern history and contemporary processes and in-
cludes the contribution by **Andrey V. Korotayev, Sergey Yu. Malkov,** and
**Leonid E. Grinin** ('A Trap at the Escape from the Trap? Some Demographic
Structural Factors of Political Instability in Modernizing Social Systems') con-
tinuing the discussion on the issues of the Malthusian and post-Malthusian
traps. This issue is also touched upon in the contributions by **Arno Tausch**
and Almas Heshmati ('Labour Migration and "Smart Public Health"'), **Anthony Howell** ('Is Geography "Dead" or "Destiny" in a Globalizing World? A Network Analysis and Latent Space Modeling Approach of the World Trade Network'), **Kent R. Crawford** and **Nicholas W. Mitiukov** ('The British-Italian Performance in the Mediterranean from the Artillery Perspective'), as well as **Alisa R. Shishkina**, **Leonid M. Issaev**, **Konstantin M. Truevtsev**, and **Andrey V. Korotayev** ('The Shield of Islam? Islamic Factor of HIV Prevalence in Africa').

Articles in this section are devoted to some rather interesting aspects and
events from the Second World War to the prospects for change of the age com-
position of the Earth's population in the coming decades. What appears valu-
able is that the contributors have managed to somehow formalize these proc-
esses, and to apply various mathematical techniques to the analysis of the re-
cent historical processes.

References

Chu C. Y. C., and Lee R. D. 1994. Famine, Revolt, and the Dynastic Cycle: Population Dynamics in Historic China. *Journal of Population Economics* 7: 351–378.

Grinin L. E. 2007. The Correlation between the Size of Society and Evolutionary Type of Polity. *History and Mathematics: The Analysis and Modeling of Socio-historical Processes* / Ed. by A. V. Korotayev, S. Yu. Malkov, and L. E. Grinin, pp. 263–303. Moscow: KomKniga/URSS. *In Russian* (Гринин Л. Е. Зависимость между размерами общества и эволюционным типом политии. *История и математика: анализ и моделирование социально-исторических процессов* / ред. А. В. Коротаев, С. Ю. Малков, Л. Е. Гринин, с. 263–303. М.: КомКнига).

Grinin L. E. 2012a. *From Confucius to Comte. Formation of the Theory, Methodology and Philosophy of History* / ed. by A. V. Korotayev. Moscow: LIBROKOM. *In Russian* (Гринин. Л. Е. *От Конфуция до Конта. Становление теории, методологии и философии истории* / отв. ред. А. В. Коротаев. М.: ЛИБРОКОМ).

Grinin L. E. 2012b. State and Socio-Political Crises in the Process of Modernization. *Cliodynamics* 3: 124–157.

Grinin L. E., and Korotayev A. V. 2009. Social Macroevolution: Growth of the World System Integrity and a System of Phase Transitions. *World Futures* 65(7): 477–506.

Grinin L., Korotayev A., and Malkov S. 2010. A Mathematical Model of Juglar Cycles and the Current Global Crisis. *History & Mathematics. Processes and Models of Global Dynamics* / Ed. by L. Grinin, P. Herrmann, A. Korotayev, and A. Tausch, pp. 138–187. Volgograd: Uchitel.

Ibn Khaldūn 'Abd al-Rahman. 1958 [1377]. *The Muqaddimah: An Introduction to History*. New York, NY: Pantheon Books (Bollingen Series, 43).

Korotayev A. 2007. Secular Cycles and Millennial Trends: A Mathematical Model. *Mathematical Modeling of Social and Economic Dynamics* / Ed. by M. G. Dmitriev, A. P. Petrov, and N. P. Tretyakov, pp. 118–125. Moscow: RUDN.
**Korotayev A., and Khaltourina D. 2006.** *Introduction to Social Macrodynamics: Secular Cycles and Millennial Trends in Africa*. Moscow: KomKniga/URSS.

**Korotayev A., and Komarova N. 2004.** A New Mathematical Model of Pre-Industrial Demographic Cycle. *Mathematical Modeling of Social and Economic Dynamics* / Ed. by M. G. Dmitriev, and A. P. Petrov, pp. 157–163. Moscow: Russian State Social University.

**Korotayev A., Malkov A., and Khaltourina D. 2006.** *Introduction to Social Macrodynamics: Secular Cycles and Millennial Trends*. Moscow: KomKniga/URSS.

**Korotayev A., Zinkina J., Kobzeva S., Bogevolnov J., Khaltourina D., Malkov A., and Malkov S. 2011.** A Trap at the Escape from the Trap? Demographic-Structural Factors of Political Instability in Modern Africa and West Asia. *Cliodynamics: The Journal of Theoretical and Mathematical History* 2(2): 276–303.

**Machiavelli N. 1996 [1531].** *Discourses on Livy*. Chicago, IL: University of Chicago Press.

**Nefedov S. A. 2004.** A Model of Demographic Cycles in Traditional Societies: The Case of Ancient China. *Social Evolution & History* 3(1): 69–80.

**Turchin P. 2003.** *Historical Dynamics: Why States Rise and Fall*. Princeton, NJ: Princeton University Press.

**Turchin P. 2005a.** Dynamical Feedbacks between Population Growth and Sociopolitical Instability in Agrarian States. *Structure and Dynamics* 1: 1–19.

**Turchin P. 2005b.** *War and Peace and War: Life Cycles of Imperial Nations*. New York: Pi Press.

**Turchin P., and Korotayev A. 2006.** Population Density and Warfare: A Reconsideration. *Social Evolution & History* 5(2): 121–158.

**Turchin P., and Nefedov S. 2009.** *Secular Cycles*. Princeton, NJ: Princeton University Press.

**Usher D. 1989.** *The Dynastic Cycle and the Stationary State*. *The American Economic Review* 79: 1031–1044.
**Mathematical Modeling of Biological and**
**Social Evolutionary Macrotrends***

Leonid E. Grinin, Alexander V. Markov,
and Andrey V. Korotayev

**Abstract**

In the first part of this article we survey general similarities and differences between biological and social macroevolution. In the second (and main) part, we consider a concrete mathematical model capable of describing important features of both biological and social macroevolution. In mathematical models of historical macrodynamics, a hyperbolic pattern of world population growth arises from non-linear, second-order positive feedback between demographic growth and technological development. Based on diverse paleontological data and an analogy with macrosociological models, we suggest that the hyperbolic character of biodiversity growth can be similarly accounted for by non-linear, second-order positive feedback between diversity growth and the complexity of community structure. We discuss how such positive feedback mechanisms can be modelled mathematically.

**Keywords:** social evolution, biological evolution, mathematical model, bio-diversity, population growth, positive feedback, hyperbolic growth.

**Introduction**

The present article represents an attempt to move further in our research on the similarities and differences between social and biological evolution (see Grinin, Markov et al. 2008, 2009a, 2009b, 2011, 2012). We have endeavored to make a systematic comparison between biological and social evolution at different levels of analysis and in various aspects. We have formulated a considerable number of general principles and rules of evolution, and worked to develop a common terminology to describe some key processes in biological and social evolution. In particular, we have introduced the notion of ‘social aromorphosis’

<footnote>* This research has been supported by the Russian Science Foundation (Project No 14-11-00634).</footnote>
to describe the process of widely diffused social innovation that enhances the complexity, adaptability, integrity, and interconnectedness of a society or social system (Grinin, Markov *et al.* 2008, 2009a, 2009b). This work has convinced us that it might be possible to find mathematical models that can describe important features of both biological and social macroevolution. In the first part of this article we survey general similarities and differences between the two types of macroevolution. In the second (and main) part, we consider a concrete mathematical model that we deem capable of describing important features of both biological and social macroevolution.

The comparison of biological and social evolution is an important but (unfortunately) understudied subject. Students of culture still vigorously debate the applicability of Darwinian evolutionary theory to social/cultural evolution. Unfortunately, the result is largely a polarization of views. On the one hand, there is a total rejection of Darwin's theory of social evolution (see, *e.g.*, Hallpike 1986). On the other hand, there are arguments that cultural evolution demonstrates all of the key characteristics of Darwinian evolution (Mesoudi *et al.* 2006).

We believe that, instead of following the outdated objectivist principle of ‘either – or’, we should concentrate on the search for methods that could allow us to apply the achievements of evolutionary biology to understanding social evolution and *vice versa*. In other words, we should search for productive generalizations and analogies for the analysis of evolutionary mechanisms in both contexts. The Universal Evolution approach aims for the inclusion of all mega-evolution within a single paradigm (discussed in Grinin, Carneiro, *et al.* 2011). Thus, this approach provides an effective means by which to address the above-mentioned task.

It is not only systems that evolve, but also mechanisms of evolution (see Grinin, Markov, and Korotayev 2008). Each sequential phase of macroevolution is accompanied by the emergence of new evolutionary mechanisms. Certain prerequisites and preadaptations can, therefore, be detected within the previous phase, and the development of new mechanisms does not invalidate the evolutionary mechanisms that were active during earlier phases. As a result, one can observe the emergence of a complex system of interaction composed of the forces and mechanisms that work together to shape the evolution of new forms.

Biological organisms operate in the framework of certain physical, chemical and geological laws. Likewise, the behaviors of social systems and people have certain biological limitations (naturally, in addition to various social-structural, historical, and infrastructural limitations). From the standpoint of Universal Evolution, new forms of evolution that determine phase transitions may result from activities going in different directions. Some forms that are similar in principle may emerge at breakthrough points, but may also result in evolutionary dead-ends. For example, social forms of life emerged among
many biological phyla and classes, including bacteria, insects, birds, and mammals. Among insects, in particular, one finds rather highly developed forms of socialization (see, e.g., Robson and Traniello 2002; Ryabko and Reznikova 2009; Reznikova 2011). Yet, despite the seemingly common trajectory and interrelation of social behaviors among these various life forms, the impacts that each have had on the Earth are remarkably different.

Further, regarding information transmission mechanisms, it appears possible to speak about certain ‘evolutionary freaks’. Some of these mechanisms were relatively widespread in the biological evolution of simple organisms, but later became less so. Consider, for example, the horizontal exchange of genetic information (genes) among microorganisms, which makes many useful genetic ‘inventions’ available in a sort of ‘commons’ for microbe communities. Among bacteria, the horizontal transmission of genes contributes to the rapid development of antibiotic resistance (e.g., Markov and Naymark 2009). By contrast, this mechanism of information transmission became obsolete or was transformed into highly specialized mechanisms (e.g., sexual reproduction) in the evolution of more complex organisms. Today, horizontal transmission is mostly confined to the simplest forms of life.

These examples suggest that an analysis of the similarities and differences between the mechanisms of biological and social evolution may help us to understand the general principles of megaevolution<fnref n="1" /> in a much fuller way. These similarities and differences may also reveal the driving forces and supra-phase mechanisms (i.e., mechanisms that operate in two or more phases) of megaevolution. One of our previous articles was devoted to the analysis of one such mechanism: *aromorphosis*, the process of widely diffused social innovation that enhances the complexity, adaptability, integrity, and interconnectedness of a society or social system (Grinin, Markov, and Korotayev 2011; see also Grinin and Korotayev 2008, 2009a, 2009b; Grinin, Markov, and Korotayev 2009a, 2009b).

It is important to carefully compare the two types of macroevolution (i.e., biological and social) at various levels and in various aspects. This is necessary because such comparisons often tend to be incomplete and deformed by conceptual extremes. These limitations are evident, for example, in the above-referenced paper by Mesoudi *et al*. (2006), which attempts to apply a Darwinian method to the study of social evolution. Unfortunately, a failure to recognize or accept important differences between biological and social evolution reduces the overall value of the method that these authors propose. Christopher Hallpike's rather thorough monograph, *Principles of Social Evolution* (1986), provides another illustration of these limitations. Here, Hallpike offers a fairly complete analysis of the similarities and differences between social and bio-

<footnote n="1">¹ We denote as *megaevolution* all the process of evolution throughout the whole of Big History, whereas we denote as *macroevolution* the process of evolution during one of its particular phases.</footnote>
logical organisms, but does not provide a clear and systematic comparison be-
tween social and biological evolution. In what follows, we hope to avoid simi-
lar pitfalls.

**Biological and Social Evolution: A Comparison at Various**
**Levels**

There are a few important differences between biological and social macroevo-
lution. Nonetheless, it is possible to identify a number of fundamental similari-
ties, including at least three basic sets of shared factors. First, we are discussing
very complex, non-equilibrium, but stable systems whose function and evolu-
tion can be described by General Systems Theory, as well as by a number of
cybernetic principles and laws. Second, we are not dealing with isolated sys-
tems, but with the complex interactions between organisms and their external
environments. As a result, the reactions of systems to ‘external’ challenges can
be described in terms of general principles that express themselves within
a biological reality and a social reality. Third (and finally), a direct ‘genetic’
link exists between the two types of macroevolution and their mutual influence.

We believe that the laws and forces driving the biological and social phas-
es of Big History can be comprehended more effectively if we apply the con-
cept of biological and social aromorphosis (Grinin, Markov, and Korotayev
2011). There are some important similarities between the evolutionary algo-
rithms of biological and social aromorphoses. Thus, it has been noticed that
the basis of biological aromorphosis

is usually formed by some partial evolutionary change that... creates sig-
nificant advantages for an organism, puts it in more favorable conditions
for reproduction, multiplies its numbers and its changeability..., thus ac-
celerating the speed of its further evolution. In those favorable condi-
tions, the total restructurization of the whole organization takes place af-
terwards (Shmal'gauzen 1969: 410; see also Severtsov 1987: 64–76).

During the course of adaptive radiation, such changes in organization dif-
fuse more or less widely (frequently with significant variations).

A similar pattern is observed within social macroevolution. An example is
the invention and diffusion of iron metallurgy. Iron production was practiced
sporadically in the 3rd millennium BCE, but regular production of low-grade
steel did not begin until the mid-2nd millennium BCE in Asia Minor (see, e.g.,
Chubarov 1991: 109). At this point, the Hittite kingdom guarded its monopoly
over the new technology. The diffusion of iron technology led to revolutionary
changes in different spheres of life, including a significant advancement in
plough agriculture and, consequently, in the agrarian system as a whole (Grinin
and Korotayev 2006); an intensive development of crafts; an increase in urban-
ism; the formation of new types of militaries, armed with relatively cheap but
effective iron weapons; and the emergence of significantly more developed
systems of taxation, as well as information collection and processing systems,
that were necessary to support these armies (e.g., Grinin and Korotayev 2007a,
2007b). Ironically, by introducing cheaply made weapons and other tools into
the hands of people who might resist the Hittite state, this aromorphosis not
only supported the growth of that kingdom, it also laid the groundwork for his-
torical phase shifts.

Considering such cases through the lens of aromorphosis has helped us to
detect a number of regularities and rules that appear to be common to biological
and social evolution (Grinin, Markov, and Korotayev 2011). Such rules and
regularities (e.g., payment for arogenic progress, special conditions for the
emergence of aromorphosis, etc.) are similar for both biological and social
macroevolution. It is important to emphasize, however, that similarity between
the two types of macroevolution does not imply commonality. Rather, signifi-
cant similarities are frequently accompanied by enormous differences. For ex-
ample, the genomes of chimpanzees and the humans are 98 per cent similar, yet
there are enormous intellectual and social differences between chimpanzees and
humans that arise from the apparently ‘insignificant’ variations between the
two genomes (see Markov and Naymark 2009).

Despite its aforementioned limitations, it appears reasonable to continue
the comparison between the two types of macroevolution following the analysis
offered by Hallpike (1986). Therefore, it may prove useful to revisit the perti-
nent observations of this analysis here. Table 1 summarizes the similarities and
differences that Hallpike (1986: 33–34) finds between social and biological
organisms.

While we do not entirely agree with all of his observations – for example,
the establishment of colonies could be seen as a kind of social reproduction
akin to organic reproduction – we do feel that Hallpike comes to a sound con-
clusion: that similarities between social and biological organisms are, in gen-
eral, determined by similarities in organization and structure (we would say
similarities between different types of systems). As a result, Hallpike believes
that one can use certain analogies in which institutions are similar to some or-
gans. In this way, cells may be regarded as similar to individuals, central gov-
ernment similar to the brain, and so on. Examples of this kind of thinking can
be found in the classic texts of social theory (see, e.g., Spencer 1898 and Durk-
heim 1991 [1893]), as well as in more recent work (see, e.g., Heylighen 2011).
<table><caption>**Table 1.** Similarities and differences between social and biological organisms, as described by Hallpike (1986)</caption><thead><tr><td>Similarities</td><td>Differences</td></tr></thead><tbody><tr><td>Social institutions are interrelated in a manner analogous to the organs of the body.</td><td>Individual societies do not have clear boundaries. For example, two societies may be distinct politically, but not culturally or religiously.</td></tr><tr><td>Despite changes in membership, social institutions maintain continuity, as do biological organs when individual cells are replaced.</td><td>Unlike organic cells, the individuals within a society have agency and are capable of learning from experience.</td></tr><tr><td>The social division of labor is analogous to the specialization of organic functions.</td><td>Social structure and function are far less closely related than in organic structure and function.</td></tr><tr><td>Self-maintenance and feedback processes characterize both kinds of system.</td><td>Societies do not reproduce. Cultural transmission between generations cannot be distinguished from the processes of system maintenance.</td></tr><tr><td>Adaptive responses to the physical environment characterize both kinds of system.</td><td>Societies are more mutable than organisms, displaying a capacity for metamorphosis only seen in organic phylogeny.</td></tr><tr><td>The trade, communication, and other transmission processes that characterize social systems are analogous to the processes that transmit matter, energy, and information in biological organisms.</td><td>Societies are not physical entities, rather their individual members are linked by information bonds.</td></tr></tbody></table>

When comparing biological *species* and societies, Hallpike (1986: 34) singles out the following similarities:

(1) that, like societies, species do not reproduce,

(2) that both have phylogenies reflecting change over time, and

(3) that both are made up of individuals who compete against one another.

