Q-exponential City Size Hierarchies, 430 BCE to 1950, Nomadism, and Population Fluctuations1
Douglas R. White,2 Céline Rozenblat,3 and(?) Nataša Keyžar4
Our principal interests here are first to determine city size distributions for the period 430 BCE to 1950
and whether these distributions are variable or constant (scale-free), and, second, to trace the
fluctuations of population sizes. Secondarily, are there biases evident in pre-1800 city population survey
data? Would these make significant alterations in findings relevant to our principal interest?
One of the crucial historiographic problems is whether city size hierarchies, as evidenced by superlinear
size distributions, are found in world civilizations prior to the industrial revolution. The question is
especially difficult given the rarity of census data prior to 1800. We are fortunate to have a database of
city sizes with which to provide initial tests of hypotheses. Although city sizes over 200K show some
possible evidence of a break in these data for world cities after 1800, which display the well-known by
superlinear distributions in seven contiguous periods, and those before, which might differ in having
linear distributions that characterize constant average group rates at all levels, i.e., these may lack
superlinear size distributions. We show, however, that distributions with different distributions at some
historical breakpoint (e.g., pre- and post-1800) entail a discontinuity in the total world population before
and after the break, which is not supported by the available evidence. Thus, urban size distributions must
have distributions that match the historical continuities in world population.
When we take all the available data for to city sizes over 50K (or even down in some periods where data
are available down to 35K or lower), the problem is resolved: Comparisons among these distributions
show relative uniformity, but these broader-range curves fall somewhere between superlinear and
linear/exponential curves in their functional form. (These similarities also argue for the absence of
significant biases for the pre-1800 period.)
The problems we must address, then, are these: What is the nature of the city size hierarchies and why
would we find distributions that are neither superlinear nor linear/exponential, but something in
between? What kind of distributions these might be? We argue from first principles that there are qexponentials that occur in network growth processes, and we fit each of our 29 datasets for successive
historial periods to the q-exponential parameters, back to 430BCE.
Methods
Céline Rosenblat provided the data and will explain how it was acquired. Fig. 1 shows the outcome of
the coding procedure using the processes described here. Each of the 29 periods 430 BCE to 1950 was
described in a text file with columns for city names, empires, city population, latitude and longitude.
1
This paper is the outcome of a collaboration under the auspices of EU project ISCOM, Information Society as a Complex
System.
2
Santa Fe Institute and Institute for Mathematical Behavioral Sciences, UC Irvine. drwhite@uci.edu.
3
Université de Montpelier, Maison de la Géographie. Celine.rozenblat@mgm.fr http://mgm.fr.
4
Santa Fe Institute and University of Ljubljana, Department of Mathematics.
Cities were sorted by size and the sorted stacks of city sizes compiled in a single file. Because sizes from
200K upwards were coded in all the files from 230BCE, sizes above 199K were log binned for
cumulative populations from sizes of 9 million down to 200K in multiples of 1/√2. This resulted in
number of people in each of 12 city size bins that were equal interval on a log scale. The X axis in in
Fig. 1 represents the log scale for the bins. The top line represents the population distribution across the
12 bins shown by the points along the line. The red line represents the upper bins which, because they
are partially empty, are always of lower slope than the full bins.
For measures of oscillation, we use
1 Average city population (for our purposes, >200 because this is lowest city size coded for all of our 29
historical periods.
2 Turnover of cities at the average (up/down, including out-of-rank)
3 Rank-size size migration rates by bins.
(Céline: How the ‘empire’ coded, and why do some of the files cease coding empires below a certain
size? Where “China” is given, for example, with 5-8 cities, is there only one empire involved? Might the
pre-1800 distributions be biased by the way that the pre-1800 city size data were estimated? If so, China
has census data prior to that period, and in the next stage of this research we might test the superlinear
distribution hypothesis for various historical periods.)
