A Compact Macromodel of World System Evolution

journal of world-systems research, xi, , july , – http://jwsr.ucr.edu/ issn 1076–156x © 2005 Andrey Korotayev The fact that up to the 1960s1 world population growth had been characterized by a hyperbolic2 trend was discovered quite some time ago (see, e.g., von Foerster, Mora, and Amiot 1960; von Hoerner 1975; Kremer 1993; Kapitza 1992, 1999). A number of mathematical models describing this trend have already been proposed (see, besides the above references, Cohen 1995; Johansen and Sornette 2001; Tsirel 2004; Podlazov 2004). Some of these models (e.g. von Foerster, Mora, and Amiot 1960; or Kapitza 1992) are rather compact but do not account for the mechanisms of this trend; some others (fi rst of all Kremer 1993) account for this trend in a very convincing way, but are rather complex. In fact, the general shape of world population growth dynamics could be accounted for with strikingly simple models like the one which we would like to propose ourselves below (or the one proposed by Tsirel [2004]). With Kremer (1993), Komlos and Nefedov (2002), and others (Habakkuk 1953; Postan 1950, 1972; Braudel 1973; Abel 1974, 1980; Cameron 1989; Artzrouni and Komlos 1985), we make “the Malthusian (1978) assumption that popula-

The fact that up to the 1960s world population growth had been characterized by a hyperbolic trend was discovered quite some time ago.A number of mathematical models describing this trend have already been proposed.Some of these models are rather compact but do not account for the mechanisms of this trend; others account for this trend in a very convincing way, but are rather complex.In fact, the general shape of world population growth dynamics could be accounted for with strikingly simple models like the one which we would like to propose ourselves: dN/dt = a (bK -N) N (1); dK/dt = cNK (2), where N is the world population, K is the level of technology/knowledge, bK corresponds to the number of people (N), which the earth can support with the given level of technology (K).Empirical tests performed by us suggest that the proposed set of two differential equations account for 96.2-99.78 of all the variation in demographic macrodynamics of the world in the last 12,000 years.We believe that the patterns observed in pre-modern world population growth are not coincidental at all.In fact, they reflect population dynamics of quite a real entity, the world system.Note that the presence of a more or less well integrated world system comprising most of the world population is a necessary pre-condition, without which the correlation between the world population numbers generated by hyperbolic growth models and the observed ones would not be especially high.In fact, our findings could be regarded as a striking illustration of the fact well known in complexity studies -that chaotic dynamics at the microlevel can generate a highly deterministic macrolevel behavior.Against this background it is hardly surprising to find that the simplest regularities accounting for extremely high proportions of all the macrovariation can be found just for the largest possible social system -the world system.

