# The Characteristic Time Scale of Cultural Evolution

Tobias Wand1,2∗ Daniel Hoyer3,4

(1Westfälische Wilhelms-Universität Münster, Insitut für Theoretische Physik
2Center for Nonlinear Science, Münster
3 George Brown College, Toronto
4 Evolution Institute, San Antonio
$*$ Corresponding Author<EMAIL_ADDRESS>
December 2022, Last Revision July 2023 )

###### Abstract

Numerous researchers from various disciplines have explored commonalities and
divergences in the evolution of complex social formations. Here, we explore
whether there is a ’characteristic’ time-course for the evolution of social
complexity in a handful of different geographic areas. Data from the Seshat:
Global History Databank is shifted so that the overlapping time series can be
fitted to a single logistic regression model for all 23 geographic areas under
consideration. The resulting regression shows convincing out-of-sample
predictions and its period of extensive growth in social complexity can be
identified via bootstrapping as a time interval of roughly 2500 years. To
analyse the endogenous growth of social complexity, each time series is
restricted to a central time interval without major disruptions in cultural or
institutional continuity and both approaches result in a similar logistic
regression curve. Our results suggest that these different areas have indeed
experienced a similar course in the their evolution of social complexity, but
that this is a lengthy process involving both internal developments and
external influences.

##### Keywords:

Cliodynamics, Cultural Evolution, Time Scale

##  1 Introduction

###  1.1 Motivation to Find Characteristic Time Scales

Researchers from various disciplines have analysed commonalities and
divergences in the evolution of complex social systems [1, 2, 3, 4, 5, 6, 7].
The recent emergence of Cliodynamics as a discipline has started the analysis
of the dynamics of human societies and states with data-driven scrutiny and
modelling approaches from natural sciences [8, 9]. Previous work established
that a common set of factors associated with complex social formations
typically moved in tandem across a wide variety of regions and time-periods;
factors such as social scale, the use of informational media, administrative
hierarchies, monetary instruments, and others [6]. These were interpreted as
comprising the primary dimension of what could be called ’social complexity’
across cultures, though other dimensions can be adduced as well [3].

Various studies have already discussed or tried to identify the causal drivers
of cultural evolution and evaluated the evidence for different theories of why
cultures become more complex [7, 10, 11, 12, 13]. Beyond the causal
similarities behind cultural evolution across cultures, researchers have also
found evidence for temporal similarities and seemingly parallel time scales in
the dynamics of various social structures. For example, models for societal
collapse have been derived from demographic and fiscal data that show
characteristic oscillation periods of a few centuries and a fine structure
with a faster periodicity of approximately two human generations [4]. Other
theories suggest that cultural evolution leads to the emergence of similar
political institutions and schools of thought at roughly identical time
intervals across different geographic regions [14, 15, 16]. Another recent
study has evaluated the connection between the first emergence of complex
societies in different world regions and the age of widespread reliance on
agriculture in those areas [17], supporting the theory that agriculture is a
necessary condition for the evolution of complex societies. While the time lag
between the primary reliance on agriculture and the emergence of states was
found to decrease over time, an average time lag of roughly 3,400 years for
pristine states suggested the existence of a characteristic time scale, though
this was not the explicit focus of that study. Similarly, the study on causal
drivers in [7] also found that the time since the adoption of agriculture had
a statistically significant effect as a linear predictor variable (called
’AgriLag’) for sociopolitical complexity, providing additional evidence for
temporal regularities in the growth of complexity across different cultures
and civilisations. Finally, using the same data as our article (cf. section
1.2), it was possible to identify periods of cultural macroevolution with
either slow or rapid change in social complexity [5].

Nevertheless, as yet there is no consensus on whether there is a ’typical’
time scale for socio-political development cross-culturally, let alone what
that time-course might be. Such characteristic time scales of dynamic systems
are, however, well documented in different areas of the natural sciences such
as physics and chemistry [18, 19]. Differentiating between fast and slow time
scales in a dynamical system can lead to useful insights and can inform
modelling assumptions for data analysis [20, 21]. In particular, Haken’s
theory of the ’enslaving principle’ [22], according to which the dynamics of
fast-relaxing modes are dominated (enslaved) by the behaviour of slowly
relaxing modes in a dynamical system, pioneered the research on how dynamics
on different time scales influence each other in the same observed system. The
existence of temporal regularities among societal dynamics would suggest that
cultural evolution not only occurs in similar developmental stages across
geographic regions and time periods, but also in similar time intervals. This
would add an important dimension to our understanding of how complex social
formations evolve, and raise a number of critical questions about what drives
these cross-cultural patterns.