Importantly, he also indicates the following *difference*: '[S]ocieties are organized systems, whereas species are simply collections of individual organisms' (Hallpike 1986: 34).

Hallpike tries to demonstrate that, because of the differences between biological and social organisms, the very idea of natural selection does not appear to apply to social evolution. However, we do not find his proofs very convincing on this account, although they do make sense in certain respects. Further,
his analysis is confined mainly to the level of the individual organism and the
individual society. He rarely considers interactions at the supra-organism level
(though he does, of course, discuss the evolution of species). His desire to dem-
onstrate the sterility of Darwinian theory to discussions of social evolution not-
withstanding, it seems that Hallpike involuntarily highlights the similarity be-
tween biological and social evolution. As he, himself, admits, the analogy
between the biological organism and society is quite noteworthy.

Just as he fails to discuss interactions and developments at the level of the
supra-organism in great detail, Hallpike does not take into account the point in
social evolution where new supra-societal developments emerge (up to the lev-
el of the emergence of the World System [e.g., Korotayev 2005, 2007, 2008,
2012; Grinin and Korotayev 2009b]). We contend that it is very important to
consider not only evolution at the level of a society but also at the level above
individual societies, as well as the point at which both levels are interconnected.
The supra-organism level is very important to understanding biological evolution,
though the differences between organisms and societies make the importance of
this supra-level to understanding social evolution unclear. Thus, it might be more
productive to compare societies with ecosystems rather than with organisms or
species. However, this would demand the development of special methods, as it
would be necessary to consider the society not as a social organism, but as a part
of a wider system, which includes the natural and social environment (cf., Leke-
vičius 2009, 2011).

In our own analysis, we seek to build on the observations of Hallpike
while, at the same time, providing a bit more nuance and different scales of
analysis. Viewing each as a process involving selection (natural, social, or
both), we identify the differences between social and biological evolution at the
level of the individual biological organism and individual society, as well as at
the supra-organismic and supra-societal level.

Natural and Social Selection

Biological evolution is more additive (cumulative) than substitutive. Put an-
other way: the new is added to the old. By contrast, social evolution (especially
over the two recent centuries) is more substitutive than additive: the new re-
places the old (Grinin, Markov *et al.* 2008, 2011).

Further, the mechanisms that control the emergence, fixation, and diffusion
of evolutionary breakthroughs (aromorphoses) differ between biological and
social evolution. These differences lead to long-term restructuring in the size
and complexity of social organisms. Unlike biological evolution, where some
growth of complexity is also observed, social reorganization becomes continu-
ous. In recent decades, societies that do not experience a constant and signifi-
cant evolution look inadequate and risk extinction.
In addition, the size of societies (and systems of societies) tends to grow constantly through more and more tightly integrative links (this trend has become especially salient in recent millennia), whereas the trend towards increase in the size of biological organisms in nature is rather limited and far from general. At another level of analysis, one can observe the formation of special supra-societal systems that also tend to grow in size. This is one of the results of social evolution and serves as a method of aromorphosis fixation and diffusion.

## The Individual Biological Organism and the Individual Society

It is very important to note that, although biological and social organisms are significantly (actually 'systemically') similar, they are radically different in their capacities to evolve. For example, as indicated by Hallpike (see above), societies are capable of rapid evolutionary metamorphoses that were not observed in the pre-human organic world. In biological evolution, the characteristics acquired by an individual are not inherited by its offspring; thus, they do not influence the very slow process of change.

There are critical differences in how biological and social information are transmitted during the process of evolution. Social systems are not only capable of rapid transformation, they are also able to borrow innovations and new elements from other societies. Social systems may also be transformed consciously and with a certain purpose. Such characteristics are absent in natural biological evolution.

The biological organism does not evolve by itself: evolution may only take place at a higher level (e.g., population, species, etc.). By contrast, social evolution can often be traced at the level of the individual social organism (i.e., society). Moreover, it is frequently possible to trace the evolution of particular institutions and subsystems within a social organism. In the process of social evolution the same social organism or institution may experience radical transformation more than once.

## The Supra-organic and Supra-societal Level

Given the above-mentioned differences, within the process of social evolution we observe the formation of two types of special supra-societal entity: (1) amalgamations of societies with varieties of complexity that have analogues in biological evolution, and (2) elements and systems that do not belong to any particular society and lack many analogues in biological evolution.

The first type of amalgamation is rather typical, not only in social but also in biological evolution. There is, however, a major difference between the two kinds of evolution. Any large society usually consists of a whole hierarchy of social systems. For example, a typical agrarian empire might include nuclear families, extended families, clan communities, village communities, primary districts, secondary districts, and provinces, each operating with their own rules
of interaction but at the same time interconnected. This kind of supra-societal
amalgamation can hardly be compared with a single biological organism
(though both systems can still be compared functionally, as is correctly noted
by Hallpike [1986]). Within biological evolution, amalgamations of organisms
with more than one level of organization (as found in a pack or herd) are usu-
ally very unstable and are especially unstable among highly organized animals.
Of course, analogues do exist within the communities of some social animals
(e.g., social insects, primates). Neither should we forget that scale is important:
while we might compare a society with an individual biological organism we
must also consider groups of organisms bound by cooperative relationships
(see, e.g., Boyd and Richerson 1996; Reeve and Hölldobler 2007). Such groups
are quite common among bacteria and even among viruses. These caveats
aside, it remains the case that within social evolution, one observes the emer-
gence of more and more levels: from groups of small societies to humankind as
a whole.

The multiplication of these levels rapidly produces the second kind of
amalgamation. It is clear that the level of analysis is very important for com-
parison of biological and social evolution. Which systems should be compared?
Analogues appear to be more frequent when a society (a social organism) is
compared to a biological organism or species. However, in many cases, it may
turn out to be more productive to compare societies with other levels of the
biota's systemic organization. This might entail comparisons with populations,
ecosystems and communities; with particular structural elements or blocks of
communities (e.g., with particular fragments of trophic networks or with par-
ticular symbiotic complexes); with colonies; or with groups of highly organized
animals (e.g., cetaceans, primates, and other social mammals or termites, ants,
bees and other social insects).

Thus, here we confront a rather complex and rarely studied methodological
problem: which levels of biological and social process are most congruent?
What are the levels whose comparison could produce the most interesting re-
sults? In general, it seems clear that such an approach should not be a mechani-
cal equation of ‘social organism = biological organism’ at all times and in every
situation. The comparisons should be operational and instrumental. This means
that we should choose the scale and level of social and biological phenomena,
forms, and processes that are adequate for and appropriate to our intended
comparisons.

Again, it is sometimes more appropriate to compare a society with an indi-
vidual biological organism, whereas in other cases it could well be more appro-
priate to compare the society with a community, a colony, a population, or
a species. At yet another scale, as we will see below, in some cases it appears
rather fruitful to compare the evolution of the biosphere with the evolution of
the anthroposphere.
**Mathematical Modeling of Biological and Social Macroevolution**

The authors of this article met for the first time in 2005, in the town of Dubna (near Moscow), at what seems to have been the first ever international conference dedicated specifically to Big History studies. Without advance knowledge of one another, we found ourselves in a single session. During the course of the session, we presented two different diagrams. One illustrated population dynamics in China between 700 BCE and 1851 CE, the other illustrated the dynamics of marine Phanerozoic biodiversity over the past 542 million years (Fig. 1).

<figure><img src="image_3.png" /><figcaption>**Fig. 1.** Similarity between the long-term population dynamics of China (top: millions of people, following Korotayev, Malkov, et al. 2006b: 47–88) and the dynamics of Phanerozoic marine biodiversity (bottom: number of genera, *N*, following Markov and Korotayev 2007)</figcaption></figure>
The similarity between the two diagrams was striking. This, despite the fact that they depicted the development of very different systems (human population vs. biota) at different time scales (hundreds of years vs. millions of years), and had been generated using the methods of different sciences (historical demography vs. paleontology) with different sources (demographic estimates vs. paleontological data). What could have caused similarity of developmental dynamics in very different systems?

* * *

In 1960, von Foerster et al. published a striking discovery in the journal *Science*. They showed that between 1 and 1958 CE, the world's population (N) dynamics could be described in an extremely accurate way with an astonishingly simple equation:<fnref n="2" />

$$N_t = \frac{C}{(t_0 - t)}, \qquad (\text{Eq. 1})$$

where $N_t$ is the world population at time t, and C and $t_0$ are constants, with $t_0$ corresponding to an absolute limit ('singularity' point) at which N would become infinite. Parameter $t_0$ was estimated by von Foerster and his colleagues as 2026.87, which corresponds to November 13, 2026; this made it possible for them to supply their article with a title that was a public-relations masterpiece: 'Doomsday: Friday, 13 November, A.D. 2026'.

Of course, von Foerster and his colleagues did not imply that the world population on that day could actually become infinite. The real implication was that the world population growth pattern that operated for many centuries prior to 1960 was about to end and be transformed into a radically different pattern. This prediction began to be fulfilled only a few years after the 'Doomsday' paper was published as World System growth (and world population growth in particular) began to diverge more and more from the previous blow-up regime. Now no longer hyperbolic, the world population growth pattern is closer to a logistic one (see, e.g., Korotayev, Malkov et al. 2006a; Korotayev 2009).

Fig. 2 presents the overall correlation between the curve generated by von Foerster et al.'s equation and the most detailed series of empirical estimates of world population (McEvedy and Jones 1978, for the period 1000–1950; U.S. Bureau of the Census 2013, for 1950–1970). The formal characteristics are: $R = 0.998$; $R^2 = 0.996$; $p = 9.4 \times 10^{-17} \approx 1 \times 10^{-16}$. For readers unfamiliar with mathematical statistics: $R^2$ can be regarded as a measure of the fit between

<footnote n="2">² To be exact, the equation proposed by von Foerster and his colleagues looked as follows: $$N_t = \frac{C}{(t_0 - t)^{0.99}}$$. However, as von Hoerner (1975) and Kapitza (1999) showed, it can be simplified as $$N_t = \frac{C}{t_0 - t}.$$</footnote>
the dynamics generated by a mathematical model and the empirically observed
situation, and can be interpreted as the proportion of the variation accounted for
by the respective equation. Note that 0.996 also can be expressed as 99.6 %.<sup>3</sup>
Thus, von Foerster *et al.'s* equation accounts for an astonishing 99.6 % of all
the macrovariation in world population, from 1000 CE through 1970, as esti-
mated by McEvedy and Jones (1978) and the U.S. Bureau of the Census
(2013).<sup>4</sup> Note also that the empirical estimates of world population find them-
selves aligned in an extremely neat way along the hyperbolic curve, which con-
vincingly justifies the designation of the pre-1970s world population growth
pattern as ‘hyperbolic’.

<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Correlation between empirical estimates of world population (black, in millions of people, 1000–1970) and the curve generated by von Foerster *et al.'s* equation (grey)</figcaption></figure>

<footnote n="3"><sup>3</sup> The second characteristic (*p*, standing for 'probability') is a measure of the correlation's statistical significance. A bit counter-intuitively, the lower the value of *p*, the higher the statistical significance of the respective correlation. This is because *p* indicates the probability that the observed correlation could be accounted solely by chance. Thus, *p* = 0.99 indicates an extremely low statistical significance, as it means that there are 99 chances out of 100 that the observed correlation is the result of a coincidence, and, thus, we can be quite confident that there is no systematic relationship (at least, of the kind that we study) between the two respective variables. On the other hand, $p = 1 \times 10^{-16}$ indicates an extremely high statistical significance for the correlation, as it means that there is only one chance out of 10,000,000,000,000,000 that the observed correlation is the result of pure coincidence (a correlation is usually considered statistically significant once $p < 0.05$).</footnote>

<footnote n="4"><sup>4</sup> In fact, with slightly different parameters (C = 164890.45; t<sub>0</sub> = 2014) the fit (R<sup>2</sup>) between the dynamics generated by von Foerster's equation and the macrovariation of world population for 1000–1970 CE as estimated by McEvedy and Jones (1978) and the U.S. Bureau of the Census (2013) reaches 0.9992 (99.92 %); for 500 BCE – 1970 CE this fit increases to 0.9993 (99.93 %) with the following parameters: C = 171042.78; t<sub>0</sub> = 2016.</footnote>
The von Foerster et al.'s equation, $N_t = \frac{C}{t_0 - t}$, is the solution for the following differential equation (see, e.g., Korotayev, Malkov et al. 2006a: 119–120):

$$ \frac{dN}{dt} = \frac{N^2}{C}. \qquad (\text{Eq. 2}) $$

This equation can be also written as:

$$ \frac{dN}{dt} = aN^2, \qquad (\text{Eq. 3}) $$

where $a = \frac{1}{C}$.

What is the meaning of this mathematical expression? In our context, $dN/dt$ denotes the absolute population growth rate at a certain moment in time. Hence, this equation states that the absolute population growth rate at any moment in time should be proportional to the square of world population at this moment. This significantly demystifies the problem of hyperbolic growth. To explain this hyperbolic growth, one need only explain why for many millennia the world population's absolute growth rate tended to be proportional to the square of the population.

The main mathematical models of hyperbolic growth in the world population (Taagapera 1976, 1979; Kremer 1993; Cohen 1995; Podlazov 2004; Tsirel 2004; Korotayev 2005, 2007, 2008, 2009, 2012; Korotayev, Malkov et al. 2006a: 21–36; Golosovsky 2010; Korotayev and Malkov 2012) are based on the following two assumptions:

(1) ‘the Malthusian (Malthus 1978 [1798]) assumption that population is limited by the available technology, so that the growth rate of population is proportional to the growth rate of technology’ (Kremer 1993: 681–682),<fnref n="5" /> and

(2) the idea that ‘[h]igh population spurs technological change because it increases the number of potential inventors… In a larger population there will be proportionally more people lucky or smart enough to come up with new ideas’, thus, ‘the growth rate of technology is proportional to total population’(Kremer 1993: 685).<fnref n="6" />

Here Kremer uses the main assumption of Endogenous Technological Growth theory (see, e.g., Kuznets 1960; Grossman and Helpman 1991; Aghion

<footnote n="5">⁵ In addition to this, the absolute growth rate is proportional to the population itself. With a given relative growth rate, a larger population will increase more in absolute number than a smaller one.</footnote>

<footnote n="6">⁶ Note that ‘the growth rate of technology’ here means the relative growth rate (i.e., the level to which technology will grow in a given unit of time in proportion to the level observed at the beginning of this period).</footnote>
and Howitt 1998; Simon 1977, 2000; Komlos and Nefedov 2002; Jones 1995, 2005).

The first assumption looks quite convincing. Indeed, throughout most of human history the world population was limited by the technologically determined ceiling of the carrying capacity of land. For example, with foraging subsistence technologies the Earth could not support more than 8 million people because the amount of naturally available useful biomass on this planet is limited. The world population could only grow over this limit when people started to apply various means to artificially increase the amount of available biomass that is with the transition from foraging to food production. Extensive agriculture is also limited in terms of the number of people that it can support. Thus, further growth of the world population only became possible with the intensification of agriculture and other technological improvements (see, e.g., Turchin 2003; Korotayev, Malkov et al. 2006a, 2006b; Korotayev and Khaltourina 2006). However, as is well known, the technological level is not constant, but variable (see, e.g., Grinin 2007a, 2007b, 2012), and in order to describe its dynamics the second basic assumption is employed.

As this second supposition was, to our knowledge, first proposed by Simon Kuznets (1960), we shall denote the corresponding type of dynamics as ‘Kuznetsian’. (The systems in which the Kuznetsian population-technological dynamics are combined with Malthusian demographic dynamics will be denoted as ‘Malthusian-Kuznetsian’.) In general, we find this assumption rather plausible – in fact, it is quite probable that, other things being equal, within a given period of time, five million people will make approximately five times more inventions than one million people.

This assumption was expressed mathematically by Kremer in the following way:

$$ \frac{dT}{dt} = kNT. \qquad (\text{Eq. 4}) $$

This equation simply says that the absolute technological growth rate at a given moment in time ($dT/dt$) is proportional to the technological level ($T$) observed at this moment (the wider the technological base, the higher the number of inventions that can be made on its basis). On the other hand, this growth rate is also proportional to the population ($N$): the larger the population, the larger the number of potential inventors.<fnref n="7" />

When united in one system, Malthusian and Kuznetsian equations account quite well for the hyperbolic growth of the world population observed before the early 1990s (see, e.g., Korotayev 2005, 2007, 2008, 2012; Korotayev, Malkov, et al. 2006a). The resultant models provide a rather convincing explanation

<footnote n="7">⁷ Kremer did not test this hypothesis empirically in a direct way. Note, however, that our own empirical test of this hypothesis has supported it (see Korotayev, Malkov et al. 2006b: 141–146).</footnote>
nation of *why*, throughout most of human history, the world population fol-
lowed the hyperbolic pattern with the absolute population growth rate tending
to be proportional to $N^2$. For example, why would the growth of population
from, say, 10 million to 100 million, result in the growth of $dN/dt$ 100 times?
The above mentioned models explain this rather convincingly. The point is that
the growth of world population from 10 to 100 million implies that human sub-
sistence technologies also grew approximately 10 times (given that it will have
proven, after all, to be able to support a population ten times larger). On the
other hand, the tenfold population growth also implies a tenfold growth in the
number of potential inventors, and, hence, a tenfold increase in the relative
technological growth rate. Thus, the absolute technological growth rate would
expand $10 \times 10 = 100$ times as, in accordance with Eq. 4, an order of magnitude
higher number of people having at their disposal an order of magnitude wider
technological base would tend to make two orders of magnitude more inven-
tions. If, as throughout the Malthusian epoch, the world population ($N$) tended
toward the technologically determined carrying capacity of the Earth, we have
good reason to expect that $dN/dt$ should also grow just by about 100 times.