abstract:
¹. Actually, up to /.After -, as a result of the world demographic transition the actual world population dynamics began to diverge more and more from the hyperbolic curve, and by the present moment world population growth rates have declined dramatically as compared with .It appears possible to develop a mathematical model describing both pre- hyperbolic growth and the subsequent global demographic transition; however, this would go out of the scope of the present article.
². Hyperbolic population growth implies that absolute population growth is proportional to the square of population (unlike exponential growth in which absolute tion is limited by the available technology, so that the growth rate of population is proportional to the growth rate of technology" (Kremer 1993: 681-2),³ and that, on the other hand, "high population spurs technological change because it increases the number of potential inventors…⁴ In a larger population there will be proportionally more people lucky or smart enough to come up with new ideas"⁵ (Kremer 1993: 685), thus, "the growth rate of technology is proportional to total population"⁶ (Kremer 1993: 682;see also, e.g., Kuznets 1960;Grossman and Helpman 1991;Aghion andHowitt 1992, 1998;Simon 1977Simon , 1981Simon , 2000;;Komlos and Nefedov 2002).
Th e simplest way to model mathematically the relationships between these two subsystems (which, given the current state of our knowledge, has not been proposed yet) is to use the following set of diff erential equations: Where N is the world population, K is the level of technology/knowledge, bK corresponds to the number of people (N), which the earth can support with the given level of technology (K).
With such a compact model we are able to reproduce rather well the longrun hyperbolic growth of world population before 1962-3.
With our two-equation model we start our fi rst simulation in the year 1650 and do annual iterations with diff erence equations derived from the diff erential ones: We choose the following values of the constants and initial conditions: N = 0.0545 of tens of billions (i.e.545 million);⁷ a = 1; b = 1; K = 0.0545;⁸ c = .05135.⁹Th e outcome of the simulation, presented in Figures 1-2 indicates that irrespective of all its simplicity the model is actually capable of replicating quite reasonably the population estimates of Kremer (1993), the US Bureau of the Census ( 2004), and other sources (Th omlinson 1975;Durand 1977;McEvedy and Jones 1978: 342-51;Biraben 1980;Haub 1995: 5;UN Population Division 2004;World Bank 2004) in most of their characteristics (see Figure 1) Th e correlation between the predicted and observed values for this imitation looks as follows: R = .9989,R² = .9978,p << .0001,which, of course, indicates an unusually high fi t for such a simple macromodel designed to account for the demographic macrodynamics of the most complex social system (see Figure 2).
We start our second simulation in the year 500 bce.In this case we choose the following values of the constants and initial conditions: N = 0.01 of tens of growth is lineally proportional to population).Th us, with exponential growth, if at the world population level of  million the absolute annual growth was , people a year, at the  billion level it will be  million people each year (the tenfold growth of population leads to the same tenfold increase in the absolute population growth rate).With hyperbolic growth, if at the world population level of  million the absolute annual growth was , people a year, at the  billion level it will be  million people a year (the tenfold growth of population leads to the -fold increase in the absolute population growth rate).Note that the relative population growth rate will remain constant with exponential growth (. in our example), whereas it will be lineally proportional to the absolute population level with hyperbolic growth (in our example, population growth by a factor of  leads to the tenfold increase in the relative annual growth rate, from . to ).
³.In addition to this, the absolute growth rate is proportional to population itselfwith the given relative growth rate a larger population will increase more in absolute numbers than a smaller one.
⁴. "Th is implication fl ows naturally from the nonrivalry of technology….Th e cost of inventing a new technology is independent of the number of people who use it.Th us, holding constant the share of resources devoted to research, an increase in population leads to an increase in technological change" (Kremer : ).
⁵. Th e second assumption is in fact Boserupian rather than Malthusian (Boserup ; Lee ).
⁶.Note that "the growth rate of technology" means here relative growth rate (that is to which level the technology will grow in the given unit of time in proportion to the level observed at the beginning of this period).Th is, of course, implies that the absolute speed of technological growth in the given period of time will be proportional not only to the population size, but also to the absolute level of technology at the beginning of this period.
⁷.We chose to calculate world population in tens of billions (rather than, say, in millions) to minimize the rounding error (which was to be taken most seriously into account in our case, as the object of modeling had evident characteristics of a blow-up regime).
⁸.To simplify the calculations we chose value "" for both a and b; thus, K in our simulations was measured directly as the number of people which can be supported by the Earth with the given level of technology (K), and the population was allowed to adjust almost instantaneously to the growth of the Earth' s carrying capacity.
⁹.Given the initial values of N and K, here (as well as in the subsequent simulations) we chose constant c in such a way as to minimize the sum of squared residuals between the observed values and those predicted by our model.