The q-exponential. Nonextensive or q-entropy Sq (Soares, Tsallis, Mariz and da Silva 2004) is a
generalization of Boltzmann-Gibbs entropy SBG = - ∫ dk P(k) ln (k), where Sq = (1- ∫ dk P(k)q )/(q-1). Sq is
associated with a plethora of scale-free phenomena for hierarchical and (multi)fractal structures (see
extensive references in [15-18] in Soares et al. 2004). Nothing has as yet been discovered about Sq that
is inconsistent with the entropy for thermostatistical systems, and Sq = SBG in the case where q=1 for
which pairwise interactions are independent. Values of q ≈ 1.33 ± 0.15, for example (which might be 5/4
or 4/3 ratios with geographic or network biases or higher with other nonindependence biases), are
consistent with the nonindependent and scale-free processes of preferential attachment models of social
networks (Soares et al. 2004; White, Keyžar, Tsallis, Farmer and White 2005). Also of interest are the
particular parameters of Sq that govern the q-exponential, which is typically a distribution somewhere
between the superlinear (power-law) and the linear/exponential, but which also accounts for the
inflections of log-log distributions both at the lower and upper ends of the distribution. One of the
parameters of the q-exponential eq (besides q) is kappa (κ), which is an expression of the largest
coherent unit of the distribution (or for a network, a characteristic number of links. If k is the variable
measured, and P(k) is the cumulative probability of k or higher values, then P(k) = P(0) eq-k/ κ, where, for
x=k, eqx ≡ [1 + (1 – q) x]1/(1-q) , and in the case where q = 1, eqx = ex , for the natural log base e.
If we assume that cities are nodes in networks in which there is preferential attachment towards ruralurban migration, then the q-exponential is consistent with the scale-free characteristics of urban
hierarchies.In fitting the q-exponential we solve for q (the exponent of scale-free nonindependence), for
α (the slope of the q-exp/log graph), and for κ (the base unit size), where α will be a function of κ and q
and possibly other parameters. If there is a preferential attachment for rural-to-urban migration, this will
be reflected in the α parameter and thus in q.
Why would we expect, from interaction theory, a preferential attraction to cities proportional (or qproportional, and thus α-proportional) to city size? The answer is simple: larger cities typically have
more functions and thus more attractions. Cities are limited, however, by the productive and trading
technologies which set as characteristic scale κ (the base unit size), which will also affect α, the scalefree exponent. We can expect that κ (the base unit size), as a limiting function, will tend to rise with the
evolution of urban functional complexity, but there will be limits at which it will stabilize or fall.
Cities grow, however, within the context of the full population, including the noncity population that
produces foodstuffs and raw materials. If we find full populations N that are relatively closed, such as
China or England during certain historical periods, for example, we should find some interaction
between the full population size and the carrying capacity Nc of the system of production and trade
(distribution, organized transports for consumption). As the ratio of N/Nc rises  1, structural
demographic theory predicts, first,
“persistent price inflation, falling real wages, rural misery, urban migration, and increased
frequency of food riots and wage protests. Second, rapid expansion of population results in an
increased number of aspirants for elite positions. Increased intraelite competition leads to the
formation of rival patronage networks vying for state rewards. As a result, elites become riven by
increasing rivalry and factionalism. Third, population growth leads to expansion of the army and
the bureaucracy and rising real costs. States have no choice but to seek to expand taxation,
despite resistance from the elites and the general populace. Yet, attempts to increase revenues
cannot offset the spiraling state expenses. Thus, even if the state succeeds in raising taxes, it is
still headed for fiscal crisis. As all these trends intensify, the end result is state bankruptcy and
consequent loss of the military control; elite movements of regional and national rebellion; and a
combination of elite-mobilized and popular uprisings that manifest the breakdown of central
authority (Goldstone 1991).” (Turchin 2005:1)
Network theory predicts that cities that are multiconnected will have lesser degrees of monopolistic or
oligarcic suppliers than those cities or rural areas that are connected in the trading system by a single
link (or fewer links). Thus, as N/Nc rises  1, inflation will be higher in the disadvantaged (rural) areas
and urban migration will be higher towards more multiconnected cities, which are typically in the larger
trading zones. Cities, then, are not only attractors, they are also part of a system in which there are
oscillations that displaces people from the land due to pressures of rural inflation. The q- and α- values
of a q-exponential urban hierarchy, then, ought to increase with monetization and trade, and fall when
monetization and trade decline, and, more specifically, vary with N/Nc ratios in closed areas.