In fact, it can be shown (see, e.g., Korotayev, Malkov et al. 2006a, 2006b; Korotayev and Khaltourina 2006) that the hyperbolic pattern of the world's population growth could be accounted for by a nonlinear second-order positive feedback mechanism that was long ago shown to generate just the hyperbolic growth, also known as the 'blow-up regime' (see, e.g., Kurdyumov 1999). In our case, this nonlinear second-order positive feedback looks as follows: more people – more potential inventors – faster technological growth – faster growth of the Earth's carrying capacity – faster population growth – more people allow for more potential inventors – faster technological growth, and so on (see Fig. 3).

<figure><img src="image_5.png" /><figcaption>**Fig. 3.** Cognitive scheme of the nonlinear second order positive feedback between technological development and demographic growth</figcaption></figure>

Note that the relationship between technological development and demographic growth cannot be analyzed through any simple cause-and-effect model, as we
observe a true dynamic relationship between these two processes – each of them is both the cause and the effect of the other.

The feedback system described here should be identified with the process of ‘collective learning’ described, principally, by Christian (2005: 146–148). The mathematical models of World System development discussed in this article can be interpreted as models of the influence that collective learning has on global social evolution (i.e., the evolution of the World System). Thus, the rather peculiar hyperbolic shape of accelerated global development prior to the early 1970s may be regarded as a product of global collective learning. We have also shown (Korotayev, Malkov et al. 2006a: 34–66) that, for the period prior to the 1970s, World System economic and demographic macrodynamics, driven by the above-mentioned positive feedback loops, can simply and accurately be described with the following model:

$$ \frac{dN}{dt} = aSN, \qquad (\text{Eq. 5}) $$

$$ \frac{dS}{dt} = bNS. \qquad (\text{Eq. 6}) $$

The world GDP (G) can be calculated using the following equation:

$$ G = mN + SN, \qquad (\text{Eq. 7}) $$

where G is the world GDP, N is population, and S is the produced surplus per capita, over the subsistence amount (m) that is minimally necessary to reproduce the population with a zero growth rate in a Malthusian system (thus, S = g – m, where g denotes per capita GDP); a and b are parameters.

The mathematical analysis of the basic model (not described here) suggests that up to the 1970s, the amount of S should be proportional, in the long run, to the World System's population: S = kN. Our statistical analysis of available empirical data has confirmed this theoretical proportionality (Korotayev, Malkov et al. 2006a: 49–50). Thus, in the right-hand side of Eq. 6, S can be replaced with kN, resulting in the following equation:

$$ \frac{dN}{dt} = kaN^2. $$

Recall that the solution of this type of differential equations is:

$$ N_t = \frac{C}{(t_0 - t)}, $$

which produces a simple hyperbolic curve.

As, according to our model, S can be approximated as kN, its long-term dynamics can be approximated with the following equation:

$$ S = \frac{kC}{t_0 - t}. \qquad (\text{Eq. 8}) $$
Thus, the long-term dynamics of the most dynamic component of the world GDP, SN, the ‘world surplus product’, can be approximated as follows:

$$SN = \frac{kC^2}{(t_0 - t)^2}. \qquad (\text{Eq. 9})$$

This suggests that the long-term world GDP dynamics up to the early 1970s must be approximated better by a quadratic hyperbola, rather than by a simple one. As shown in Fig. 4, this approximation works very effectively indeed.

<figure><img src="image_6.png" /><figcaption>**Fig. 4.** The fit between predictions of a quadratic-hyperbolic model and observed world GDP dynamics, 1-1973 CE (in billions of 1990 international dollars, PPP)</figcaption></figure>

*Note:* $R = .9993$, $R^2 = .9986$, $p \ll .0001$. The black markers correspond to Maddison's (2001) estimates (Maddison's estimates of the world per capita GDP for 1000 CE has been corrected on the basis of [Meliantsev 2004]). The grey solid line has been generated by the following equation:

$$G = \frac{17749573.1}{(2006 - t)^2}.$$

Thus, up to the 1970s the hyperbolic growth of the world population was accompanied by the quadratic-hyperbolic growth of the world GDP, as suggested by our model. Note that the hyperbolic growth of the world population and the quadratic-hyperbolic growth of the world GDP are very tightly connected processes, actually two sides of the same coin, two dimensions of one process propelled by nonlinear second-order positive feedback loops between the technological development and demographic growth (see Fig. 5).
<figure><img src="image_7.png" /><figcaption>**Fig. 5.** Cognitive scheme of the world economic growth generated by nonlinear second-order positive feedback between technological development and demographic growth</figcaption></figure>

We have also demonstrated (Korotayev, Malkov *et al.* 2006a: 67–80) that the World System population's literacy ($l$) dynamics are rather accurately described by the following differential equation:

$$ \frac{dl}{dt} = aSl (1-l), \qquad (\text{Eq. 10}) $$

where $l$ is the proportion of the population that is literate, $S$ is per capita surplus, and $a$ is a constant. In fact, this is a version of the autocatalytic model. Literacy growth is proportional to the fraction of the population that is literate, $l$ (potential teachers), to the fraction of the population that is illiterate, $(1-l)$ (potential pupils), and to the amount of per capita surplus $S$, since it can be used to support educational programs. (Additionally, $S$ reflects the technological level $T$ that implies, among other things, the level of development of educational technologies.) From a mathematical point of view, Eq. 9 can be regarded as logistic where saturation is reached at literacy level $l = 1$. $S$ is responsible for the speed with which this level is being approached.

It is important to stress that with low values of $l$ (which correspond to most of human history, with recent decades being the exception), the rate of increase in world literacy generated by this model (against the background of hyperbolic growth of $S$) can be approximated rather accurately as hyperbolic (see Fig. 6).
<figure><img src="image_8.png" /><figcaption>**Fig. 6.** The fit between predictions of the hyperbolic model and observed world literacy dynamics, 1-1980 CE (%)</figcaption></figure>

Note: R = 0.997, R² = 0.994, p << 0.0001. Black dots correspond to World Bank (2013) estimates for the period since 1970, and to Meliantsev's (2004) estimates for the earlier period. The grey solid line has been generated by the following equation:

$$l_t = \frac{3769.264}{(2040 - t)^2}$$

The best-fit values of parameters C (3769.264) and t₀ (2040) have been calculated with the least squares method.

The overall number of literate people is proportional both to the literacy level and to the overall population. As both of these variables experienced hyperbolic growth until the 1960s/1970s, one has sufficient grounds to expect that until recently the overall number of literate people in the world (L)<fnref n="8" /> was growing not just hyperbolically, but rather in a quadratic-hyperbolic way (as was world GDP). Our empirical test has confirmed this – the quadratic-hyperbolic model describes the growth of the literate population of this planet with an extremely good fit indeed (see Fig. 7).

<footnote n="8">⁸ Since literacy appeared, almost all of the Earth's literate population has lived within the World System; hence, the literate population of the Earth and the literate population of the World System have been almost perfectly synonymous.</footnote>
<figure><img src="image_9.png" /><figcaption>**Fig. 7.** The fit between predictions of the quadratic-hyperbolic model and observed world literate population dynamics, 1-1980 CE (L, millions)</figcaption></figure>

Note: R = 0.9997, R² = 0.9994, p << 0.0001. The black dots correspond to UNESCO/World Bank (2014) estimates for the period since 1970, and to Meliantsev's (2004) estimates for the earlier period; we have also taken into account the changes of age structure on the basis of UN Population Division (2014) data. The grey solid line has been generated by the following equation:

$$L_t = \frac{4958551}{(2033 - t)^2}$$

The best-fit values of parameters C (4958551) and t₀ (2033) have been calculated with the least squares method.

Similar processes are observed with respect to world urbanization, the macro-dynamics of which appear to be described by the differential equation:

$$\frac{du}{dt} = bSu(u_{\text{lim}} - u), \qquad (\text{Eq. 11})$$

where u is the proportion of the population that is urban, S is per capita surplus produced with the given level of the World System's technological development, b is a constant, and uₜₘₐₜ is the maximum possible proportion of the population that can be urban. Note that this model implies that during the Malthusian-Kuznetsian era of the blow-up regime, the hyperbolic growth of world urbanization must have been accompanied by a quadratic-hyperbolic growth of the urban population of the world, as supported by our empirical tests (see Figs 8-9).
<figure><img src="image_10.png" /><figcaption>**Fig. 8.** The fit between predictions of the hyperbolic model and empirical estimates of world megaurbanization dynamics (% of the world population living in cities with > 250,000 inhabitants), 10,000 BCE – 1960 CE</figcaption></figure>

*Note:* R = 0.987, R² = 0.974, p << 0.0001. The black dots correspond to Chandler's (1987) estimates, UN Population Division (2014), Modelski (2003), and Gruebler (2006). The grey solid line has been generated by the following equation:

$$u_t = \frac{403.012}{(1990 - t)}$$

The best-fit values of parameters C (403.012) and t₀ (1990) have been calculated with the least squares method. For comparison, the best fit (R²) obtained here for the exponential model is 0.492.
<figure><img src="image_11.png" /><figcaption>**Fig. 9.** The fit between predictions of the quadratic-hyperbolic model and the observed dynamics of world urban population living in cities with > 250,000 inhabitants (millions), 10,000 BCE – 1960 CE</figcaption></figure>

Note: R = 0.998, R² = 0.996, p << 0.0001. The black markers correspond to estimates of Chandler (1987) and UN Population Division (2014). The grey solid line has been generated by the following equation:

$$U_t = \frac{912057.9}{(2008 - t)^2}$$

The best-fit values of parameters C (912057.9) and t₀ (2008) have been calculated with the least squares method. For comparison, the best fit (R²) obtained here for the exponential model is 0.637.

Within this context it is hardly surprising to find that the general macrodynamics of largest settlements within the World System are also quadratic-hyperbolic (see Fig. 10).

As has been demonstrated by socio-cultural anthropologists working across cultures (see, e.g., Naroll and Divale 1976; Levinson and Malone 1980: 34), for pre-agrarian, agrarian, and early industrial cultures the size of the largest settlement is a rather effective indicator of the general sociocultural complexity of a social system. This, of course, suggests that the World System's general sociocultural complexity also grew, in the Malthusian-Kuznetsian era, in a generally quadratic-hyperbolic way.
<figure><img src="image_12.png" /><figcaption>**Fig. 10.** The fit between predictions of the quadratic-hyperbolic model and the observed dynamics of size of the largest settlement of the world (thousands of inhabitants), 10,000 BCE – 1950 CE</figcaption></figure>

Note: R = 0.992, R² = 0.984, p << 0.0001. The black markers correspond to estimates of Modelski (2003) and Chandler (1987). The grey solid line has been generated by the following equation:

$$U_{\max t} = \frac{104020618.573}{(2040 - t)^2}$$

The best-fit values of parameters C (104020618.5) and t₀ (2040) have been calculated with the least squares method. For comparison, the best fit (R²) obtained here for the exponential model is 0.747.

Turning to a more concrete case study, as suggested at the beginning of this section, the hyperbolic model is particularly effective for describing the long-term population dynamics of China, the country with the best-known demographic history. The Chinese population curve reflects not only a hyperbolic trend, but also cyclical and stochastic dynamics. These components of long-term population dynamics in China, as well as in other complex agrarian societies, have been discussed extensively (see, e.g., Braudel 1973; Abel 1980; Usher 1989; Goldstone 1991; Chu and Lee 1994; Komlos and Nefedov 2002; Turchin 2003, 2005a, 2005b; Nefedov 2004; Korotayev 2006; Korotayev and Khaltourina 2006; Korotayev, Malkov et al. 2006b; Turchin and Korotayev 2006; Korotayev, Komarova et al. 2007; Grinin, Korotayev et al. 2008; Grinin, Malkov et al. 2009; Turchin and Nefedov 2009; van Kessel-Hagesteijn 2009; Korotayev, Khaltourina, Malkov et al. 2010; Korotayev, Khaltourina et al. 2010; Grinin and Korotayev 2012).

As we have observed with respect to world population dynamics, even before the start of its intensive modernization, the population dynamics of China were characterized by a pronounced hyperbolic trend (Figs 11 and 12).
The hyperbolic model describes traditional Chinese population dynamics much more accurately than either linear or exponential models.

<figure><img src="image_13.png" /><figcaption>**Fig. 11.** Population dynamics of China (million people, following Koro-tayev, Malkov et al. 2006b: 47–88), 57–1851 CE. Fit with Linear and Exponential Models</figcaption></figure>

*Note*: Linear model: $R^2$ = 0.469. Exponential model: $R^2$ = 0.600.

<figure><img src="image_14.png" /><figcaption>**Fig. 12.** Fit between a hyperbolic model and observed population dynamics of China (million people), 57–1851 CE</figcaption></figure>

*Note*: $R^2$ = 0.884. The grey solid line has been generated by the following equation:

$$N_t = \frac{33431}{1915 - t}.$$
The hyperbolic model describes the population dynamics of China in an especially accurate way if we take the modern period into account (Fig. 13).

<figure><img src="image_15.png" /><figcaption>**Fig. 13.** Fit between a hyperbolic model and observed population dynamics of China (million people, following Korotayev, Malkov et al. 2006b: 47–88), 57–2003 CE</figcaption></figure>

Note: R² = 0.968. The grey solid line has been generated by the following equation:

$$N_t = \frac{63150}{2050 - t}.$$

It is curious that, as we noted above, the dynamics of marine biodiversity are strikingly similar to the population dynamics of China. The similarity probably derives from the fact that both curves are produced by the interference of the same three components (the general hyperbolic trend, as well as cyclical and stochastic dynamics). In fact, there is a lot of evidence that some aspects of biodiversity dynamics are stochastic (Raup et al. 1973; Sepkoski 1994; Markov 2001; Cornette and Lieberman 2004), while others are periodic (Raup and Sepkoski 1984; Rohde and Muller 2005). In any event, the hyperbolic model describes marine biodiversity (measured by number of genera) through the Phanerozoic much more accurately than an exponential model (Fig. 14).

When measured by number of species, the fit between the empirically observed marine biodiversity dynamics and the hyperbolic model becomes even better (Fig. 15).
<figure><img src="image_16.png" /><figcaption>**Fig. 14.** Global change in marine biodiversity (number of genera, N) through the Phanerozoic (following Markov and Korotayev 2007)</figcaption></figure>

*Note*: Exponential model: R² = 0.463. Hyperbolic model: R² = 0.854. The hyperbolic line has been generated by the following equation:

$$N_t = \frac{183320}{37 - t}.$$

<figure><img src="image_17.png" /><figcaption>**Fig. 15.** Global change in marine biodiversity (number of species, N) through the Phanerozoic (following Markov and Korotayev 2008)</figcaption></figure>

*Note*: Exponential model: R² = 0.51. Hyperbolic model: R² = 0.91. The hyperbolic line has been generated by the following equation:

$$N_t = \frac{892874}{35 - t}.$$

Likewise, the hyperbolic model describes continental biodiversity in an especially accurate way (Fig. 16).
<figure><img src="image_18.png" /><figcaption>**Fig. 16.** Global change in continental biodiversity (number of genera, N) through the Phanerozoic (following Markov and Korotayev 2008)</figcaption></figure>

*Note*: Exponential model: $R^2 = 0.86$. Hyperbolic model: $R^2 = 0.94$. The hyperbolic line has been generated by the following equation:

$$N_t = \frac{272095}{29-t}.$$

However, the best fit between the hyperbolic model and the empirical data is observed when the hyperbolic model is used to describe the dynamics of total (marine and continental) global biodiversity (Fig. 17).

<figure><img src="image_19.png" /><figcaption>**Fig. 17.** Global change in total biodiversity (number of genera, N) through the Phanerozoic (following Markov and Korotayev 2008)</figcaption></figure>

*Note*: Exponential model: $R^2 = 0.67$. Hyperbolic model: $R^2 = 0.95$. The hyperbolic line has been generated by the following equation:

$$N_t = \frac{434635}{30-t}.$$
The hyperbolic dynamics are most prominent when both marine and continental biotas are considered together. This fact can be interpreted as a proof of the integrated nature of the biosphere. But why, throughout the Phanerozoic, did global biodiversity tend to follow a hyperbolic trend similar to that which we observed for the World System in general and China in particular?