Here, we adapt some of the methods employed in the natural sciences in an
attempt to identify characteristic time scales in the evolution of complex
societies. We utilise data collected by the Seshat: Global History Databank
[23, 24, 25], a large repository of information about the dynamics of social
complexity across world regions from the Neolithic to the early modern period
[26]. We find that, despite significant differences in the timing and
intensity of major increases in social complexity reached by polities across
the Seshat sample, there is a typical, quantitatively identifiable time course
recognisable in the data. This result is robust to a variety of checks and
covers polities from all major world regions and across thousands of years of
history. Our findings offer a novel contribution to the study of cultural
evolution, indicating the existence of a general, cross-cultural pattern in
both the scale as well as the pace of social complexity development.

###  1.2 Seshat Databank

The Seshat: Global History Databank includes systematically coded information
on over 35 geographic areas and over 200 variables across up to 10,000 years
in time steps of 100 years ([23, 26]; see also publicly available data at
http://www.seshatdatabank.info/databrowser/). During the time interval
captured by the Seshat databank, these NGAs are occupied by over 370 different
identifiable polities, defined as an "independent political unit". This sample
is constructed by identifying all known polities that occupied part or all of
each NGA over time (see [23, 24, 25] for details). The recorded variables are
aggregated into nine complexity characteristics (CCs) and a principal
component analysis shows that 77% of the variation in the data can be
explained by the first principal component (SPC1), which has almost equal
contributions from all nine CCs [6]. In the case of missing data or expert
disagreement in [6], multiple imputation [27] was used to create several data
sets with the differently imputed values which were aggregated into the
principle component analysis. The NGAs in the Seshat data cover a wide
geographical range and different levels of social complexity, though it is
important to note that the Seshat sample is focused laregly on relatively
complex, sedentary societies (but not exclusively). Data on the CCs is sampled
at century intervals, giving a time series of each polity’s estimated social
complexity measure throughout its duration.

Seshat data has allowed researchers to quantitatively test hypotheses on
cultural evolution such as identifying drivers of social complexity and
predictors of change in military technology, for example gauging the effect of
moralising religions on cultural evolution or predicting historical grain
yields [7, 28, 29, 30]. Further analysis of the Seshat data includes a
discussion of ideas from biological evolutionary theory with respect to the
tempo of cultural macroevolution, defined as "rates of change, including their
acceleration and deceleration", concluding that "cultural macroevolution is
characterized by periods of apparent stasis interspersed by rapid change" [5].
These results strongly relate to the question of the present article, whether
there is some generality in the time scale of cultural evolution in the Seshat
data.

###  1.3 Data on Culture/Polity Boundaries and Duration

Each NGA’s time series can contain data about very different polities that
succeeded each other. Sometimes, a gradual and continuous change between the
polities justifies treating predecessor and successor polities as closely
related; for instance, in the Latium NGA (modern-day central Italy), Seshat
records three separate polities for the Roman Republic, indicating the Early,
Middle, and Late phases. These phases are culturally and (to a signfiicant
degree) institutionally continuous, so can be treated as a single polity-
sequence. In other cases, there may have been an invasion or mass migration as
a clear break-point between the two polity’s continuity; for instance, between
the Ptolemaic Kingdom and Roman Principate polities in the Upper Egypt NGA.
Data from [26] and other information recorded in the Seshat sample, notably
information on the relationship between polities, is here used to establish a
list of continuous polities. The continuity is evaluated either as cultural
continuity or as political-institutional continuity and our cutout data for
both approaches is published on [31].

###  1.4 Organisation of this Article

Section 2 explains how we transformed the time series data on each NGA in the
Seshat sample to establish a common reference point to investigate the time
course of changes in social complexity across NGAs. In short, we shift each
NGA’s time series with respect to a single anchor time such that the
transformed time variable RelTime shows major overlap between the RelTime-vs-
SPC1-curves of all NGAs. Exploratory data analysis for the whole dataset
reveals that there is a logistic relationship between RelTime and the SPC1
response variable. Section 3 identifies the time scale of growth from the
lower to the upper plateau of the logistic curve via bootstrapping. The
logistic curve is compared to a regression using only either the culturally or
institutionally continuous time series and moreover, the duration of those
continuous time series is compared to the estimated characteristic time scale.
Finally, the results of the analyses are summarised and discussed in section
4. The mathematical methods and technical details are discussed in the
appendices A and B and appendix C gives more details on the used data.