In different historical periods, in consequence, as N/Nc ratios rise, there will be higher urban attraction
when the trading systems are viable, but if trading systems crash for whatever reason (which can include
the violence resulting from high N/Nc ratios), there will be urban-rural flight for economic reasons
balanced by urban attraction in spite of poverty if there are strong protective polities. Thus, we would
expect considerable urban hierarchy instability.
Cities as attractors, however, to the extent to which the form multiconnected trade networks, in
developing urban hierarchies of specialized functions and skills. In this context they become
disproportionate centers for invention and the diffusion of innovation, for education and literacy, of
governance, of elite concentration and elite consumption, medical improvements and (up to a limit)
increased longevity.
There are, then, a considerable number of dimensions on which city hierarchies have fractal scaling
patterns, ones which when summed in the aggregate, such as the role of cities in infusing the world
population with growth, create not just oscillations but long-term (and presumably reversable) trends.
Besides a growing city (and world) population, another seems to be the (increase) in α values (varying
inversely to q), in which the city hierarchy becomes ever steeper in slope. The tendencies for long-term
growth in q-entropic curves such as urban hierarchies do not seem to have a critical value of α, although
in scale-free attachment networks (Barabási 2000), an attachment steepness of α > 3 is the threshold at
which disease transmission can become uncontrollable if the item diffusing surpasses an initial
frequency threshold. This seems to have happened in human history in examples such as the Black
Plague of the Early Renaissance. Another such threshold appears to be the fertility transition. When
fertility decline (which is limited by zero) more than offsets longevity increase (which is limited to 90
years or so), then (controlling for migration) urban population will decline. And as rural-urban migration
has proportional limits, world population may decline as well. There is, in short, no guarantee of
increasing or stabilizing world population, and no secure guarantee of a rising standard of living given
rural/urban dynamics, although this is an areas of great contention.
Empirical Results for City Sizes
1000000
1950
1925
100000
1900
1875
1850
10000
1000
y = 5E+06x-0.630
R2 = 0.9684
1825
1800
1750
1700
1650
1600
1550
1500
1450
1400
1350
1300
1250
1200
1150
1100
1000
900
800
622
361
100
-200
-430
-0.7687
y = 6E+06x
2
R = 0.9762
-0.8127
y = 3E+06x
2
R = 0.9757
-0.8701
y = 2E+06x
2
R = 0.9837
y = 15813e-0.0035x
R2 = 0.9969
y = 819291x-0.796
R2 = 0.9747
y = 1E+06x-0.9109
R2 = 0.9869
-0.0058x
100
100
y = 10051e
R2 = 0.9621
1000
10000
totPowerLaw.xls
Figure 1: Binned Population Sizes, from 1950 back to 430 BC. Prior to 1800, the distributions are fitted
by an exponent (linear growth R2 ≈ .97), although this may be a function of the estimation methods and
missing data. From 1800 forward the distributions follow superlinear distributions with coefficients in
the range of -.91 to -.63 tending to fall with time (R2 ≈ .98). The population bins of 200 282 400 566 800
1131 1600 2263 3200 4526 6400 9052 are for number of population supported at or above that level.
When the regression lines are projected back to an axis of N=1 the intercept, as shown in Figure 1a (axes
in thousands), gives a good estimation of the global population, e.g., circa 1 billion in 1950. In this
figure, some of the upper bins with incomplete data are deleted. Now, however, we see a major problem
of discontinuity – if there were a major break between the pre- and post- 1800 curves, there would be a
huge discontinuity between the estimate world population figures for the pre-1800 period as well. We
can conclude that both sets of data are governed by the same curve.