As we have noted above, in sociological models of macrohistorical dynamics, the hyperbolic pattern of world population growth arises from non-linear second-order positive feedback (more or less identical with the mechanism of collective learning) between demographic growth and technological development. Based on analogy with these sociological models and diverse paleontological data, we suggest that the hyperbolic character of biodiversity growth can be similarly accounted for by non-linear second-order positive feedback between diversity growth and the complexity of community structure: more genera – higher alpha diversity – enhanced stability and ‘buffering’ of communities – lengthening of average life span of genera, accompanied by a decrease in the extinction rate – faster diversity growth – more genera – higher alpha diversity, and so on. Indeed, this begins to appear as a (rather imperfect) analogue of the collective learning mechanism active in social macroevolution.

The growth of genus richness throughout the Phanerozoic was mainly due to an increase in the average longevity of genera and a gradual accumulation of long-lived (stable) genera in the biota. This pattern reveals itself in a decrease in the extinction rate. Interestingly, in both biota and humanity, growth was facilitated by a decrease in mortality rather than by an increase in the birth rate. The longevity of newly arising genera was growing in a stepwise manner. The most short-lived genera appeared during the Cambrian; more long-lived genera appeared in Ordovician to Permian; the next two stages correspond to the Mesozoic and Cenozoic (Markov 2001, 2002). We suggest that diversity growth can facilitate the increase in genus longevity via progressive stepwise changes in the structure of communities.

Most authors agree that three major biotic changes resulted in the fundamental reorganization of community structure during the Phanerozoic: Ordovician radiation, end-Permian extinction, and end-Cretaceous extinction (Bambach 1977; Sepkoski *et al.* 1981; Sepkoski 1988, 1992; Markov 2001; Bambach *et al.* 2002). Generally, after each major crisis, the communities became more complex, diverse, and stable. The stepwise increase of alpha diversity (i.e., the average number of species or genera in a community) through the Phanerozoic was demonstrated by Bambach (1977) and Sepkoski (1988). Although Powell and Kowalewski (2002) have argued that the observed increase in alpha diversity might be an artifact caused by several specific biases that influenced the taxonomic richness of different parts of the fossil record, there is evidence that these biases largely compensated for one another so that the observed increase
in alpha diversity was probably underestimated rather than overestimated (Bush and Bambach 2004).

Another important symptom of progressive development of communities is an increase in the evenness of species (or genus) abundance distribution. In primitive, pioneer, or suppressed communities, this distribution is strongly uneven: the community is overwhelmingly dominated by a few very abundant species. In more advanced, climax, or flourishing communities, this distribution is more even (Magurran 1988). The former type of community is generally more vulnerable. The evenness of species richness distribution in communities increased substantially during the Phanerozoic (Powell and Kowalewski 2002; Bush and Bambach 2004). It is most likely there was also an increase in habitat utilization, total biomass, and the rate of trophic flow in biota through the Phanerozoic (Powell and Kowalewski 2002).

The more complex the community, the more stable it is due to the development of effective interspecies interactions and homeostatic mechanisms based on the negative feedback principle. In a complex community, when the abundance of a species decreases, many factors arise that facilitate its recovery (e.g., food resources rebound while predator populations decline). Even if the species becomes extinct, its vacant niche may ‘recruit’ another species, most probably a related one that may acquire morphological similarity with its predecessor and thus will be assigned to the same genus by taxonomists. So a complex community can facilitate the stability (and longevity) of its components, such as niches, taxa and morphotypes. This effect reveals itself in the phenomenon of ‘coordinated stasis’. The fossil record contains many examples in which particular communities persist for million years while the rates of extinction and taxonomic turnover are minimized (Brett *et al.* 1996, 2007).

Selective extinction leads to the accumulation of ‘extinction-tolerant’ taxa in the biota (Sepkoski 1991b). Although there is evidence that mass extinctions can be nonselective in some aspects (Jablonski 2005), they are obviously highly selective with respect to the ability of taxa to endure unpredictable environmental changes. This can be seen, for instance, in the selectivity of the end-Cretaceous mass extinction with respect to the time of the first occurrence of genera. In younger cohorts, the extinction level was higher than that of the older cohorts (see Markov and Korotayev 2007: fig. 2). The same pattern can be observed during the periods of ‘background’ extinction as well. This means that genera differ in their ability to survive extinction events, and that extinction-tolerant genera accumulate in each cohort over the course of time. Thus, taxa generally become more stable and long-lived through the course of evolution, apart from the effects of communities. The communities composed of more stable taxa would be, in turn, more stable themselves, thus creating positive feedback.
The stepwise change of dominant taxa plays a major role in biotic evolution. This pattern is maintained not only by the selectivity of extinction (discussed above), but also by the selectivity of the recovery after crises (Bambach et al. 2002). The taxonomic structure of the Phanerozoic biota was changing in a stepwise way, as demonstrated by the concept of three sequential ‘evolutionary faunas’ (Sepkoski 1992). There were also stepwise changes in the proportion of major groups of animals with different ecological and physiological parameters. There was stepwise growth in the proportion of motile genera to non-motile, ‘physiologically buffered’ genera to ‘unbuffered’, and predators to prey (Bambach et al. 2002). All these trends should have facilitated the stability of communities. For example, the diversification of predators implies that they became more specialized. A specialized predator regulates its prey's abundance more effectively than a non-specialized predator.

There is also another possible mechanism of second-order positive feedback between diversity and its growth rate. Recent research has demonstrated a shift in typical relative-abundance distributions in paleocommunities after the Paleozoic (Wagner et al. 2006). One possible interpretation of this shift is that community structure and the interactions between species in the communities became more complex. In post-Paleozoic communities, new species probably increased ecospace more efficiently, either by facilitating opportunities for additional species or by niche construction (Wagner et al. 2006; Solé et al. 2002; Laland et al. 1999). This possibility makes the mechanisms underlying the hyperbolic growth of biodiversity and human population even more similar, because the total ecospace of the biota is analogous to the ‘carrying capacity of the Earth’ in demography. As far as new species can increase ecospace and facilitate opportunities for additional species entering the community, they are analogous to the ‘inventors’ of the demographic models whose inventions increase the carrying capacity of the Earth.

Exponential and logistic models of biodiversity imply several possible ways in which the rates of origination and extinction may change through time (Sepkoski 1991a). For instance, exponential growth can be derived from constant per-taxon extinction and origination rates, the latter being higher than the former. However, actual paleontological data suggest that origination and extinction rates did not follow any distinct trend through the Phanerozoic, and their changes through time look very much like chaotic fluctuations (Cornette and Lieberman 2004). Therefore, it is more difficult to find a simple mathematical approximation for the origination and extinction rates than for the total diversity. In fact, the only critical requirement of the exponential model is that the difference between the origination and extinction through time should be proportional to the current diversity level:

$$ (N_o - N_c)/\Delta t \approx kN, \qquad (\text{Eq. 12}) $$
where $N_o$ and $N_c$ are the numbers of genera with, respectively, first and last occurrences within the time interval $\Delta t$, and $N$ is the mean diversity level during the interval. The same is true for the hyperbolic model. It does not predict the exact way in which origination and extinction should change, but it does predict that their difference should be roughly proportional to the square of the current diversity level:

$$(N_o - N_c)/\Delta t \approx kN^2. \qquad (\text{Eq. 13})$$

In the demographic models discussed above, the hyperbolic growth of the world population was not decomposed into separate trends of birth and death rates. The main driving force of this growth was presumably an increase in the carrying capacity of the Earth. The way in which this capacity was realized – either by decreasing death rate or by increasing birth rate, or both – depended upon many factors and may varied from time to time.

The same is probably true for biodiversity. The overall shape of the diversity curve depends mostly on the differences in the mean rates of diversity growth in the Paleozoic (low), Mesozoic (moderate), and Cenozoic (high). The Mesozoic increase was mainly due to a lower extinction rate (compared to the Paleozoic), while the Cenozoic increase was largely due to a higher origination rate (compared to the Mesozoic) (see Markov and Korotayev 2007: 316, fig. 3a and b). This probably means that the acceleration of diversity growth during the last two eras was driven by different mechanisms of positive feedback between diversity and its growth rate. Generally, the increment rate $((N_o - N_c)/\Delta t)$ was changing in a more regular way than the origination rate $N_o/\Delta t$ and extinction rate $N_c/\Delta t$. The large-scale changes in the increment rate correlate better with $N^2$ than with $N$ (see Markov and Korotayev 2007: 316, fig. 3c and d), thus supporting the hyperbolic rather than the exponential model.

## Conclusion

In mathematical models of historical macrodynamics, a hyperbolic pattern of world population growth arises from non-linear second-order positive feedback between the demographic growth and technological development. Based on the analogy with macrosociological models and diverse paleontological data, we suggest that the hyperbolic character of biodiversity growth can be similarly accounted for by non-linear second-order positive feedback between the diversity growth and the complexity of community structure. This hints at the presence, within the biosphere, of a certain analogue to the collective learning mechanism. The feedback can work via two parallel mechanisms: (1) a decreasing extinction rate (more surviving taxa – higher alpha diversity – communities become more complex and stable – extinction rate decreases – more taxa, and so on), and (2) an increasing origination rate (new taxa – niche construction – newly formed niches occupied by the next ‘generation’ of taxa – new taxa, and so on). The latter possibility makes the mechanisms underlying
the hyperbolic growth of biodiversity and human population even more similar, because the total ecospace of the biota is analogous to the ‘carrying capacity of the Earth’ in demography. As far as new species can increase ecospace and facilitate opportunities for additional species entering the community, they are analogous to the ‘inventors’ of the demographic models whose inventions increase the carrying capacity of the Earth.

The hyperbolic growth of Phanerozoic biodiversity suggests that ‘cooperative’ interactions between taxa can play an important role in evolution, along with generally accepted competitive interactions. Due to this ‘cooperation’ (which may be roughly analogous to ‘collective learning’), the evolution of biodiversity acquires some features of a self-accelerating process. The same is naturally true of cooperation/collective learning in global social evolution. This analysis suggests that we can trace rather similar macropatterns within both the biological and social phases of Big History. These macropatterns can be represented by relatively similar curves and described accurately with very simple mathematical models.

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2

**The World System Trajectory:**
**The Reality of Constraints**
**and the Potential for Prediction**

Tony Harper

**Abstract**

This paper discusses a number of constraints imposed on the trajectory of the World-System. Chief among these are the overall trend of the trajectory of the world system, the existence of lines of equal maximum urban area, that is iso-urban lines, constraints revealed by converting the trajectory to polar coordinates, periodic relationships within that representation, and the use of the ratio of observed maximum urban area to idealized maximum urban area to establish significant similarity with regard to two separate phases of urbanization including similarity in fractal dimensions of each phase. Beyond recognizing this suite of constraints, their potential for prediction of the state or position of the world system is discussed.

**Keywords:** de-urbanization, fractal dimension, iso-urban lines, optimum urban area, parametric equation.

In the employing of mathematical methods,
however, biological facts must be reduced to
a mere abstract of their real complexity. It is im-
portant to understand that this simplification is
recognized for what it is: a working method. It
does not mean to ignore the complexity and to-
tality of biological relationships. These clearly
remain the foundation upon which any mathe-
matical model may be built.

C. C. Li, 1955

**Acknowledgements**

I wish to thank the following people for their comments and constructive criti-
cisms: Andrey Korotayev, Sergey Tsirel, and Alexey Fomin. However, any
errors or misinterpretations within the text of this paper are mine and mine
alone. I wish to thank Kseniya Ukhova for her expert editorial hand.
**Dedication:** I wish to dedicate this paper to four scholars who have helped shape my thinking: the late Dr. Richard V. Bovbjerg, Dr. Robert DeMar, Dr. Elliott Speiss, and Dr. Dennis Bramble.

## Introduction

Constraints on living systems are ubiquitous, and, if anything, the World-System should certainly be considered a living system. Constraints at the population level of biological organization have been documented formally for over 200 years à la Malthus, and mathematical formalism has been used to represent those constraints for almost as long. Verhulst in 1838 (as noted in Hutchinson 1978) proposed the logistic equation,

$$ \frac{dN}{dt} = rN(1 - \frac{N}{K}), \qquad (\text{Eq. 1}) $$

as a reasonable model of population growth occurring within limits, that is within constraints, the constraint being represented by K, the carrying capacity. Further, although K was considered a constant by Verhulst and by many to follow who applied this equation to a variety of problems in population biology (for a review of the utility of this equation see Hutchinson 1978), the constraint itself can be seen to change, and in his insightful book, *How Many People Can the Earth Support*, Joel Cohen presents this extended model of population growth,

$$ \frac{dN}{dt} = rN(1 - \frac{N}{K}) \text{ and } \frac{dK}{dt} = \frac{[L/N]dN}{dt}, \qquad (\text{Eq. 2}) $$

where L is some threshold below which K grows by a factor greater than 1 and above which K grows by a factor less than one. Note that the growth of K does decrease continuously from N < L through N > L. In this more complicated model L represents then a second constraint.

The World System, at its most fundamental level, is a system of populations and is therefore it should be expected that the World System would be subject to the constraints imposed on population growth. In this light Korotayev, Malkov, and Khaltourina (2006a, 2006b, and 2006c) use a system of equations with the core of the system dependent on logistic-like constraints. Specifically, their system of equations,

$$ \frac{dN}{dt} = aS(1 - L)N, \qquad (\text{Eq. 3a}) $$

$$ \frac{dS}{dt} = bSN, \text{ and} \qquad (\text{Eq. 3b}) $$

$$ \frac{dL}{dt} = cS(1 - L)L, \qquad (\text{Eq. 3c}) $$

and the coefficients a, b, and c can also be seen as constraints on the system. The end result of applying this system is that as the World System grows it will reach an equilibrium established by L = 1. How rapidly the system reaches equilibrium will be determined by initial conditions and by the values of the three coefficients mentioned above. A further paper by Grinin and Korotayev (2006) demonstrated that the process of mega-urbanization exhibits similar constrained behavior. See their Diagrams 1, 3, 4, 7, and 10 which clearly suggest a strong and constraining relationship between the process of (mega-) urbanization, state formation, and associated territory and in fact show logistic-
like behavior with the existence of plateaus punctuating periods of rapid, hyperbolic growth. Inspection of Diagrams 3, 4, 7, and 10 reveal not only logistic-like behavior but also more complex behavior than might otherwise be expected. (Also see papers by Korotayev [2010] and by Grinin and Korotayev [2006] for similar analyses.) Such complexities are in all probability emergent, that is in the manner of Anderson (1972) and of Mayr (1988), in that they are characteristic of the level of organization of the world system and not of its component parts.

It is on this level of organization of the World System, highly complex and urbanized, that this paper will be focused. The pattern of the trajectory of the World System, previously reported on by Harper (2010a, 2010b), exhibits both general trends and more complex finer structure, both levels of which exhibit constrained behavior. It will be the intent of this paper first to investigate these constraints and then to consider their potential for predicting the behavior of the system. Constraints will be considered from three different perspectives, that of the morphology of the graph in Fig. 1, that of a modified polar representation of the same data, and that of a comparison of observed maximum urban area and idealized maximum urban area. Once these different sets of constraints have been analysed, then the information that these constraints contribute to predicting the behavior of the system will be considered.

Finally, the research that this paper is based on was done in the spirit of the lead quotation from C. C. Li (1955). It is important for the reader, particularly those with little or no serious mathematical training, to understand that the simplifications used to make the Mathematics even somewhat tractable do not in any way devalue the detailed knowledge of the historian, but at once are both a consequence of that detailed work and also a context in which it is hoped the work of the historian and social scientist can be given new perspective. There is one more thing: mathematical reasoning may in many instances be the only way in which the emergent properties of complex systems can be revealed and understood.

## The Organization of the Trajectory of the World System

This section will show that the organization of the world system trajectory as first described in Harper (2010a, 2010b) exhibits both gross structural trends and finer trends, all explainable in terms of urbanization, de-urbanization, and the constraint that maximum urban area places on the trajectory. There are clear trends at both the macro-level of organization of the trajectory and at finer scales that all suggest constraint and, as previously suggested, ultimately the potential for prediction.

Fig. 1 is a rotated version of Fig. 7 in Harper (2010b) in which the variable, γ, has been rotated to the y-axis and the variable, lnT, that is the natural log transform of the global population of the world system, has then obviously been rotated to the x-axis. This was done so that certain trends and characteristics of this trajectory would become more obvious than in the way these variables
were originally displayed. In particular, some of the fine(r) scale characteristics
are easier to understand, and this graphical representation also displays a non-
obvious relationship between the two variables graphed and the natural log of
maximum urban area size.

<figure><img src="image_20.png" /><figcaption>**Fig. 1.** lnT on the X-axis is graphed against γ on the y-axis</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>**Fig. 2.** The graph above is as in Fig. 1 but with a linear regression imposed on the data. A linear regression of the data yields, $\gamma = -.0465\ln T + 2.2607$, representing an inverse relationship between $\gamma$ and $\ln T$. $R^2 = .5225$</figcaption></figure>

It can be seen by inspection of Fig. 1 that there is an inverse relationship be-
tween γ and lnT, since over the full range of lnT γ decreases from an initial
value of 1.4851 to a final value of 1.2099, a difference of .2752 or a decrease of
18.53 % of the initial value of γ. Further, the largest values of γ occur with low
values of lnT, for example, when γ = 1.5674, its largest value, the value of lnT
is 17.1731, which is an increase of 4.37 % over the initial value of lnT, while the largest value of lnT, 22.5478, is an increase of 37.03 % of the initial value of lnT. Perhaps, an easier and more general method of demonstrating this trend of decreasing γ with respect to lnT would be to generate a linear regression of this data. This is done in Fig. 2, yielding the regression equation,

$$ \gamma = 2.23 - .0449\ln T, \qquad (\text{Eq. 4}) $$

and the fit of this equation is quite good with r = -.708 and the RMSE = .0593. The fact that this equation has a negative slope clearly shows that the overall trend of the world system is one of decreasing γ with increasing lnT. What does this general trend imply?