82
A Compact Macromodel of World System Evolution  Th us, it turns out that the set of two diff erential equations specifi ed above accounts for 96.2 of all the variation in the demographic macrodynamics of the world in the last 25 millennia; it also accounts for 99.66 of this macrovariation in 500 bce-1962 ce, and for 99.78 in 1650-1962 ce.
In fact, we believe this may not be a coincidence that the compact macromodel shows such a high correlation between the predicted and observed data just for 500 bce-1962 ce.But why does the correlation signifi cantly decline if the pre-500 bce period is taken into account?
To start with, when we fi rst encountered models of world population growth, we felt a strong suspicion about them.Indeed, such models imply that the world population can be treated as a system.However, at a certain level of analysis one may doubt if this makes any sense whatsoever.For up until recently (especially before 1492) humankind did not constitute any real system, as, for example, the growth of the populations of the Old World, the New World, Australia, Tasmania, or Hawaii took place almost perfectly independently from each other.It seems quite clear, for example, that demographic processes in, say, West Eurasia billions (i.e. 100 million); a = 1; b = 1; K = 0.01; c = 0.04093.Th e outcome of the simulation, presented in Figures 3-4 indicates that irrespective of all its simplicity the model is still quite capable of replicating rather reasonably the population estimates of Kremer (1993), US Bureau of the Census ( 2004) and other sources in most of their characteristics and in terms of the important turning points even for such a long period of time (see Figures 3 and 4).
Th e correlation between the predicted and observed values for this imitation looks as follows: R = .9983,R² = .9966,p << .0001,which, of course, again indicates an unusually high fi t for such a simple macromodel designed to account for demographic macrodynamics of the most complex social system for c. 2500 years (see Figure 4).
Note that even the simulation started c. 25000 bce still produced a fi t with observed data as high as .981(R² = .962,p << 0.0001).¹⁰  in the fi rst millennium ce did not have the slightest possible impact on the ones in Tasmania in the same time period.However, we believe that the patterns observed in pre-modern world population growth are not coincidental at all.In fact, they refl ect population dynamics of a very real entity, the world system.We are inclined to speak together with Andre Gunder Frank (e.g., Frank and Gills 1994; but not with Wallerstein 1974) about the single world system, which originated long before the "long sixteenth century." Note that the presence of a more or less well integrated world system comprising most of the world' s population is a necessary pre-condition, without which the correlation between world population numbers generated by our model and the observed ones would not be particularly high.For example, suppose we encounter a case where the world population of N grew fourfold but was split into four perfectly isolated regional populations comprising N persons each.Of course, our model predicts that a fourfold increase in world population would tend to lead to a fourfold increase in the relative world technological growth rate.But have we any grounds to expect to fi nd this in the case specifi ed above?Of course not.Yes, even in this case four times the number of people will likely produce four times more innovations.However, the eff ect predicted by our model would be observed only if innovations produced by any of the four regional populations were shared among all the other populations.However, we assumed that the four respective populations lived in perfect isolation from each other.Hence, as such a sharing would not take place, and the expected increase in the technological growth rate would not be observed, which would produce a huge gap between the predictions generated by our model and the actually observed data.
It seems that it was precisely the fi rst millennium bce when the world system integration reached a qualitatively new level.A strong symptom of this seems to be the "Iron Revolution," as a result of which iron metallurgy spread within a few centuries (not millennia!)throughout a huge space stretching from the Atlantic to the Pacifi c, producing (as was already supposed by Jaspers [1953]) a number of important unidirectional transformations in all the main centers of the emerging world system (the Circummediterranean region, the Middle East, South Asia,  and East Asia), after which the development of each of these centers cannot be adequately understood, described and modeled without taking into consideration the fact that they were part of a larger and perfectly real whole-the world system.
A few other points seem to be relevant here.Of course, there would be no grounds to speak about the world system stretching from the Atlantic to the Pacifi c even at the beginning of the fi rst Millennium ce if we applied the "bulkgood" criterion suggested by Wallerstein (1974), as there was no movement of bulk goods at all between, say, China and Europe at this time (as we have no grounds not to agree with Wallerstein in his classifi cation of fi rst century Chinese silk reaching Europe as a luxury, rather than a bulk good).However, the fi rst century ce (and even the fi rst millennium bce) world system would be defi nitely qualifi ed as such if we were to apply a "softer" information network criterion suggested by Chase-Dunn and Hall (1997).Note that at our level of analysis the presence of an information network covering the whole of the world system is a perfectly suffi cient condition, which makes it possible to consider this system as a single evolving entity.In the fi rst millennium bce no bulk goods could be transported from the Pacifi c coast of Eurasia to its Atlantic coast.However, the world system had reached, by that time, such a level of integration that iron metallurgy could spread across the whole of Eurasia within a few centuries.
Th e other point is that even in the fi rst century ce the world system still covered far less than 50 of all the terrain of the Earth.However, what seems to be far more important is that already by the beginning of the fi rst century ce more than 90 of total world population lived just in those regions, which were constituent parts of the fi rst century ce world system (the Circummediterranean region, the Middle East, and South, Central and East Asia) (see, e.g., Durand 1977: 256).Hence, since the fi rst millennium bce the dynamics of world population refl ects very closely the dynamics of the world system population.
On the one hand, it might not be coincidental that the hyperbolic growth trend may still be traced back to 25000 bce.Of course, I am not going to insist on the existence of anything like the world system, say, around 15000 bp.Note, however, that there does not seem to be any evidence for hyperbolic world population growth in 40000-10000 bce.In fact the hyperbolic eff ect within the 25 millennia bce is produced by the world population dynamics in the last 10 millennia of this period that fi ts the mathematical model specifi ed above rather well (though not as well as the world population dynamics in 500 bce-1962 ce [let alone 1650-1962 ce]).
Th e simulation for 10000-500 bce was done with the following constants and initial conditions: N = 0.0004 of tens of billions (i.e. 4 million); a = 1; b = 1; K = 0.0004; c = 0.32.Th e outcome of the simulation, presented in Figure 5 indicates that the model is still quite capable of replicating rather reasonably the population estimates of McEvedy and Jones (1978) and Kremer (1993) for the 10000-500 bce period (see Figure 5).
Th e correlation between the predicted and observed values for this imitation looks as follows: R = .982,R² = .964,p = .0001.Note that, though this correlation for 10000-500 bce remains rather high, it is substantially weaker¹¹ than the one observed above for the 500 bce-1962 ce period and, especially, for 1650-1962 ce (in fact this is visible quite clearly even without special statistical analysis in Figures 1, 3, and 5).On the one hand, this result could hardly be regarded as surprising, because it appears evident that in 10000-500 bce the world system was much less tightly integrated than in 500 bce-1962 ce (let alone in Millions (10000 -500 bce) Note: The solid grey curve has been generated by the model; black markers correspond to the estimates of world population by McEvedy and Jones (1978) and Kremer (1993).
¹¹ Note, however, that even for - bce our hyperbolic growth model still demonstrates a much higher fi t with the observed data than, for example, the best-fi t exponential model (R² = .,p = .).