##  2 Approach: Data Transformation and Exploratory Analyses

First, all raw SPC1 time series are rescaled via a min-max scaling, i.e.

$SPC1=\frac{SPC1_{raw}-\min(SPC1_{raw})}{\max(SPC1_{raw})-\min(SPC1_{raw})}.$
(1)

This has the advantage of making the interpretation of high and low SPC1
values much easier as high/low correspond to close to 1 or close to 0,
respectively. It also makes the parametrisation of a logistic curve easier by
restricting the observed data to a range between 0 and 1.

###  2.1 Anchor Time

Considering that most NGAs have an SPC1 time series that starts at a low value
barely above $0$ and ends at a high value close to $1$, a logistic regression
model seems like a reasonable suggestion for the data. Although all NGAs
experience a growth in SPC1 over time, they start at very different calendar
years. Therefore, it is necessary to shift the time series via an anchor time
so that in the new "relative" time, the growth phase in each NGA’s time series
coincide. Then, one logistic regression can be used for all shifted time
series (cf. figure 1; also A.1). Hence, each NGA $i$ needs an anchor time
$T^{(i)}_{a}$ so that if all time series are shifted by $-T^{(i)}_{a}$, they
roughly overlap. The shifted time series of the NGAs and the logistic fit are
shown in the main part of figure 1.

Figure 1: Main figure: Time series of RelTime vs. SPC1 for all 23 NGAs that
cross $\textmd{SPC}1_{0}$ and the logistic regression. Marked in red is the
area of growth between the two plateaus of the curve as identified in section
3. The various time series are shown in appendix C in multiple plots to make
the identification easier. Insets: a) distribution of SPC1 for all 35 NGAs,
the associated KDE (red) and the threshold $\textmd{SPC}1_{0}$ (vertical); b)
residuals of the logistic regression.

The anchor time can be chosen as the year during which the NGA $i\textmd{'s}$
SPC1 value crosses a threshold value. It has already been shown that there is
a clear threshold $\textmd{SPC}1_{0}$ between high and low values of SPC1 in
the data, which was used to define the RelTime variable in [29]. A similar
methodology was also used in [32], but there, the authors used the emergence
of a moralising religious belief as the "year zero" to shift each NGA’s time
series. Copying the procedure from [29] to get the RelTime variable,
$\textmd{SPC}1_{0}$ is chosen as the minimum between the two maxima in the
kernel density estimation (KDE; explained in A.2) of the SPC1 values (figure
1, inset a). The anchor time $T^{(i)}_{a}$ is then selected as the first
recorded data point when the NGA $i$ exceeds $\textmd{SPC}1_{0}$. An
illustration of the anchor time shift is provided in the appendix B. Thus, the
12 NGAs that never exceed $\textmd{SPC}1_{0}$ are discarded from this
analysis. On the one hand, this is not too problematic because their limited
growth in SPC1 means that they would have only contributed little information
to the estimation of SPC1’s characteristic growth time, but on the other hand,
this discards all NGAs from the world region Oceania-Australia in the Seshat
sample, meaning that it might introduce a bias. We discuss this and other
possible limitations of the approach further in section 4 below.

###  2.2 Logistic Regression

The RelTime-vs-SPC1 data is fitted to a logistic regression curve (cf. A.1)
via the optimisation algorithm scipy.optimize.curve_fit from [33]. The quality
of the regression curve is evaluated with the methods from A.3. With the
exception of a few outlier observations occurring in several of the NGAs, all
time series qualitatively agree with the regression curve fairly well. Also,
the majority of the residuals shown in figure 1 (inset b) are distributed
roughly symmetrically in a neighbourhood of zero. The distribution of the
residuals and the rather low value of the root-mean-square-error $RMSE\approx
0.11$ both indicate that the logistic regression is a suitable model for the
shifted SPC1 data.
To further increase our trust in the quality of the regression, it is
evaluated via the coefficient of determination $\rho^{2}$ in an out-of-sample
prediction. The data is split randomly into equally sized training and testing
data sets and a logistic regression curve $f_{i}$ is estimated by only using
the training data. Then, $f_{i}$ is used to predict the values for the test
data and the prediction is evaluated via the $\rho^{2}$ metric in A.4. The
random training-test-split is repeated $i=1,\dots,100$ times, each time using
the estimated parameters from the full time series as initial values, and the
resulting $\rho^{2}$ values have an average of $\rho^{2}=0.81\pm 0.01$ far
above 0 and therefore further strengthens our trust in the logistic model.