1000000000
100000000
10000000
1000000
100000
y = 5E+06x
10000
-0. 6302
R 2 = 0.9684
y = 6E+06x
-0. 7687
R 2 = 0.9762
-0. 8127
y = 3E+06x
2
R = 0.9757
y = 2E+06x-0. 8701
1000
2
R = 0.9837
y = 819291x
-0. 7966
2
R = 0.9747
-0. 9109
y = 1E+06x
100
0.001
0.01
0.1
1
10
100
1000
10000
R 2 = 0.9869
100000
Figure 1a.
When more of the lower-sized city data are added, as in Figure 1b, the two sets of curves resemble one
another. Clearly, these are a family of curves that are not power-law or superlinear. Nor, however, are
they linear/exponential.
1000000
100000
10000
1000
100
0.001
0.01
0.1
1
10
100
1000
10000
Figure 1b. City size data down to sizes of 50K where available and excluding bins with incomplete data
(tot.xls)
Fig. 2 (not yet constructed) shows the previous data (Fig. 1b) fitted with q-exponentials that are
extrapolated back to the Y axis at .001 (estimated cumulative population including units down to 1).
1000000
100000
10000
1000
100
0.001
0.01
0.1
1
10
100
1000
10000
Figure 2. (tot.xls)
(One question at the end of q-exp curve fitting is how well the extrapolated total populations for each
time period match the actual (independently estimated) total world population)
Macro-Theory and Indices of Oscillation
Estimating total population from fitted city size distributions makes the unwarranted assumption that
human population is governed by urban dynamics. To look at urban and population dynamics from the
perspective of other processes, we look first at indices of population oscillations, both urban and
nonurban.
Fig. 3 compares the time series for total size of urban population in over 200K and World Population
(top and bottom graphs, with graphs for two oscillatory variables in the middle: Average World City
Size over 200K (upper middle), and Changes in World Population (lower middle). The upper and the
lower graphs show what appear to be similar superlinear growth trajectories, but with a flattening after
1950. The middle two graphs show strong oscillations, as expected for urbanism. These two graphs –
one for city and one for total population oscillations – are very different. The downward arrow at the
year 630CE for average world city size shows a depression that is not reflected in total population
oscillations. An second such arrow at 1200CE shows a second depression of this type that runs directly
opposite to oscillations in total world population. What is occurring to make these graphs so different?
Similarly, the upward arrows at the years 800 (also 1000 although not an independent peak) and 1350CE
for average world city size show a high points that are not reflected in total population oscillations.
Historically, these two downward deviations are the result of invasions by nomads of urban Eurasian
heartlands, one after the fall of classical empires (e.g., Rome, China?), and the other with the Mongol
conquests of the MidEast and China, which aimed at an Empire (nearly the largest in history) in which
cities were crushed if they resisted submission. The upward arrows show periods of urban recuperation
following nomad conquest, when trade also flourished.
In the lower middle graph, the two downward and one upward arrows between 1400 and 1700 represent
world population oscillations that are not reflected in urban hierarchy shifts – a period (after 1350) when
further nomad conquests (such as the Qing conquest of China) are symbiotic with urbanism rather than
destructive of urban resistance.