Andrey Korotayev 88
A Compact Macromodel of World System Evolution  to fi nd the evolving unit.Th e basic idea was that it is impossible to account for the evolution of a single society without taking into consideration that it was a part of a larger whole.However, traditional world-systems analysis concentrated on bulk-good movements, and core-periphery exploitation, thoroughly neglecting the above-mentioned dimension.However, the information network turns out to be the oldest mechanism of world system integration, and remained extremely important throughout its whole history, remaining as important up to the present.It seems to be even more important than core-periphery relations of exploitation (for example, without taking this mechanism into consideration it appears impossible to account for such things as, the demographic explosion in the 20t century, whose proximate cause was the dramatic decline of mortality, but whose main ultimate cause was the diff usion of innovations produced almost exclusively within the world system core).Th is also suggests a redefi nition of the world system (ws) core.Th e core is not the ws zone, which exploits other zones, but rather the ws core is the zone with the highest innovation donor/recipient (D/R) ratio, the principal innovation donor.¹²Th us, it turns out that the sets of two diff erential equations specifi ed above account for 96.2-99.78 of all the variation in demographic macrodynamics of the world in the last 12,000 years.Indeed, our fi ndings could be regarded as a striking illustration of the fact well known in complexity studies -namely, that in 1650-1962 ce).What seems more remarkable is that for 10000-500 bce the best fi t is achieved with a substantially diff erent value of the coeffi cient c, which appears to indicate that the world system development pattern in the pre-500 bce epoch was substantially diff erent from the one observed in the 500 bce-1962 ce era, and thus implies a radical transformation of the world system in the fi rst millennium bce.
We believe that, among other things, the compact macromodel analysis seems to suggest a rather novel approach to world system analysis.Th e hyperbolic trend observed for world population growth after 10000 bce appears to be mostly a product of the growth of the world system, which seems to have originated in West Asia around that time in direct connection with the Neolithic Revolution.Th e presence of the hyperbolic trend indicates that the major part of the entity in question had some systemic unity, and, we believe we have evidence for this unity.Indeed, we have evidence for the systematic spread of major innovations (domesticated cereals, cattle, sheep, goats, horses, the plow, the wheel, copper, bronze, and later iron technology, and so on) throughout the whole North African-Eurasian Oikumene for a few millennia bce.As a result, the evolution of societies in this part of the world already at this time cannot be regarded as truly independent.By the end of the fi rst millennium bce we observe a belt of cultures stretching from the Atlantic to the Pacifi c with an astonishingly similar level of cultural complexity based on agriculture involving production of wheat and other specifi c cereals, cattle, sheep, goats, the plow, iron metallurgy, professional armies with rather similar weapons, cavalries, developed bureaucracies, and so on -this list could be extended for pages.A few millennia before we would fi nd a belt of societies with a similarly strikingly close level and character of cultural complexity stretching from the Balkans to the Indus Valley (note that in both cases the respective entities included the major part of the contemporary world population).We would interpret this as a tangible result of the functioning of the world system.Th e alternative explanations would involve a sort of miraculous scenario -that the cultures with strikingly similar levels and characters of complexity somehow developed independently from each other in a very large but continuous zone, whereas nothing like that appeared in the other parts of the world, which were not parts of the world system.We fi nd such an alternative explanation highly implausible.
It could be suggested that within this new approach the main emphasis should be moved to innovation generation and diff usion.If a society borrows systematically important technological innovations, its evolution already cannot be considered as really independent, but should rather be considered as a part of a larger evolving entity, within which such innovations are systematically produced and diff used.Th e main idea of a world-systems approach was ¹².Earlier we regarded an "information network" as a suffi cient condition to consider the entity covered by it as a "world-system."However, some examples seem to be rather telling in this respect.For example, Gudmund Hatt (: ) found evidence on not less than  Japanese ships accidentally brought by the Kurosio and North Pacifi c currents to the New World coast between  and .Against this background it appears remarkable that the "Japanese [mythology] hardly contains any motifs that are not found in America (which was noticed by Levi-Strauss long ago)" (Berezkin : -).Already this fact makes it impossible to exclude entirely the possibility of some information fi nding its way to the New World from the Old World in the pre-Columbian era, information that could even infl uence the evolution of some Amerindian mythologies.However, we do not think this is suffi cient to consider the New World as a part of the pre-Columbian world system.Th e Japanese might have even told Amerindians about such wonderful animals as horses, or cows (and some scholars even claim that a few pre-Columbian Amerindian images depict Old World animals [von Heine-Geldern ; Kazankov ]), the Japanese fi shermen might even have had some idea of, say, horse breeding; but all such information would be entirely useless without some specifi c matter-actual horses.Hence, we would rather denote respective "system-creating" networks as "innovation diff usion networks" rather than just "information networks."