Decreasing γ implies increasing urbanization. By establishing a fixed value of C, that is C_a, with respect to C_max, a defined section of the triangular area bounded by the natural log transform of the equation,

$$ F = \alpha C^{-\gamma} \qquad (\text{Eq. 5}) $$

can be arbitrarily defined as a boundary between urbanized and non-urbanized regions of the world system at a given time. By then holding A, the area bounded by

$$ \ln F = \ln \alpha - \gamma \ln C, \qquad (\text{Eq. 6}) $$

constant and reducing γ it can be shown that the subsection of A bounded by C_a is greater than the subsection bounded by the original value of γ, and consequently represents an increase in urbanization. Note, that for any given time it can be shown that the ratio of urbanized to non-urbanized portions of the world system is given by:

$$ U = \frac{(\ln C_m + x - \ln C_c)^2}{(\ln C_m + x)^2}. \qquad (\text{Eq. 7}) $$

As γ decreases with increasing lnT as represented by Eq. 4, it then follows that urbanization increases with increasing lnT. Modelski (2003) and Kortotayev and Grinin (2006) have shown that urbanization has increased over time with the increase being punctuated by plateau-like phases and these phases appear to be synchronized with a similar pattern in global population, here labeled as T. It is a contention of this paper that the increasing population of the world system is dependent on increasing urbanization and would not occur without the process of urbanization and in turn its attendant process of technological progress.

Within this broad trend of decreasing gamma, increasing lnT and therefore T, and increasing urbanization are micro-trends only the average of which is reflected by these broad trends. While a detailed analysis, on a century by century basis, will not be entered into here, two significant micro-trends will be mentioned, the period from 400 BCE to 700 CE, essentially the formation, florescence, and demise of Rome, of the Han Empires and the mosaic of empires to follow the Han through to the establishment of the Tang Empire, and the origination of the Islamic Caliphates, and then the 12th to the 14th centuries CE. These two periods of time are significant, because they represent periods of
rapid urbanization followed by rapid de-urbanization. Both are also associated with significant pandemics, with epidemic warfare, and with a variety of technological changes, for example, aqueducts in the West and the Grand Canal in the East to name two.

From 400 BCE to 100 BCE both East and West experienced very rapid urbanization, and if changes in gamma can be taken as an indicator of this trend, then over that period gamma decreased by .1460, or per century, .0487. These numbers may seem insignificant, but they are associated with a three-fold change in maximum urban area size from 320,000 in 400 BCE to approximately one million in 100 BCE. The reverse trend, increasing gamma over time, began in 200 CE with $\gamma = 1.2699$ and ended in 500 CE with $\gamma = 1.3793$, a change of .1097 or .0565 per century. With respect to change in maximum urban area size the decrease was from 1.2 million to approximately one-half million, or slightly less than a three-fold decrease. In the case of the 12th to the 13th century the changes in both gamma and maximum urban area size were less in absolute terms, but per century the change in gamma was .0486 accompanied by an increase in maximum urban area size of approximately one-half million, and the following century there was an increase of .0461 with a concomitant decrease in maximum urban area size of one-half million, almost exactly reversing the trend of the century before. What occurred over six hundred years in the period of Rome, Han et al. occurred in this second episode within a period of two hundred years, that is from 1200 CE to 1400 CE. (Note that the period from 100 BCE to 200 CE was not included, because it involved relatively small changes in gamma and urbanization and the detail at present of these changes does not add to understanding of the overall process of urbanization and de-urbanization.)

What also is significant is the fact that both sets of changes occurred during periods of relatively small change, sometimes negative, in the world-system population as a whole. From 400 BCE to 100 BCE there was a net loss of two million people from the global population, that is from 162E6 to 160E6, and from 200 CE to 500 CE there was no change in the population of the world system. While from 1200 CE to 1300 CE there was also no change in population, but in the succeeding century there was a loss of ten million to the total population. This all seems to imply that the rapid changes in urbanization followed by rapid de-urbanization, are directly associated with a static or negatively growing total population. To demonstrate this point using

$$C_{\max}^{\gamma} - C_{\max} - (\gamma - 1)T = 0, \qquad (\text{Eq. 8})$$

observed values of $C_{\max}$ from 400 BCE through 100 BCE were used with held constant at 1.4239 to compute T, the total population of the world system for each century. These values are given in Table 1 below. As can be seen the observed values of T remain relatively constant, while the expected values for T as dictated by the pattern of urbanization constrained by $\gamma = 1.4239$ increase significantly. This implies that support of the pattern of urbanization with in-
creasing maximum urban area size required much greater world system popula-
tion size.

<table><caption>**Table 1.** Maximum urban area and gamma with respect to both observed and expected population sizes</caption><thead><tr><td>Century</td><td>C<sub>max</sub></td><td>γ</td><td>T<sub>EXP</sub></td><td>T<sub>OBS</sub></td></tr></thead><tbody><tr><td>400 BCE</td><td>32.E5</td><td>1.4239</td><td>162E6</td><td>162E6</td></tr><tr><td>300 BCE</td><td>5.0E5</td><td>“</td><td>306E6</td><td>156E6</td></tr><tr><td>200 BCE</td><td>6.0E5</td><td>“</td><td>397E6</td><td>150E6</td></tr><tr><td>100 BCE</td><td>1.0E6</td><td>“</td><td>822E6</td><td>160E6</td></tr></tbody></table>

A further implication of this difference between the observed and expected values of T then would be that the initial level of urbanization was somehow adjusted or adapted to an initial level or threshold of population, but when the process of urbanization continued but was not underwritten by continued total population growth, then after a time delay exceeding that threshold was followed by a process of de-urbanization that occurred until a reasonable steady state was re-established with respect to the total population.

**Constraints Imposed on the Trajectory of the World System by Maximum Urban Area Size**

Visual inspection of Fig. 1 will reveal that certain sets of points appear to be arranged linearly. As an example of a linear array a high-lighted region of the graph in Fig. 1 is depicted in Fig. 3.

<figure><img src="image_22.png" /><figcaption>**Fig. 3.** LnT v. γ. Note the section of the graph indicated with an arrow. This section represents four data points as an example of a linear array of points</figcaption></figure>

This linear array of points shares the same maximum urban area size, and it can
be shown that other less obvious linear arrays embedded within this graph also
share unique maximum urban areas; in other words, the arrangement of points in this graph is determined by and limited by associated maximum urban area sizes. Fig. 4 below shows three such sets, each with a line drawn through the set having an equation of the form,

$$ \gamma = m\ln T + b. \qquad (\text{Eq. 9}) $$

Table 1 gives the set of all slopes and y-intercept values for each maximum urban area size, however, in several instances a specific point has a unique maximum urban area value, and for those cases a linear regression of all empirically determined slopes and of all empirically determine y-intercepts was used to determine the specific values of the slope and y-intercept for these unique maximum urban area sizes.

<figure><img src="image_23.png" /><figcaption>**Fig. 4.** This graph is similar to that of Fig. 1 but with iso-urban lines representing all positions of the world system with respect to γ and lnT for three specified maximum urban area sizes. All three lines have similar but not identical parameters for their characteristic equations</figcaption></figure>

<table><caption>**Table 2**</caption><thead><tr><td>Maximum urban area size</td><td>Slope (m)</td><td>Y-intercept (b)</td></tr></thead><tbody><tr><td>1</td><td>2</td><td>3</td></tr><tr><td>4E4</td><td>.115</td><td>-.400</td></tr><tr><td>5E4</td><td>.1120</td><td>-.3940</td></tr><tr><td>6E4</td><td>.1110</td><td>-.4090</td></tr><tr><td>7E4</td><td>.1090</td><td>-.4410</td></tr><tr><td>8E4</td><td>.1100</td><td>-.4410</td></tr><tr><td>1E5</td><td>.1060</td><td>-.4110</td></tr><tr><td>1.2E5</td><td>.1055</td><td>-.4227</td></tr><tr><td>1.25E5</td><td>.1037</td><td>-.3986</td></tr></tbody></table>
<table><thead><tr><th>1</th><th>2</th><th>3</th></tr></thead><tbody><tr><td>2E5</td><td>.1006</td><td>-.4104</td></tr><tr><td><strong>3.2E5</strong></td><td><strong>.0980</strong></td><td><strong>-.4393</strong></td></tr><tr><td><strong>4E5</strong></td><td><strong>.0967</strong></td><td><strong>-.4440</strong></td></tr><tr><td>5E5</td><td>.0948</td><td>-.4532</td></tr><tr><td>6E5</td><td>.0946</td><td>-.4532</td></tr><tr><td><strong>7E5</strong></td><td><strong>.0938</strong></td><td><strong>-.4560</strong></td></tr><tr><td>8E5</td><td>.0944</td><td>-.4801</td></tr><tr><td><strong>9E5</strong></td><td><strong>.0927</strong></td><td><strong>-.4614</strong></td></tr><tr><td>1E6</td><td>.0910</td><td>-.4430</td></tr><tr><td><strong>1.1E6</strong></td><td><strong>.0918</strong></td><td><strong>-.4614</strong></td></tr><tr><td>1.2E6</td><td>.0926</td><td>-.4946</td></tr><tr><td><strong>1.5E6</strong></td><td><strong>.0906</strong></td><td><strong>-.4723</strong></td></tr><tr><td><strong>6.5E6</strong></td><td><strong>.0858</strong></td><td><strong>-.5037</strong></td></tr><tr><td><strong>35E6</strong></td><td><strong>.0828</strong></td><td><strong>-.5308</strong></td></tr></tbody></table>

*Note:* Numbers in bold face type were produced by linear regression of empirically determined slopes and y-intercepts. All slope and intercept values can be substituted into an equation of the form, $\gamma = m\ln T + b$.

There are several consequences of this finding. First, it is entirely possible to determine the position of an ordered pair of values of $\ln T$ and $\gamma$ as the position of that point must lie on the appropriate linear array, here referred to as an iso-urban line, as determined by specific maximum urban area size. Second, the set of all linear arrays falls on a complex curved surface as determined by the range of values of both slope and y-intercept. Third, this surface must be undulatory as slope and y-intercept values change with respect to maximum urban area size, while slope appears to decrease with respect to maximum urban area size, y-intercept values fluctuate, for example, the sequence from 4E4 through 2E5 exhibits continually decreasing slope values but decreasing then increasing then decreasing values of $b$. Fourth, the distance over which the world system moves is directly determined by maximum urban area. Fifth, since change in position of the world system from century to century can be represented by angular change with respect to a given iso-urban line and the actual distance moved, the world system trajectory can be represented by polar coordinates. This last aspect will be investigated in the next section.

## A Polar Plot Representation of the World System Trajectory

As was previously mentioned iso-urban lines can be used to generate a polar-plot of the world system trajectory. This was done in the following way. Local distance, defined as the distance from one point representing the position of the world system to a consecutive point was determined by the formula

$$d = [(\gamma_1 - \gamma_0)^2 + (\ln T_1 - \ln T_0)^2]^{\frac{1}{5}}. \quad (\text{Eq. 10})$$
The continuous summation of d is represented by S = Σd. The angle of the distance vector, θ, was determined by finding the cosine of the angle between the local distance and the vertical distance from the previous iso-urban line to the following iso-urban line as determined be the value of the maximum urban area. Once this was done, the ratio of the vertical distance divided by the local distance is then used to determine the arc cosine. This angle, θ, is then summed continuously and is represented by

$$ \Sigma\theta = \Psi. \qquad (\text{Eq. 11}) $$

These parameters, S and Ψ, are then used in the parametric equations:

$$ X = \text{Scos}\Psi, \qquad (\text{Eq. 12a}) $$

$$ Y = \text{Ssin}\Psi, \qquad (\text{Eq. 12b}) $$

to determine the polar coordinates for each position of the world system.

Using the above procedure a polar plot of the world system was generated and represented in Fig. 5 below.

<figure><img src="image_24.png" /><figcaption>Fig. 5. A polar plot of the World System Trajectory. X = ScosΨ. Y = SsinΨ. This graph extends from 3000 BCE at the center most position to 2000 CE at the far right</figcaption></figure>

Clearly, the overall form of the plot is that of a spiral with the initiation of the system at the origin which then spirals outward in a counter-clockwise fashion. Changes in the distance of the outwardly spiraling plot are a consequence of both changes in Ψ and in S, however, there are instances in which the magnitude of the change in one of these parameters over-rides the change in the other, that is where there is greater change in either Ψ or more likely, S, which would imply an increase or decrease in urbanization. It should also be noted that the distances from individual position (of the world-system) to individual position vary and are a consequence of the relative changes in both Ψ and S, and it is these positions collectively that bring about changes in Ψ and S. Further, change in Ψ represents change in lnT, and change in S represents change in the
distribution of urban area sizes as represented by $\gamma$. It should also be quite apparent that the spiral plot is not a smooth one but is punctuated by a number of segments which unquestionably represent a shifting forward of the entire position of the spiral and, further, the changes in segment length cause the spiral at times to fold over on itself. From the perspective of Fig. 5 the trajectory of more recent spirals falls beneath the point on a previous spiral. This is due primarily to the paucity of data and the specific perspective of Fig. 5.

If the data used to produce the previous graph are supplemented with a time axis, that is the x-axis represents time, the y-axis represents $X = \text{Scos}\Psi$, and the z-axis represents $Y = \text{Ssin}\Psi$, then a three dimensional polar plot of the world system trajectory can be created. Such a graph is represented immediately below and consists of a slight rotation of the graph in Fig. 5 to give the impression of depth.

<figure><img src="image_25.png" /><figcaption>Fig. 6. This graph gives a 3D representation of the polar plot of the world system trajectory over the last 5000 years. The graph has been rotated to give the perception of depth with 3000 BCE being represented in the upper left of the graph and 2000 CE being represented on the far right hand side of the graph</figcaption></figure>

There are three aspects of the graph in Fig. 6 that are immediately striking, the regularity of the spirals, the expansion of each spiral over the previous one, and the extended position of the last point, that of 2000 CE. As previously mentioned, however, further inspection of this graph will reveal that there are any number of instances in which the trajectory does not spiral but moves directly forward, for example, the sharp spike immediately to the left of the 2000 CE position. These periods of forward movement (in time) without rotation are best represented by rotation of this graph so that the y-axis is minimized as in Fig. 7 below.
<figure><img src="image_26.png" /><figcaption>**Fig. 7.** The z-axis has been converted to a y-axis and plotted against time. This is equivalent to rotating the graph in Fig. 6 so that the y-axis has been eliminated. The regions of the graph in which there is no rotation but only forward movement in time are represented by the (near) parallel setions to the x-axis</figcaption></figure>

As is clearly apparent, the maximum and minimum y-axis values appear to be approximately linearly distributed, and, as to be expected, expand with time, since S is a summation of each movement, and there is clear regularity or periodicity exhibited by the maxima and minima. In turn, if the z-axis in Fig. 6 is eliminated so that $Y = S\sin\Psi$ is represented on the y-axis, the graph is Fig. 8 is produced, which differs little in over-all form from that in Fig. 7. It exhibits the periodically spaced maxima and minima exhibited by Fig. 7, all of which appear to be approximately linearly ordered with the exception one point, the last maximum, which is unquestionably positioned beyond any reasonable linear extension of the previous points. In other words, the current position of the world system with respect to its previous linear expansion is much greater than would otherwise be expected.