##  3 Analysis of Time Scales

###  3.1 Finding a Characteristic Time Scale

Having established that the data can be accurately captured by a logistic
curve, we can investigate our research question; namely, how many years did it
typically take in these different regions to transition from a polity with low
SPC1 to one with high SPC1? Or to reformulate the question: when does the
curve leave the low plateau and when does it reach the high plateau? We
attempt to answer these questions by estimating the heights of the plateaus
and their respective uncertainties and by checking when the regression curve
crosses these thresholds.
We performed 1000 steps of bootstrapping by sampling from the list of NGAs and
by estimating the regression parameters
$(a_{i},b_{i},c_{i},d_{i})_{i=1,\dots,1000}$ for each sample (cf. A.5).
According to the asymptotic behaviour in A.1, the plateaus are given by
$b_{i}$ and $a_{i}+b_{i}$. In order to make conservative estimates instead of
being influenced by noise, an upper boundary for the lower plateau’s value
$Th_{1}$ and a lower boundary for the upper plateau’s value $Th_{2}$ are used
as the thresholds. $Th_{1}$ is chosen as $Th_{1}=\mu(b)+3\sigma(b)$ of the
bootstrapped distribution of $b$, $Th_{2}$ as $Th_{2}=\mu(a+b)-3\sigma(a+b)$.
For each bootstrapped logistic curve $f_{i}(t)$, it is then determined at
which RelTime values $t^{(i)}_{1}$ and $t^{(i)}_{2}$ it crosses the lower and
upper thresholds $Th_{1}$ and $Th_{2}$. We can then understand the mean value

$\mu\left(t^{(i)}_{2}-t^{(i)}_{1}\right)=\mu\left(t^{(i)}_{2}\right)-\mu\left(t^{(i)}_{1}\right)\approx$2500\text{\,}\mathrm{yr}$$
(2)

as the characteristic time scale for the period of rapid cultural evolution
between low and high plateaus of socio-political complexity, across geography
and not in reference to any specific time period. Note that one can also
choose less restrictive thresholds via $Th_{1}=\mu(b)+\sigma(b)$ and
$Th_{2}=\mu(a+b)-\sigma(a+b)$. With these thresholds, the regression curve
leaves the vicinity of the lower plateau rather quickly but needs much longer
until it is close enough to the upper plateau to be considered as having
reached the upper plateau. These $1\sigma$ thresholds would result in a longer
time scale of roughly

$\mu\left(t^{(i)}_{2}-t^{(i)}_{1}\middle|1\sigma\right)\approx$4000\text{\,}\mathrm{yr}$.$
(3)

We can check the general validity of these results by explicitly identifying
for each NGA $i$ the first time $\tau_{1}^{(i)}$ that their SPC1 value exceeds
$Th_{1}$ and the first time $\tau_{2}^{(i)}$ they exceed $Th_{2}$. With the
exception of the Ghanaian Coast, all NGAs cross $Th_{2}$ and therefore, this
procedure yields 22 time durations $d^{(i)}=\tau_{2}^{(i)}-\tau_{1}^{(i)}$.
Only for the two NGAs Kachi Plain and Middle Yellow River Valley (two
’pristine’ states, cf. the discussion in section 4.2) does the duration
$d^{(i)}$ exceed the 4000 years estimated as an upper boundary in (3). Both
the mean (approximately 2200 years) and median (2100 years) are in line with
the main estimation in (2).