>200
350000
>200
300000
250000
200000
150000
100000
50000
0
-500 -400 -300 -200 -100 0 100 200 300 400 500 600 700 800 900 100 110 120 130 140 150 160 170 180 190
0 0 0 0 0 0 0 0 0 0
800
700
600
500
400
300
150
200
3500
100
100
3000 0
-500 -400 -300 -200 -100
50
0
100
200
300
400
0
100
200 300 400
500
600
700
800
900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900
2500
2000 0
- 200 - 100
500 600
700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000
1500
-50
1000
-100
500
-150
0
-200
0
200
400
600
800 1000 1200 1400 1600 1800 2000
-200
Fig. 3: Total and Average World City Size over 200K (top two graphs), Changes in World Population
(lower middle) and-250
Total World Population (bottom) Check that the Mongols lopped off several large
cities before 1150 (e.g., Baghdad)
-300
In sum, then, nomadic
invasions and empires need to be considered as part of urban dynamics. What of
the more detailed changes of relative numbers of cities at different levels in the urban hierarchy? Are
these affected by the seven events marked by arrows in fig. 3? Fig. 4-6 provide different views of these
oscillations. Oscillations are evident in fig. 4 but cannot be clearly distinguished. The same is true for
fig. 5, which shows the same data as percentages. One might argue that the oscillations are a function of
noise from having binned our city sizes to narrowly by multiples of √2. We will see that this is not the
case.
25000
>200
>282
>400
20000
>566
>800
15000
>1131
>1600
10000
>2263
>3200
>4526
5000
>6400
0
-500
0
500
1000
1500
2000
Figure 4: Historical Changes for the Binned Data. Percents in Logged Population Bins for 200K, 282K,
400K, 565K, and 800K are the relevant categories up to 1800.
>200
>282
>400
>566
>800
>1131
>1600
>2263
>3200
>4526
>6400
>9052
1.000
0.900
0.800
0.700
0.600
0.500
0.400
0.300
0.200
0.100
-500
-300
0.000
-100
100
300
500
700
900
1100
1300
1500
1700
1900
Figure 5: Historical Changes for the Binned Data. Percents in Logged Population Bins for 200K, 282K,
400K, 565K, and 800K are the relevant categories up to 1800.
Fig. 6 enhances fig. 4 by raising each of the log-binned population to reflect numbers of the population
supported at or above that level, as in fig. 1. All the arrows in fig. 3 are now reflected in urban dynamics,
and the data are not binned too narrowly: The red arrows in fig. 6 correspond to the two major Eurasian
nomad conquest events identified in fig. 3 (upper middle graph). The green arrows show the subsequent
urban renaissance periods (upper middle graph, fig.3 ). The blue arrows correlate with the world
population shifts in the lower middle graph of fig. 3. All of these effect are in the predicted direction.
100
>200
>282
90
>400
>566
80
>800
>1131
70
>1600
>2263
60
>3200
>4526
50
>6400
>9052
40
30
20
10
0
-500 -400 -300 -200 -100
0
100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900
Figure 6: Historical Changes for the Binned Data. Percents in Logged Population Bins for 200K, 282K,
400K, 565K, and 800K are the relevant categories up to 1800. Fluctations reflect peaks in the period of
Early Empires (230BCE-300CE), Urban renaissance (1000CE-1150CE), Early and Late Renaissance
(1300-1550) and Industralization (post 1800); low points follow the fall of the Roman and Chinese
empires (600CE) and the Mongol Conquest 1200CE). Growth periods are long and have unusual wave
patterns; declines are more precipitous. Key to colored arrows: red = two major Eurasian nomad
conquest events identified in fig. 3, upper middle graph; green = subsequent urban renaissance periods
(upper middle graph, fig. 3); blue = world population shifts in the lower middle graph of fig. 3.
Still need to do
2 Turnover of cities at the average (up/down, including out-of-rank)
3 Rank-size size migration rates by bins.
Run pairs of these variables by one another.
Do Phase Diagrams.