The regularity of the graphs in Figs 7 and 8 can be further investigated by considering the periodicity exhibited by these curves. The magnitudes of the maxima and minima of each are represented in Table 3 on the following page and their periodicities are represented in Table 4. It is quite clear from an inspection of the data in Table 3 that the maximum values for $S\cos\Psi$ and $S\sin\Psi$ are offset by either 200 or 300 years (av. = 240 years), while the periodicities in Table 4 range from 700 to 900 years (av. = 833.3 years) and 800 to 900 years (820 years) respectively.
<figure><img src="image_27.png" /><figcaption>**Fig. 8.** Time v. Y = SsinΨ. Those regions of the trajectory without rotation are less apparent. However, the periodicity of the trajectory is quite obvious, as is the linearity of maximum and minimum y-axis values. Of significance is the extended position of the last point on the graph</figcaption></figure>

<table><caption>**Table 3**</caption><thead><tr><th>Time</th><th>ScosΨ Max</th><th>Time</th><th>ScosΨ Min</th><th>Time</th><th>SsinΨ Max</th><th>Time</th><th>SsinΨ Min</th></tr></thead><tbody><tr><td>3000 BCE</td><td>.0671</td><td>2600 BCE</td><td>-.2488</td><td>2800 BCE</td><td>.1577</td><td>2400 BCE</td><td>-.4392</td></tr><tr><td>2200 BCE</td><td>.6504</td><td>1800 BCE</td><td>-1.0405</td><td>1900 BCE</td><td>.8303</td><td>1500 BCE</td><td>-1.2398</td></tr><tr><td>1300 BCE</td><td>1.3846</td><td>1000 BCE</td><td>-1.6646</td><td>1100 BCE</td><td>1.5547</td><td>800 BCE</td><td>-1.9387</td></tr><tr><td>500 BCE</td><td>2.3495</td><td>100 BCE</td><td>-3.1177</td><td>300 BCE</td><td>2.9417</td><td>1 CE</td><td>-2.8417</td></tr><tr><td>200 CE</td><td>3.1995</td><td>700 CE</td><td>-3.5077</td><td>500 CE</td><td>3.4192</td><td>900 CE</td><td>-3.6111</td></tr><tr><td>1100 CE</td><td>3.4538</td><td>1800 CE</td><td>-4.3881</td><td>1300 CE</td><td>4.1896</td><td>1900 CE</td><td>-5.4817</td></tr><tr><td>2000 CE</td><td>7.1432</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td></tr></tbody></table>
<table><caption>Table 4</caption><thead><tr><th>ΔTime</th><th>ΔScosΨ Max</th><th>ΔTime</th><th>ΔScosΨ Min</th><th>ΔTime</th><th>ΔSsinΨ Max</th><th>ΔTime</th><th>ΔSsinΨ Min</th></tr></thead><tbody><tr><td>800</td><td>.5833</td><td>800</td><td>.7917</td><td>900</td><td>.6726</td><td>900</td><td>.8006</td></tr><tr><td>900</td><td>.7342</td><td>800</td><td>.6241</td><td>800</td><td>.7244</td><td>700</td><td>.6989</td></tr><tr><td>800</td><td>.9649</td><td>900</td><td>1.4531</td><td>800</td><td>1.3870</td><td>800</td><td>.903</td></tr><tr><td>700</td><td>.8500</td><td>800</td><td>.39</td><td>800</td><td>.4775</td><td>900</td><td>.7694</td></tr><tr><td>900</td><td>.2545</td><td>1100</td><td>.8804</td><td>800</td><td>.7704</td><td>1000</td><td>1.8706</td></tr><tr><td>900</td><td>3.6894</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td><td>X</td></tr></tbody></table>

Considering the crudeness of the data, these respective periodicities are approximately the same. Further, both the periodicities and the regularity of pattern of polar trajectory of the world system contribute to the support of the notion that the world system trajectory is highly constrained.

The data represented in Tables 3 and 4 can be visualized by putting both plots, those of Figs 7 and 8 on the same axes as in Fig. 9. In this graph it can be seen that the maxima of the solid curve that is $X = ScosΨ$, precedes the maxima of the dotted curve, $Y = SsinΨ$, by approximately 200 years.

<figure><img src="image_28.png" /><figcaption>**Fig. 9.** Time on the x-axis and both $X = ScosΨ$, solid curve, and $Y = SsinΨ$, dotted curve, on the y-axis. While these oscillations are offset by approximately one-fourth of a period, they are not analogous to predator-prey oscillations, as both curves reach approximately the same maxima and minima per period, whereas the prey species would maintain a population level far below the prey species population, that is approximately one-tenth of the prey species population. The expanding oscillations imply that the world system as it is represented here is a non-equilibrium system</figcaption></figure>
It is also apparent that the spacing of minima is more variable as noted in Table 4. If the overall pattern in Fig. 9 is compared with that of Fig. 5 it can be seen that the amplitude of the spiral in each graph increases over time and can be expected to continue increasing over time, the single and momentary exception of increase in S without increase in Ψ which will be addressed shortly. Given that in general the amplitude of the world system will continue to increase with some regularity, it can be accepted that this is supporting evidence for the world system being a non-equilibrium system.

If we now return to the graph in Fig. 9 and the data in Tables 3 and 4 and focus only on the maxima and minima in that graph and those tables, it becomes quite apparent that the maxima and minima occur periodically. What is the relationship over time of these maxima and minima? The periodicity of these maxima and minima is plotted against time in Fig. 10 below.

<figure><img src="image_29.png" /><figcaption>**Fig. 10.** Time is represented on the x-axis, and the magnitude of the maxima of both $X = \text{Scos}\Psi$ and $Y = \text{Ssin}\Psi$ is represented on the y-axis. The periodicity of the maxima of both X and Y is apparent. This graph was created with the use of spline interpolation</figcaption></figure>

Clearly, with the exceptions of the first and last point, the maxima for both $X = \text{Scos}\Psi$ and $Y = \text{Ssin}\Psi$ occur periodically as is visually represented below and noted in Tables 3 and 4. It is also interesting, and it is understood that extrapolation is a far riskier business than interpolation, that if this periodicity holds, then the second point beyond the one representing the maximum of 2000 CE, that is 50 on the x-axis below, will occur some 200 years into the future, and if one observes the overall pattern of the graph and notes that we have now entered a demographic transition, perhaps a comparison can be made between the current position of the world system and its position at 500 BCE, that is point
25 on the x-axis of Fig. 10. If this is a valid comparison, then it can be expected that some 200 years into the future the world system will be approaching a demographic plateau. It is noted here that this speculation goes counter to the thinking of a number of scholars, for example, von Forester *et al*. (1960), that plateau will be achieved this century.

If attention is now turned to the pattern of minima as represented in Fig. 11, it is again clear that these pairs of minima occur periodically and therefore predictably. The fact that there are six complete pairs of minima, as opposed to the five pairs of maxima and two singleton points is simply a function of the range of the data and the domain of the respective maxima and minima with respect to the current position of the world system. It is quite clear that while the most recent pair of minima will be succeeded by the next pair some 200 years into the future, there is no way to associate the most recent data directly with any demographic transition, as all minima are paired and as the most recent have no previously corresponding position for comparison.

<figure><img src="image_30.png" /><figcaption>Fig. 11. Time is represented on the x-axis, and the magnitude of minima for both $X = \text{Scos}\Psi$ and $Y = \text{Ssin}\Psi$ is represented on the y-axis. This graph was created with the use of spline interpolation</figcaption></figure>

The overall regularity of both polar plots can also be visualized by considering a plot of S v. $\Psi$, as is represented in Fig. 12.
<figure><img src="image_31.png" /><figcaption>Fig. 12. $\Psi$ is graphed on the x-axis, and S is graphed on the y-axis. Note the near linearity of this graph with a linear regression yielding, $\Psi = .0026S - .3128$. $R^2 = .9521$. Each vertical segment of the graph, and there are seven of them, represents a period of time when there was no change in $\Psi$ but only in S</figcaption></figure>

It can be seen that this plot, as represented by

$$ \Psi = .0026S - .3128, \qquad (\text{Eq. 13}) $$

is relatively linear, $R^2 = .9521$. However, there are pronounced non-linear aspects to this graph, and this amount to considerable positive changes in S with respect to $\Psi$, most probably phase changes in the system itself. Further, these phase changes occur in a two-step process, and there are three such two-step phase changes, 2000 BCE – 1700 BCE, 900 BCE – 300 BCE, and 900 CE – 2000 CE, this last amounting to 1100 years of change and the total amount of time associated with these changes is 2300 or approximately 46 % of the total time of recorded history. It should be noted that a single step change occurred from 2500 BCE – 2200 BCE and is the earliest example of such change.

If the axes in Fig. 12 are rotated so that S is represented on the x-axis, as in Fig. 13, an interesting pattern is revealed which is not apparent in the previous figure. Here it is quite apparent that periods of heightened increase in $\Psi$ alternate with periods of increase in S. Specifically, from 3000 BCE to 900 BCE and from 400 BCE to 1400 CE, $\Psi$ changes rapidly, but S changes moderately, while from 900 BCE to 400 BCE and from 1400 CE to 2000 CE represents periods of relatively rapid change in S, again with only moderate change in $\Psi$. These data are represented numerically in Table 5.
<figure><img src="image_32.png" /><figcaption>**Fig. 13.** S is represented on the x-axis, and Ψ is represented on the y-axis. Of note in this graph is the fact that periods of rapid change in S alternate with periods of rapid change in Ψ</figcaption></figure>

In this table the per cent rates of change for both Ψ and S over their respective periods of change. For the first 2100 years of world system history the rate of change in Ψ is almost twice that of S, while in the next 500 years are reversed with S having twice the rate of change per century that Ψ does, and, again, for the next 1000 years the rate of change for Ψ is now approximately twice that of S. Finally, for the last 600 years the rate of S becomes three times that of Ψ. Of significant interest is the fact that the two time periods for rapid change in each variable are approximately the same, that is both periods of rapid change for Ψ are approximately a millennium in length, while both periods of change for S are approximately one-half a millennium in length. Of some significance is the fact that if increased changes in S occur over a period of one-half a millennium, then the world system should be entering another period of rapid change in Ψ which should last approximately 1000 years.<fnref n="1" />

<table><caption>**Table 5**</caption><thead><tr><td>Time Period</td><td>% ΔΨ/Time</td><td>% ΔS/Time</td></tr></thead><tbody><tr><td>1</td><td>2</td><td>3</td></tr><tr><td>3000 BCE – 900 BCE</td><td>2.1511</td><td>1.2335</td></tr><tr><td>900 BCE– 400 BCE</td><td>1.5251</td><td>2.9833</td></tr></tbody></table>

<footnote n="1">¹ The data set on which this last statement is made is not exceptionally large, and, consequently, making a statement such as this last one comes at some risk. My salvation is that none of us will be here 1000 years from now, however, hopefully the world system will be!</footnote>
<table><thead><tr><th>1</th><th>2</th><th>3</th></tr></thead><tbody><tr><td>400 BCE–1400 CE</td><td>2.0104</td><td>1.0351</td></tr><tr><td>1400 CE–2000 CE</td><td>1.8356</td><td>6.7747</td></tr></tbody></table>

The focus of this paper will now be turned to the abstract distance traveled from point to point on the polar representation of the world system trajectory. These changes in spiral distance with respect to time are represented in Fig. 14. Segment length was determined by Pythagorean Theorem as in Eq. 8, where the parametric equations were used to determine X and Y values.

<figure><img src="image_33.png" /><figcaption>**Fig. 14.** Time is represented on the x-axis and spiral segment length as determined by the Pythagorean Theorem is represented on the y-axis. Note that the maximum size of segment length increases slightly with time. Also note that there are a number of instances in which segment length almost drops to zero. In fact, this is an artifact of the data and will be explained in the text</figcaption></figure>

As can be seen below, there is a slight increase in average size of individual spiral segments over the 5000 years of world system history, and a wide range of values from almost zero to almost 90 is apparent.<fnref n="2" /> Given that these near zero instances of spiral segment length exist, what is implied by their existence? Are they unique in world system history or simply artifacts of the data themselves? The idiosyncrasies of the graph in Fig. 14 are to some extent an artifact of the procedure to create the graph as opposed to the data themselves. Due to the fact that the data for S range from approximately .02 to 1.40, while the data for Ψ

<footnote n="2">² The reader should be aware that the data in Fig. 14 have not been normalized.</footnote>
range from approximately 6.00 to almost 90.00, the procedure for computing segment length results in heavily weighting the values for Ψ over those of S. Even so, the question must be asked, to what extent does the graph reflect reality? In order to address this question both sets of data, those of Ψ and those of S, were normalized by their largest value. Using this normalized data the graph in Fig. 12 was produced. In fact, this graph exhibits the same general characteristics of the graph in Fig. 11, that is as light increase in average segment length over time and periods in which the size of segment length decreases precipitously. It should be noted that these minima represent instances in which there is no change from century to century in maximum urban area size, and as a result the component, S, is the only component to contribute to the summative process producing successive new positions of the world system. Returning to the comparison of the graphs in Figs 11 and 12, both graphs reveal the same general pattern.

<figure><img src="image_34.png" /><figcaption>**Fig. 15.** Time is represented on the x-axis, and segment length is represented on the y-axis. The average size of the largest segment lengths remains relatively stable across time with a slight increase over that same period of time. The instances of minimal segment length are pronounced as they are in Fig. 14.</figcaption></figure>

Considering only the component, S, the summation of the distance component in the polar plot of the world system trajectory, and plot it against time a near linear graph is produced as can be seen in Fig. 15. This graph shows clearly that on average S grows incrementally in a regular fashion, but with two pronounced exceptions where the slope of the graph changes abruptly. These are the period of time from 500 BCE to 400 BCE and also from 1800 CE to 2000 CE.
In both instances the slope of the graph becomes markedly more positive, that is the rate of change of S increases. With regard to the most recent instance this change can be associated with the industrial revolution, however, with regard to the first instance of increased slope, one can only guess at a variety of technological changes of the time, a time when the Roman Empire was in its formative stage and shortly before the genesis and development of the Han Empire in China. If the slopes of the four respective periods, 5000 BCE – 500 BCE, 500 BCE – 400 BCE, 400 BCE – 1800 CE, and 1800 CE – 2000 CE, they are respectively: .0938, .4828, .0956, and 1.0261.

<figure><img src="image_35.png" /><figcaption>Fig. 16. Time is represented on the x-axis, and S is represented on the y-axis</figcaption></figure>

This plot is effectively linear with the exception of the last 200 years of world system history, that is points 49 and 50. Even so, linear regression of this data gives:

$$S = .1102t - .2654 \qquad (\text{Eq. 14})$$

with R² = .9568.

Interestingly, the slopes of the two extended periods are approximately the same, while the slopes of the two transition periods are both significantly greater than the slopes of the extended periods, by a factor of 5.1471 for the phase change from 500 BCE to 400 BCE and by a factor of 10.7333 for the phase change from 1800 CE to 2000 CE, a period of change that may not yet be over. One further comment here, if the phase change is in fact over, do these data imply that the change in S will return to a more stately ~.09?

When a similar analysis is performed on Ψ, the summation of θ, the individual angles normal to specific iso-urban lines. Fig. 16 represents the graph of
this data, and, most remarkably, the plot is effectively linear yielding the equation,

$$ \Psi = 42.6751t + 22.3262, \qquad (\text{Eq. 15}) $$

with R² = .9968. This graph is quite different from that of Fig. 15, in that it has a closer linear fit and that while there are divergences from linearity there are no significant instances of what have been labeled as phase changes but rather the divergences from linearity are distributed throughout the range of the graph. Appropriately, the next step in analysis is to compare the differences between observed and expected values of both S and Ψ, the latter as predicted by linear regression.

<figure><img src="image_36.png" /><figcaption>**Fig. 17.** Time is represented on the x-axis and Ψ is represented on the y-axis. Linear regression gives: Ψ = 42.6751t + 22.3263, R² = .9968</figcaption></figure>

If the differences are taken between the observed values for both S and Ψ based on their respective linear regressions and plotted against time, the following two graphs reveal a periodic character to both plots. These graphs are represented in Figs 18 and 19. The first of these two graphs representing the difference between observed and expected for the variable, S, shows approximate periodicity with peaks at 100 BCE and (apparently) at 2000 CE. Both of these peaks represent times of rapid urbanization, the first matching the urban area growth approximating 1,000,000 succeeding the previous maximum of 600,000 one century prior and the second representing the incredible growth from 6,500,000 in 1900 CE to an approximately 35,000,000 in 2000 CE. Both of these instances represent events in which the maximum urban area size increases by approximately an order of magnitude, with the second maximum urban area increase associated with the change from the industrial to the post-industrial world system. The pattern of the graph in Fig. 18, that of the difference between observed and expected in the variable, Ψ, is more problematic
with nine clear maxima with values greater than zero but excluding both the first and last points, 3000 BCE and 2000 CE.

<figure><img src="image_37.png" /><figcaption>**Fig. 18.** Time is represented on the x-axis and the difference between the observed value for S and the regressed value for S is represented on the y-axis</figcaption></figure>

These maxima correspond to 2600 BCE, 2100 BCE, 1000 BCE, 800 BCE, 500 BCE, 100 CE, 600 CE, 900 CE, and 1200 CE. With the exception of 2100 BCE, which represents a new maxima for urban area size in the world system of the time, no other times represent unique maxima, although the maxima at both 800 BCE and 500 BCE represent instances of sequential new maxima, that is instances when the new maxima was maintained over a period of 100 years.

<figure><img src="image_38.png" /><figcaption>**Fig. 19.** Time is represented on the x-axis, and the differences between the observed values for ψ and the regressed values for ψ are represented on the y-axis</figcaption></figure>
There is another unique characteristic of this particular graph, the minima at 1700 CE, which is the smallest minima, that is the greatest negative difference exhibited over the entire 5000 year history of the world system. This minima also precedes the greatest increase in maximum urban area size over the entire history of the world system.

In summary, this section has demonstrated the value of restructuring the world system data on time, log-transformed total population, and log-transformed maximum urban area population for a polar plot representation. First, the polar plot with time eliminated yields a counterclockwise outwardly spiraling graph that reveals the magnitude of contribution to the world system at any time throughout its history. When this graph is rotated appropriately its three-dimensional nature can be observed, and, on further rotation, the contribution and periodicity of each polar component can be evaluated. In turn, the periodicity of these components, S and Ψ, can be represented for both their maxima and minima. Then if S and Ψ are plotted against one another, both with S on the x-axis and Ψ on the axis, it can be seen that the last 5000 years of world system history have alternated periods in which change in Ψ and change in S were greater, usually on the order of a 2X factor, with the exception of this last period in which S was greater than Ψ by a factor of 3X. Differences between observed and expected values of both S and Ψ were calculated revealing peaks in ΔS corresponding to about a 2500 year periodicity. The record for ΔΨ is less clear.

## Optimal Urban Area Size and Observed Maximum Urban Area

In previous work there was no standard of comparison of observed maximum urban area other than the context of the temporal sequence that each maximum urban area was part of. In this section a standard of comparison will be established, which will then be the basis of comparison per time step for each maximum urban area. In turn, any relationships revealed by comparative research will be analysed.