Andrey Korotayev 90
A Compact Macromodel of World System Evolution  chaotic dynamics at the microlevel can generate a highly deterministic macrolevel behavior (e.g.Chernavskij 2004).
To describe the behavior of a few dozen gas molecules in a closed vessel we need very complex mathematical models, which will remain incapable of predicting long-run dynamics of such a system due to an inevitably irreducible chaotic component.However, the behavior of zillions of gas molecules can be described with extremely simple sets of equations, which are capable of predicting almost perfectly the macrodynamics of all the basic parameters (and just because of chaotic behavior at the microlevel).
It appears that a similar set of regularities is observed in the human world too.To predict demographic behavior of a concrete family we would need extremely complex mathematical models, which would still predict a very small fraction of actual variation just due to inevitable irreducible chaotic components.For systems including large numbers of people (cities, states, civilizations) we would need simpler mathematical models having much higher predictive capacity.Against this background it is hardly surprising to fi nd that the simplest regularities accounting for extremely high proportions of all the macrovariation can be found just for the largest possible social system -the world system.
Th is, of course, suggests a novel approach to the formation of the general theory of social macroevolution.Th e approach prevalent in social evolutionism is based on the assumption that evolutionary regularities of simple systems are signifi cantly simpler than the ones characteristic for complex systems.A rather logical outcome from this almost self-evident assumption is that one should study fi rst evolutionary regularities of simple systems and only after understanding them move on to more complex ones.¹³We believe this approach was somehow misleading and led to an almost total disenchantment in the evolutionary approach in the social sciences all together.¹⁴

Figure 1 -
Figure 1 -Predicted and Observed Dynamics of World Population Growth, in Millions (1650-1962 ce)

Figure 3 -
Figure 3 -Predicted and Observed Dynamics of World Population Growth,

Figure
Figure 4 -Correlation between Predicted and Observed Values

Figure 5 -
Figure 5 -Predicted and Observed Dynamics of World Population Growth,