Conclusion
Phenomenologically, all the variables we have used in this study are world aggregates, and hence macro
variables for urban hierarchies and population sizes, plus several external shocks to the urban system
(such as nomad invasions) and presumed responses to those shocks (such as urban recovery), as well as
observed oscillations that have been studied in terms of dynamical interactions with population
variables. The latter dynamics have typically been studied in smaller and, by design, relatively closed
regions (Turchin 2005). In our period of study, there are also network interactions within two regions
(Old and New World) that are interactionally separable up to the 16th century. What we have shown is
that the macro variables we have studied, including the world population size, should not be considered
at any time as a closed system that follows its own developmental laws, such as dN/dt ≈ K (relatively
constant growth rate, growing exponentially, which does not fit the observed data for world population
in this period) or dN/dt ≈ KN (relative growth increase proportional to existing population, i.e.,
superlinear or power-law growth). The second of these closed-system models is descriptively accurate
(it does fit the observed data for world population in this period, up to 1963, fairly well, as might be
evident from fig. 3), but is not a valid model of the interactions that affect our macro-variables.
Phenomenologically, our attempt to model urban dynamics at the macro level is not a closed-system or
endogenous model, as with many of the power-law or Zipfian regularites proposed for urban hierarchies,
but one that is oriented toward indentifying interaction dynamics.
Our study of urban dynamics and city size hierarchies in relation to world population changes across 29
historical periods is indicative of two contrasting rural populations that are relevant to urban dynamics:
peasants and farmers who provide the raw materials for city growth; and nomads or others outside urban
political orbits who can mobilize the resources for invasion of urban regions and establish empires. The
first of these peri-urban populations, along with urban wage-workers and proletariats, are involved in the
oscillatory dynamics of population and sociopolitical violence in agrarian polities (Turchin 2005). The
second is involved in longer-term oscillatory dynamics of warfare, conquest and the administration of
empires (Kradin 200?).
Of these three population subsystems, each has a sociopolitical organization that can be characterized as
fractal. The q-exponentials that we have fitted to urban hierarchies reflect principles that are common to
hierarchical and fractal organization. Fractal properties are evident in the food riots, wage protests,
rebellions, and revolutions of peasantries in periods of population pressure (high ratios of population to
carrying capacity). Fractal properties are found in two forms among nomads. One is the Middle Eastern
social organization characterized by White and Houseman (2002) and White and Johansen (2005)
associated with the segmentary systems of endoconical clan organization. The other is the fluidity of
leadership in Eurasian horse nomadism that is associated with exogamous patriclans and interclan
alliances.
The larger part of the world-system where the tripartite dynamics of these population subsystems is
evident is found in Eurasia. It is perhaps not an exceptionalism of Europe that allowed it to become a
core region of this system in the lattermost centuries, overtaking China and India, but the exceptionalism
of China in having twice been invaded by horse nomads who were successful in establishing empires
that in many respects crushed China’s earlier innovations and potentials for commercial hegemony
(which was present in the Early Renaissance, for example).
Our conclusion is that it is not urban characteristics per se such as technological innovation that extends
carrying capacity for urban populations, or modern medicine, that drives the expansion of world
population up to a point where slowdown is necessary through transitions in demographic behavior
(Kremer 199?) or education. What we have shown here is that urban populations are embedded in at
least a tripartite dynamics among the different types of populations that share our common habitat.
Nonetheless, the city-size hierarchy does have scale-free properties that reflect these three, and perhaps
other, nonindependent components of interacting populations. The scale-free model that best fits the
urban size data over all historical periods since 430BCE, however, is not that of the a power-law or
Zipfian hierarchy. Rather, it is q-exponential, with several parameters that change over time. One is the
manageable scale of the system of interaction (κ). Another is the attachment inequality (α), the other is
the deviation from BG equipartition of energy principles, q itself, which is a function of α and κ. We
have yet to learn how these parameters themselves might influence urban, population, and world-system
dynamics and co-evolve with other dynamic variables.
References
Barabási 2000
Goldstone 1991.
Kradin, N. 200?
Kremer 199
Soares, D. J. B., C. Tsallis, A. M. Mariz and L. R. da Silva. 2004. Preferential attachment growth model
and nonextensive statistical mechanics.
Turchin 2005.
White, D. R., and M. Houseman. 2002.
White, D. R., and U. Johansen. 2005
White, D. R., N. Keyžar, C. Tsallis, D. Farmer and S. D. White. 2005.