The theoretical maximum urban area size, here called the optimum urban area size, can be determined by recognizing that the natural log-transform of the equation, $F = \alpha C^{-\gamma}$ (Eq. 5), represents the hypotenuse of a right triangle in log-space (see Fig. 14). Consequently, the area of this triangle is given by

$$A = .5\gamma\ln C_{\max}^2. \qquad (\text{Eq. 16})$$

Further, the relationship between the sides of this triangle is an inverse one, that is if one side is increased by an amount, x, then the other is decreased by the same amount. Eq. 11 may then be rewritten as:

$$A = (\gamma - x)(1 + x).5\ln C_{\max}^2, \qquad (\text{Eq. 17a})$$

or

$$[\gamma + (\gamma - 1)x - x^2].5\ln C_{\max}^2. \qquad (\text{Eq. 17b})$$
<figure><img src="image_39.png" /><figcaption>**Fig. 20.** This is a graph of the log-transformation of $F = \alpha C^{-\gamma}$, that is $\ln F = \ln \alpha - \gamma \ln C$, for the specific values, $C = 6E5$, and $\gamma = 1.3606$</figcaption></figure>

The graph of this last equation (see Fig. 15) is a parabola, concave down, and area is maximized at the peak of the parabola by the appropriate value of x.

It should be noted that for any given set of data the observed maximum urban area remains constant, so then finding the optimum urban area becomes a maximization problem in which the partial derivative of A with respect to x is set equal to zero, and then x is solved for. This gives the very simple formula,

$$x = (\gamma - 1)/2, \qquad (\text{Eq. 18})$$

and on substituting into both $(\gamma - x)$ and $(1 + x)$ it is found that both of these terms are equal to $(\gamma + 1)/2$ (Eqs 18a and 18b). In other words, maximization with respect to x requires that both sides of the triangle be equal, and this implies that the optimal urban area is determined graphically (and analytically) by $[(\gamma + 1)/2]\ln C_{\max}$, the anti-log of which is $C_{\max}^{(\gamma + 1)/2}$, the importance of this relationship will become clear shortly.

Given that the optimum urban area can be determined from readily available data (Harper 2010a, 2010b), the observed maximum urban area magnitude can then be compared with the optimum urban area magnitude. This will be done by generating the ratio, $C_{\max}/C_{\max}^{(\gamma + 1)/2}$, for each point of the world system trajectory as it is represented by $\gamma$, $\ln C_{\max}$, and $\ln T$. This graph is given in Fig. 3. The graph clearly exhibits two distinct phases, each about 2500 years in length separated by a brief period of transition.
<figure><img src="image_40.png" /><figcaption>**Fig. 21.** The above is a graph of Eq. 2 using the specific values of $\gamma = 1.3606$ and $C_{max} = 6E5$ or $\ln C_{max} = 13.3047$</figcaption></figure>

*Note:* the x-axis is for the range of x, while the y-axis represents A in Eq. 2. Further note, as long as the values of $\gamma$ and $C_{max}$ are those determined by observation for a specific time, any (of those) values may be used (see Harper 2010a).

It is also apparent that the latter phase on average has a higher position than the former. This is a clear indication that the level of urbanization over the last 2500 years or so is greater than in the former period and implies a greater level of technology to support the greater degree of urbanization.

The graph in Fig. 16 may be divided into two equal segments, one from 3000 BCE to 500 BCE, and the other from 500 BCE to 2000 CE (see Figs 17 and 18). These separate graphs share some common characteristics. They both begin with a trend of increasing urbanization as defined by the ratio, $C_{max}/C_{max}^{(\gamma+1)/2}$, they both end with slight decreases in urbanization, and they both exhibit considerable oscillations between their initiation and termination. However, to give the reader perspective, the y-axis has been scaled to the same interval as the x-axis, and, as revealed in Figs 6 and 7, at this scaling the graphs are effectively linear.
<figure><img src="image_41.png" /><figcaption>**Fig. 22.** The above graph represents the ratio of $C_{max}/C_{max}^{(y+1)/2}$ over the last 5000 years of world system history. Very clearly there are two distinct phases, each about 2500 years in length, with a short transition period in between each</figcaption></figure>

The graph in Fig. 16 may be divided into two equal segments, one from 3000 BCE to 500 BCE, and the other from 500 BCE to 2000 CE (see Figs 17 and 18). These separate graphs share some common characteristics. They both begin with a trend of increasing urbanization as defined by the ratio, $C_{max}/C_{max}^{(y+1)/2}$, they both end with slight decreases in urbanization, and they both exhibit considerable oscillations between their initiation and termination. However, to give the reader perspective, the y-axis has been scaled to the same interval as the x-axis, and, as revealed in Figs 6 and 7, at this scaling the graphs are effectively linear.
<figure><img src="image_42.png" /><figcaption>**Fig. 23.** This graph represents the first 2500 years of the time span represented in Fig. 3 and with an imposed trend line. The slope of the line is slightly negative, -.00111, and implies a decreasing ratio of $C_{max}/C_{max}^{(Y+1)/2}$ over that period of time</figcaption></figure>

<figure><img src="image_43.png" /><figcaption>**Fig. 24.** This graph represents the second 2500 years of the graph in Fig. 3, again with an imposed trend line, also with a negative slope, -.00118, which implies decreasing $C_{max}/C_{max}^{(Y+1)/2}$ over this latter time period</figcaption></figure>
This point is being emphasized so that the reader will keep in perspective the
actual magnitude of change represented by these graphs over their respective
time periods of 2500 years.

<figure><img src="image_44.png" /><figcaption>**Fig. 25.** This graph represents the same data as in Fig. 17 but with the y-axis adjusted to the same scale as the x-axis. The intent is to give context to the actual magnitude of change of world system urbanization as represented by the ratio, $C_{max}/C_{max}^{(y+1)/2}$</figcaption></figure>
<figure><img src="image_45.png" /><figcaption>**Fig. 26.** This graph is of the data represented in Fig. 18 but adjusted so that both the x and y-axes are scaled the same as in Fig. 19</figcaption></figure>

If linear regression is applied to both sets of data, that is data from both the first and second 2500 year periods, the respective linear equations,

$$C_R = .0869 - .00111t, \qquad (\text{Eq. 19})$$

and

$$C_R = .148 - .00118t, \qquad (\text{Eq. 20}),$$

where $C_R$ represents the ratio, $C_{max}/C_{max}^{(\gamma + 1)/2}$, and t is time. The slopes of these regression equations are effectively the same, differing by only .00008, implying similar rates of average change with respect to the changing magnitude of $C_{max}/C_{max}^{(\gamma + 1)/2}$ over the two respective 2500 year periods. This can be more emphatically demonstrated by looking at a composite graph in Fig. 27 of both regression lines. In this graph the lower solid line represents the average change in the ratio, $C_{max}/C_{max}^{(\gamma + 1)/2}$, for the first 2500 years of world system history, and the higher solid line represents the same for the second 2500 years of world system history. Both lines have been extrapolated by a dashed line to represent the full extent of the trend had the trend been extended over the full 5000 years.
<figure><img src="image_46.png" /><figcaption>**Fig. 27.** This is a composite graph of the linear regressions in which the x-axis is time and the y-axis is $C_R$, and in which the trend averages for the first and second halves of world system history as represented individually in Figs 17 and 18 are combined. Emphasized is the fact that the change from the trajectory of the first half to the second half, the solid lines of the graph, occurred relatively quickly and represents a significant change in the position of the world system</figcaption></figure>

While the trend lines give the average direction of change, it is quite clear that the observed data are very different and exhibit change, as mentioned previously, that oscillates about each trend line. A two sample T-test was performed on the y-axis data of each set to see if there was any similarity in trend and the P-value for pooled data is 1.9926E-5, implying that the two sets of data are significantly different. This evidence is of course at odds with a visual comparison. For instance as mentioned previously but here more explicitly, such a comparison reveals that there is a 500 year decrease in $C_{max}/C_{max}^{(γ+1)/2}$ prior to an upturn at the end of each time period, and there are other similarities. These patterns need to be investigated more thoroughly in future research.

A remark here needs to be made about both Grinin and Korotayev's research (2006), the research of Korotayev and Grinin (2013, and that of George Modelski (2003) with respect to world system phases, periodicities, and trends. The previous graph represents a single and quite rapid phase change from that of the Ancient World to the Modern World we now live in, and it suggests that regarding urbanization the data here imply that the Modern World began 2500 years ago. I would argue that that is the case only with respect to the ratio, $C_{max}/C_{max}^{(γ+1)/2}$, however, it does also suggest that our modern world has very deep roots, and that Arrighi's *The Long Twentieth Century* and other research on historical trends by a variety of authors (see above) are clearly prescient. Mention
should also be made of L. S. Stavrianos' *The Promise of the Coming Dark Age* as a work of deep historical insight in the same vein as Arrighi's but with focus on the future.

Given that there is evidence for both dissimilarity and similarity of the data over each 2500 year subset, a very simple fractal analysis was performed to see if the fractal dimension of each sub pattern shared any similarity. This was done by determining the total distance in theoretical space of the world system trajectory for each subset, determining the length of the line segment of the regression equation for each respective time period of 2500 years, that is the lengths of the solid lines in Fig. 21, taking the natural log-transform of each, and dividing the log-transformed distance of the actual trajectory by the log-transformed regression distance. This gives the fractal dimension, here labeled D, or more explicitly,

$$D = \ln\Sigma C_o/\ln C_R. \qquad (\text{Eq. 21})$$

C will be used to represent $\Sigma C_o$ to simplify the symbolism. The anti-log of Eq. 18 is $C = C_R^D$ (Eq. 19). When the dimension, D, was calculated for each subset of data, these values were respectively, $D = 1.4853$ for the first 2500 years, and $D = 1.4870$ for the second 2500 years. The difference between these two values is .0017 or approximately two one-thousands, quite slight. Unquestionably, these are important results, as they imply fractal similarity over different time periods of the same magnitude but with different histories, which further implies significant constraints on the trajectory of the world system.

As an extension of the metric used previously to represent world system trends, that is $C_{max}/C_{max}^{(\gamma + 1)/2}$, this section will investigate the relationship between the theoretical area determined by the actual values of $\ln C_{max}$ and $\ln \alpha$ and the maximum possible area determined by $[(\gamma + 1)/2]^2 \ln C_{max}^2$. This involves producing a ratio of $\{(\gamma \ln C_{max})^2/[(\gamma + 1)^2/4) \ln C_{max}^2]\}$, which simplifies to $C_{max}^{\gamma/[(2\gamma - \gamma^2 - 1)/2]}$ in the following way: Since $(\gamma \ln C_{max})^2$ simplifies to $C_{max}^{2\gamma}$, and since $[(\gamma+1)/2]^2 \ln C_{max}^2$ simplifies to $C_{max}^{\gamma(\gamma^2 + 2\gamma + 1)/2}$, then $C_{max}^{2\gamma}/C_{max}^{\gamma(\gamma^2 + 2\gamma + 1)/2} = C_{max}^{\gamma(2\gamma - (\gamma^2 + 2\gamma + 1)/2)}$ which in turn equals $C_{max}^{\gamma(2\gamma - \gamma^2 - 1)/2}$. Calculated values for this new metric were graphed on the y-axis against time in increments of 100 years over the past 5000 years of world system history to give Fig. 22 below. This graph represents two periods of fluctuation about a mean separated by a transition phase or phase change.

This period of phase change has been included within the bounds of both portions of the graph for convenience, but should be considered to span the time period, 700 BCE to 100 BCE, including what Karl Jaspers referred to as the Axial Age. If a linear regression is fitted to these graphed data the results are as represented in Fig. 23 below. It should be noted that the slope of the regression equation is quite small, $m = .00450$, in fact almost horizontal, and also that the actual fit of the data is quite good.
<figure><img src="image_47.png" /><figcaption>**Fig. 28.** Time in increments of 100 years for the 5000 years of world system history is represented on the x-axis, while values of $C_{max}^{(2\gamma - \gamma^2 - 1)/2}$ are represented on the y-axis. Note that this graph effectively has two portions, one from 3000 BCE to 500 BCE and the other from 500 BCE to 2000 CE. These portions or segments of the graph will be analysed separately later in this section</figcaption></figure>

<figure><img src="image_48.png" /><figcaption>**Fig. 29.**</figcaption></figure>

In this graph the data of the previous graph have been linearly regressed to produce the equation,

$$L = .00450t + .264, \qquad (\text{Eq. 22})$$
having r = .564, and having an RMSE = .0971, where L represents the expres-
sion, C<sub>max</sub>^((2γ - γ<sup>2</sup> - 1)/2. The implications of this graph are that, while the
correlation is acceptable but not strong, the RMSE, indicting goodness of fit is
quite good. Further, there appear to be two distinct periods of activity of the
world system, each spanning about 2500 years.

<figure><img src="image_49.png" /><figcaption>**Fig. 30.** This graph shows the relatively linear trend exhibited by the data represented in the previous graph and also the relatively close fit of the data to the regression line</figcaption></figure>

To place the graph in a different and more appropriate context to show its line-
arity, the y-axis values have been expanded to –2, 2 in Fig. 30 above. This fig-
ure shows both the linear trend of the data plotted in Figs 22 and 23 and the
closeness of fit of the data to the regression line. However, the representation of
the distinct phases of world system activity separated by a phase change is
blurred. What the above graph does reveal is the clear liner trend of world sys-
tem history over the last 5000 years, which is, again, an indication of con-
straints operating on the system.

What is also interesting is that the graph of the entire world system history
of the relationship represented by the term, C<sub>max</sub>^((2γ - γ<sup>2</sup> - 1)/2 does not require
natural log transformation, so that changes in early world system history, say,
during the first 1000 years or so, can be compared directly with any other por-
tion of the data plot. With this in mind, it should be noted that the system
change in the phase change noted previously represents an overlapping of sys-
tem activity in that the greatest y-values of the first 2500 years are greater that
the smallest y-values of the second 2500 years. What this may imply is that the
second phase change may represent a similar overlap and can be predicted to
extend to a y-value of approximately .85, which indicates that the world system
has reached a level of activity which is 85 % of its theoretical maximum, no
small consequence to contemplate.

Turning now to a representation of the individual periods, the first
2500 years of world system activity are represented in Fig. 25 and including the
regression line, show a slightly negative slope, m = -.00227, with a weak r =
= -.244, but a significant RMSE = .0677. The second 2500 years of world sys-
tem activity are represented in Fig. 26 with a slope, m = -2.22E-4, that is an order of magnitude less negative than that of the first and an RMSE = .0916 which is not significantly different from the RMSE of the first, that is RMSE = .0971, representing a difference of .0055 between the RMSE's of the two regressions.

What is significantly different between the two regressions are the y-intercept values, the first being .264 and the second being .470. Clearly, while the slopes of both regressions are almost horizontal, the intercepts are quite different and can be represented in the composite graph in Fig. 27. This is not unlike the graph in Fig. 21, however the difference in the current y-intercept values is .206, while that in Fig. 21 is .0611, and the difference in the differences is unquestionably a function of the analysis in Fig. 21 being done at a single dimension, while that in Fig. 27 involves a two-dimensional analysis.

<figure><img src="image_50.png" /><figcaption>**Fig. 31.** This graph is of $C_{max}^{(2\gamma - \gamma^2 - 1)/2}$ plotted over the first 2500 years of world system history. The important aspects of this graph are that the slope of the linear regression is almost zero and that the RMSE is quite small</figcaption></figure>

Further, the composite graph in Fig. 27 affirms the assertion made regarding the graph in Fig. 21, that of a distinctly different level of world system activity for the last 2500 years of world system history and also that the world system is in all probability in a phase transition to a new level of activity. This last assertion is based on the change in world system position for the last two positions represented in Fig. 26 which show a remarkable increase from 1800 CE, that is point 48 on the x-axis, to the year 2000 and also the assumption that
phase changes occur approximately every 2500 years with respect to the metric, $C_{max}^{\wedge}(2\gamma - \gamma^2 - 1)/2$.

<figure><img src="image_51.png" /><figcaption>**Fig. 32.** This graph represents $C_{max}^{\wedge}(2\gamma - \gamma^2 - 1)/2$ plotted over the second 2500 years of world system history, and, as in Fig. 27, has a very acceptable RMSE value and a slope approaching zero</figcaption></figure>

Further, the composite graph in Fig. 27 affirms the assertion made regarding the graph in Fig. 21, that of a distinctly different level of world system activity for the last 2500 years of world system history and also that the world system is in all probability in a phase transition to a new level of activity. This last assertion is based on the change in world system position for the last two positions represented in Fig. 26 which show a remarkable increase from 1800 CE, that is point 48 on the x-axis, to the year 2000 and also the assumption that phase changes occur approximately every 2500 years with respect to the metric, $C_{max}^{\wedge}(2\gamma - \gamma^2 - 1)/2$.

Of further importance is the considerable change in position of the world system at the 2500 year point. As it is represented in Fig. 27, it appears as a discontinuity in history, however, the reality is that continuous change, much of it over the Axial Age, brought about the new mode (and tempo) for the world system. One has only to take a look at Fig. 23 to see the nature of this phase transition, which is quite rapid, occurring over a period of 600 years, but hardly instantaneous! Exactly how this phase transition was brought about from a cliodynamic perspective should be the focus of future research.
<figure><img src="image_52.png" /><figcaption>**Fig. 33.** This graph represents the trends of $C_{max}^{\wedge}(2\gamma - \gamma^2 - 1)/2$ for both the first and second portions of world system history, both 2500 years in extent. Of note is the significant change in both position and slope of the second 2500 years</figcaption></figure>

A rudimentary fractal analysis was done for the trends of $C_{max}/C_{max}^{(\gamma + 1)/2}$ over all of world system history, and the same will be done for the metric $C_{max}^{\wedge}(2\gamma - \gamma^2 - 1)/2$ over the same period of time using the same symbolism, see Eq. 19. When D, the fractal dimension, was calculated for both the first and second 2500 year periods and also for the entire extent of world system history the values computed were all approximately the same and did not vary significantly from linearity, that is $D \sim 1$. More specifically, $D_1 = 1.0029$, $D_2 = 1.0036$, and $D_3 = 1.0059$. These values are effectively linear and suggest that the trajectory of the world system as represented by the regressions,

$$R_1 = .318 - .0027t, \qquad (\text{Eq. 23})$$

$$R^2 = .470 - 2.22E-4, \text{ and}$$

$$R = .00450t + .264, \qquad (\text{Eq. 24})$$

Of $C_{max}^{\wedge}(2\gamma - \gamma^2 - 1)/2$ over world system history, is dimension-limited and therefore a considerable constraint on the future course of that trajectory.

## The Relationship between Maximum Urban Area and World System Population over the Last 5000 Years

The graph of $\ln T$ against $\ln C_{max}$ is given in Fig. 34 on the following page. There is a clear positive trend to the plot of this graph, albeit with some dispersion of points. It should be noted that the actual space occupied by this graph is quite restricted with respect to the phase space that it resides in. In Fig. 35, the same graph but with a best fit line, the aspect of linearity is more clearly defined, and the equation for this line is:

$$\ln C_{max} = 1.0521\ln T - 6.8404 \qquad (\text{Eq. 25})$$
with an R² = .9171, in other words, the fit of this line is quite good. It is hardly earth-shattering that there is a positive relationship between increased urbanization and the increased magnitude of the world system population over time, but the ability to give an explicit description to this relationship has some considerable utility. Even though the trend of this natural log plot is unquestionably linear, there are some significant departures from linearity, and there are also other characteristics of this graph that bear commenting on.

For the sake of communication it will be best to consider the graphs in Figs 34 and 35 to be divided into three regions, the first from 16.45 to lnT = 18.42, the second from lnT = 18.42 to lnT = 20.52, and the third from lnT = 20.52 to the right-most extension of the graph, 22.55. In Region 1 there are clear early (and tight) oscillations having periods on the order of 100 years that give way to more extended oscillations. The significance of these oscillations is that an increase above the mean position of the trajectory, as defined by the regression line, implies increasing urbanization, while a downward trend implies decreasing urbanization.

<figure><img src="image_53.png" /><figcaption>**Fig. 34.** X-axis is lnT, and Y-axis is lnCmax. The trend is linear with some dispersion of points</figcaption></figure>

It could be argued that the relationship is only with maximum urban area, however, there is a clearly defined relationship between T and Cmax as represented by the equation: Cmax<sup>γ</sup> – Cmax – (γ – 1)T = 0 (Eq. 2). So, a change in the magnitude of Cmax also implies a change in the pattern of urbanization as reflected in the variable, γ.
<figure><img src="image_54.png" /><figcaption>**Fig. 35.** Axes same as in Fig. 1. The regressed line is represented by the equation: $\ln C_{max} = 1.0521\ln T - 6.8404$. $R^2 = .9171$</figcaption></figure>

Considering the first segment of the world system trajectory from 16.45 to 18.42, which spans the time from 3000 BCE to 500 BCE, the key characteristic of this segment is the oscillatory nature of the trajectory and, in particular, the relatively width of the oscillations and their frequency. It could be argued here, especially through ~17.2, the world system was equilibrating from the emergence of the first empires. The trajectory rises quite rapidly through 17.6, a point on the graph representing the end of the Late Bronze Age, which is followed by a significant down turn, a mild rise and plateau, another slight down turn and then a rise to the end of this period. What characterizes this first half of the trajectory with respect to time, that is 2500 years, is apparent rapid urbanization which appears to overshoot the carrying capacity for that degree of urbanization and is consequently followed by decline. This segment is followed by a segment representing the next 2300 years of world system history.

The second segment of the world system trajectory is punctuated by two prominent events. The first of these begins with a very rapid phase of urbanization accompanied by stable or slightly decreasing world system population figures and begins in Figs 1 and 2 around lnT ~ 19.0. A maximum is reached after 300 years, followed by a slight decline and then a second maximum, which spans 300 years from 100 BCE to 200 CE. After this a second period of relative world system population stasis accompanies a decline in urbanization, spanning some 500 years. The total time period from the beginning of increased urbanization to the end of decreasing urbanization amounts to 1100 years. The second, although a temporally briefer event, represents a total excursion of 200 years from 1200 CE to 1400 CE, during which the maximum urban area population swung from 1.0 million to 1.5 million and then back again to
1.0 million with a slight decrease with world system population from 360 million at 1300 CE to 350 million at 1400 CE, due in large part to the ravages of pandemic plague.

The third segment extending from lnT = 20.52 at 1800 CE to 22.55 at 2000 CE represents similar change in the magnitude of the world system population but over a much shorter period of time, only 200 years. This massive excursion over a much shorter time period is unquestionably a product of the industrial revolution but more deeply a consequence of hyperbolic population growth (Korotayev et al. 2006a), which is itself a result of cooperative human interaction. If the ratio, $\Delta\ln T/\Delta t$, for each period, the values tell an interesting story. The values for the first two periods are respectively, 7.88E-4 and 9.22E-4, both within the same order of magnitude. However, the value for the third period is, $\Delta\ln T/\Delta t$ = 1E-2, two orders of magnitude greater than either of the two previous periods and represents a rate of change of the world system that is unique in world system history. Further, during this period of rapid change there are no significant oscillations, no evidence of overshoot soon to be followed by rapid deurbanization. A caveat here, this present analysis documents change but does not ascribe a cost, either environmentally or economically, to that change, consequently, the footprint of the last two centuries, either in carbon or some energy unit, is not apparent. In perusing the Fig. 34, there is only one portion of that plot that bears any resemblance to this third segment and that is the portion from lnT = 18.42 to 18.90, that is from 400 BCE to 100 BCE, which was succeeded by a period of very rapid urbanization.

If attention is now turned to Eq. 1, it will be seen that the fit is quite good with respect to correlation, $R^2$ = .9171, and the world system as reflected in a graph of $\ln C_{max}$ v. lnT has a clear primary direction that is both linear and positive. What are the implications of this positive linearity? First, it appears that each variable depends strongly on the other. It is important to emphasize that a large world system population is not simply a result of a large urban area population, just as it is important to avoid stating the opposite, that a large urban area is not a consequence of a large world system population. Both are interdependent on the other, and the fact that lnT is represented on the y-axis and not the x-axis is a result of choice by the author to emphasize the rapid changes in urbanization apparent in the second section of the graph. Second, the positive trend of the trajectory suggests that the near future state of the trajectory will in a general way continue to be positive. If the previous pattern of this trend is considered, specifically with respect to relatively significant positive change in lnT, then the section of the graph immediately prior to lnT = 18.90, that is at 400 BCE when T ~ 162E6, may be considered a reasonable analogue of the last 200 years of the World System trajectory. Following this period of increased lnT, was, as described previously, a period of rapid urbanization, followed by a period of relative stasis, and then rapid deurbanization. May the same be ex-
pected in the near future, that is relative stasis of lnT but a rapid increase in urbanization as characterized both by increasing lnC<sub>max</sub> and decreasing magnitude of γ. From my perspective, this is a distinct possibility, as the rate at which the world system population is growing has been declining since the early 1960's, which mirrors the relative stasis of the population between 400 BCE and 100 BCE.

## A Comparison of the Time Series of Both the Natural Log Transform of World System Population (lnT) and the Natural Log Transform of Maximum Urban Area (lnC<sub>max</sub>)

When the plot of points represented in both Figs 34 and 35 is decomposed into two time series plotted on the same axes (see Fig. 3), a shared characteristic of both plots becomes apparent. They are approximately parallel to each other separated by an approximately constant distance. However, the upper plot, that of lnT, is far smoother with an almost linear component over the first 2500 years of world system history, a slight increase immediately after 500 BCE followed by another relatively linear component ending about 1000 years ago, which was in turn followed by slight oscillations over the next 800 years, and terminated by an abrupt upturn representing the consequences of the industrial and post-industrial eras. When these data are plotted on rectangular axes, the classic hyperbolic growth curve as noted by von Forester et al. (1960) and by Korotayev et al. (2006a) is produced. Without the advent of the Industrial Revolution the trajectory of the world system would be almost linear, reflecting exponential growth. Second, the positive linearity of these two variables suggests that the system as a whole will continue to move in a positive direction, which of course suggests constraint on the direction of the trajectory and also suggests the potential for prediction of the future position of the trajectory.

Linear regressions of each plot give respectively,

$$ \ln T = 16.2 - 0.0868t \qquad (\text{Eq. 26}) $$

and

$$ \ln C_{\max} = 10.3 - 0.0904t \qquad (\text{Eq. 27}) $$

As can be seen by comparing values at t = 0, the equivalent of 3000 BCE and t = 2000 CE, the difference in regressed values at t = 0 is 5.9 and that at t = 50 is 6.08, so, as a general trend the difference has diverged over time. The implication being that a larger portion of the world system population lives outside the largest urban area, and in fact it could be restated as the largest urban areas, if the observed difference matches the expected difference. This, however, is not the case. The observed value of ln[T/C<sub>max</sub>] is 5.6000, which of course implies that a greater proportion of the world system population than expected due to regressed values resides in large urban areas.
<figure><img src="image_55.png" /><figcaption>**Fig. 36.** Time is represented on the x-axis as a multiple of 100 years. The natural log transform of population size is represented on the y-axis with the top plot being that of lnT, the total population of the world system through the last 5000 years and the bottom plot being the plot of the natural log transform of maximum urban size, lnCmax, over the same period of time. The essentially parallel trajectories of both plots should be quite apparent, suggesting a very clear relationship between the two.</figcaption></figure>

<figure><img src="image_56.png" /><figcaption>**Fig. 37.** The axes are the same as in Fig. 3, however each trajectory has been fitted by linear regression. The equation for the top regression is, lnT = .0826t + 16.1681, and that of the lower regression is, lnCmax = .0904t + 10.2681, where lower case 't' = time, again in units times 100 years. The parallels nature of both plots is even more apparent in Fig. 4</figcaption></figure>
If the values of γ are compared at each of the times, 1.4851 at t = 0 and 1.2490 at t = 50, it can be seen that a larger proportion of the total population would be expected to reside in (relatively) large urban areas.

## A Model of the Difference between lnT and lnC<sub>max</sub>

This parallel trend is relatively easy to model mathematically. Inspection of Figs 3 and 4 will show that lnT – lnC<sub>max</sub> ~ 5.9, or more explicitly, lnT – lnC<sub>max</sub> = 5.9 – .0078t, where t = time. The anti-log transform of this equation is

$$ \frac{T}{C_{\max}} = e^{5.9 - 0.0078t} \qquad (\text{Eq. 28}) $$

or by rearrangement,

$$ T = C_{\max}e^{5.9 - 0.0078t} \qquad (\text{Eq. 29}) $$

While this is an empirically based model, an analytical model can also be derived. Recall Eq. 2, C<sub>max</sub><sup>γ</sup> – C<sub>max</sub> – (γ – 1)T = 0. It is not too difficult to show that

$$ \frac{T}{C_{\max}} = \frac{(C_{\max}^{\gamma} - 1)}{(\gamma - 1)} \qquad (\text{Eq. 30}) $$

(see Mathematical Appendix) and therefore,

$$ \ln T - \ln C_{\max} = \ln(C_{\max}^{\gamma} - 1)/(\gamma - 1). \qquad (\text{Eq. 31}) $$

If the natural log-transformed right hand sides of Eqs 5 and 6 are equated, we have,

$$ 5.9 - 0.0078t = \ln\left[\frac{(C_{\max}^{\gamma} - 1)}{(\gamma - 1)}\right], \qquad (\text{Eq. 32}) $$

and the left and right hand sides of this last equation, representing observed and expected values of the world system, can be regressed against one another (see Fig. 40).

<figure><img src="image_57.png" /><figcaption>**Fig. 38.** Observed v. Expected re: ln[T/C<sub>max</sub>] = ln[C<sub>max</sub><sup>γ-1</sup> – 1]/[γ – 1], where the left-hand side of the equation represents the observed, which is calculated from data on both T and C<sub>max</sub>, while the right-hand side of the equation is used to predict the left-hand side</figcaption></figure>
It should be noted here that the above relationship is not unexpected, as Eq. 2 was used to generate values of $\gamma$ with observed values of both lnT and lnCmax. However, it should also be noted that the left-hand side of Eq. 32 represents an alternative way of finding ln[T/Cmax], one in which the fit is quite good.

Interestingly, the essentially parallel trend of lnT and lnCmax over the last 5000 years suggests that there is a clear link between urbanization and total population growth. This in one sense is unsurprising, however, the contention here and in previous research is that link is in fact $\gamma$, the exponent characterizing the distribution of urban size given maximum urban size and total world system population.

## The Potential for Prediction

The previous four sections of this paper have addressed the notion of constraints imposed on the trajectory, the historical trajectory, of the world system over the last 5000 years. A clear general trend with respect to the relationship between $\gamma$ and lnT has been demonstrated, one that shows a decrease in $\gamma$ and therefore an increase in the total distribution of the urban population, here defined as $N \ge 100$, as the natural logarithm of the world system population and therefore the population itself increases. It has been shown that there are specific instances of change in the world system position in reaction to previous system change, e.g., the rapid urbanization of the world system from 400 BCE to 100 BCE followed by a rapid de-urbanization from 200 CE to 500 CE, that suggest that the system is being returned to some steady-state level. The existence of iso-urban lines has been established, lines of similar maximum urban area magnitude with respect to both $\gamma$ and lnT, and the polar plot of the world system trajectory, the rather tight linearity between H and $\Psi$, H and time and $\Psi$ and time, the periodicity between the observed and expected of the last two linear relationships, and the ratio of the observed maximum urban area value and the idealized one determined by maximizing Eq. 12a, $A = (\gamma - x)(1 + x)$. $5\ln C_{max}^2$, which reveals that the urbanization of the world system appears to have occurred in two specific regimes, each with a similar rate of occurrence but with differing points of initiation, and which also demonstrates a fractal relationship for the two regimes that is effectively identical. All of these instances suggest that the world system is highly constrained, and, in turn, if this is in fact the case, that the behavior of the system, at least at the level of organization implied by the equation, $C_{max}^{\gamma} - C_{max} - (\gamma - 1)T = 0$ (Harper 2010b), is (potentially) predictable.

Each of the above constraints will now be given more detailed analysis and explanation and then some discussion will follow with respect to the potential for prediction both in the past, that is retroactively, and for the system as a whole.

To begin, the fact that the linear relationship between $\gamma$ and lnT has a negative slope suggests in general that for any given position of the world system
future positions from that point on will involve both a decrease in γ and an increase in lnT, and over an extended period of time this is unquestionably the case. For example, the position of the world system in 200 CE is one in which the value of γ has decreased as lnT has increased with respect to the world system position of, say, 2000 BCE. However, if the position of the world system from 200 CE to 300 CE is considered, this is not the case, as there the change involves an increase in γ with no attendant decrease in lnT. In Table 6 below are documented all the changes in both γ and lnT and what this data reveals is that while two relationships predominate, there are several other arrangements between γ and lnT.

<table><caption>Table 6</caption><thead><tr><th>γ/lnT</th><th>+/+</th><th>+/0</th><th>+/-</th><th>-/-</th><th>-/0</th><th>-/+</th></tr></thead><tbody><tr><th>Number</th><td>22</td><td>3</td><td>1</td><td>2</td><td>1</td><td>21</td></tr></tbody></table>

As is represented above on a century by century basis then depending where one starts the actual trend of change in γ with respect to lnT may fall into one of six categories.

What is to be made then of the negative slope of the equation

$$ \gamma = 2.23 - .0449\ln T? \qquad (\text{Eq. 33}) $$

This unquestionably has to be an emergent property of the system itself and as such is only of general predictive value. Of more importance is the fact that the dispersion of points about the trend line is that the greatest variance is approximated by ~.1γ. This fact along with the negative slope have some predictive worth in that bounds are placed on future changes in position of the world system from any previous point. A note of caution here: There are at least two instances in which lnT, and therefore T, the population itself, decrease, the first during the period of intense urbanization from 300 BCE to 200 BCE and the second from 1300 CE to 1400 CE, the (calamitous) 14th century, and these population deficits were quite small.

It is to these two specific instances, the period of rapid urbanization at the end of the first millennium BCE and the 14th century and the periods of adjustment that followed them, that attention will now be given. Both of these periods of time share the same pattern of development and response, that is both represent periods of intense urbanization directly associated with little or no negative change in total population followed by rapid de-urbanization, in the case of the first period, punctuated by a brief period of de-urbanization followed by a short period of urbanization. It appears then that rapid urbanization constrained by zero world system population growth results in equally rapid de-urbanization, either directly or followed by some relatively short period of time.

To give a numerical explanation for the above phenomenon Eq. 23 from Harper (2010b) was used to calculate the ratio of urbanized population to that of the rural population for the total population. The data are given in Table 7 below.