###  3.2 Continuous Polities

There are two reasons why it makes sense to restrict the logistic regression
only to a central part of each NGA’s time series, during which the polities in
that NGA are not disrupted by external influence or major dislocations in
socio-political structures. First, the logistic regression starts at a plateau
of low values of SPC1 close to $0$ and ends at a plateau of high values close
to $1$. Therefore, even a bad interpolation for the central part can achieve a
good $RMSE$, if the plateaus of the high and low tails are sufficiently
accurate. However, this would not be a reliable estimation to make an
inference on the growth phase in the centre of the curve. Second, if the NGA’s
polity is e.g. annexed by another, more developed polity, then it inherits the
invading polity’s high SPC1 value and may make a sudden jump in the SPC1
curve. However, the logistic regression here is intended to model steady,
uninterrupted growth like in [34] and not major transitions driven by
developments experienced elsewhere, as through annexations by an external
invader. Therefore, it makes sense to divide each NGA’s time series into
intervals which are separated by sharp, discontinuous changes within each NGA
and to restrict the analysis of the NGA to its central interval, i.e. to the
time series from the polities that cross the $\textmd{SPC}1_{0}$ threshold.
As mentioned earlier, there are two ways of identifying such discontinuous
changes: either via cultural changes of via major institutional changes of the
polity’s governance. Both approaches are analysed separately. The central time
series for both methods and their resulting logistic regressions are shown in
figure 2.

Figure 2: Main figure: estimated logistic curves for the full data and the two
cutout methods. Insets: a) each NGA’s central time series and resulting
logistic curve for the culturally continuous time series; b) the same as
subfigure a) for the institutionally continuous time series.

####  3.2.1 Cultural Continuity

One set of sequences was determined by the absence of a major cultural
dislocation; namely, the introduction of a new ideological and linguistic
system, major population displacement, or major technological advance (the
adoption of iron metallurgy, for instance). This is a very broad and lenient
definition of continuity, as it allows for very different social formations to
be part of a single sequence and can include significant developments. In
Egypt, for instance, we treat nearly the entire Pharaonic period (from the
Naqada period to the Achaemenid conquest) of over 3000 years, including the
so-called Intermediate periods when central rule was fragmented (though many
cultural and social features were retained), as a culturally continuous time
period.
For the 23 NGAs under consideration, the mean value of data points for the
culturally continuous central interval is approximately $11.7$, i.e. there is
on average a bit more than one millennium of data. While this is much shorter
than the characteristic time scale of roughly $2500$ years, the longest
continuous time series of the NGAs show a similar length to that of the
characteristic time scale (cf. the left half of table 1). Hence, the logistic
regression for these cutouts is rather close to the regression of the full
data (cf. main part of figure 2) and in particular, the regression curves’
steepness (i.e. their time of growth) is quite similar.

####  3.2.2 Institutional Continuity

For institutionally continuous time periods, we follow a similar procedure as
above, though with different criteria for continuity leading to shorter
sequences. Namely, we break each sequence at any significant
political/institutional change, even if there was much continuity in socio-
cultural forms. In Egypt, for instance, the institutional sequence starts at
the 1st Dynasty period and ends a the end of the Old Kingdom period and the
First Intermediate Period, which we call the ’Period of the Regions’. The mean
value of data points for institutionally continuous central time series is
only $5.9$ and represents approximately 500 years of data. Even the longest
continuous sequences now do not last as long as the characteristic growth time
of $2500$ years (cf. right half of table 1). Moreover, the logistic regression
has only very little data for the parameter estimation (cf. inset b of figure
2) and hence, the logistic regression has a much lower SPC1 level for the
upper plateau than the regression to the full data (main part of figure 2),
because the cutout time series are too short to reach the high-SPC1 levels.

NGA | Cultural Continuity Length | NGA | Institutional Continuity Length
---|---|---|---
Yellow River | 38 | Susiana | 17
Upper Egypt | 33 | Crete | 17
Kachi Plain | 22 | Konya Plain | 15
Susiana | 21 | Upper Egypt | 10
Table 1: For both methods of identifying continuous time sequences, the four
longest continuous central time series are shown and the amount of data points
they contain (given as their length). The data points are sampled at intervals
of one century. The culturally continuous time series are much longer than the
longest institutionally continuous sequences.

##  4 Summary and Discussion

###  4.1 Summary

Exploratory data analysis shows in figure 1 that the logistic regression is a
suitable model for the RelTime-vs-SPC1 time series. Bootstrapping allows us to
narrow down the time interval of rapid SPC1 growth to approximately 2500
years, as highlighted in figure 1. Together, these results illustrate that
there is a uniform behaviour in growth of social complexity represented by the
time evolution of SPC1.
If the data is restricted to the central part of each NGA’s time series
without any discontinuous cultural or institutional transitions, the logistic
regression is still a reasonable model and shows a similar shape to the full
data as depicted in figure 2. In particular, the regression based on the
culturally continuous time series show a very similar steepness (i.e. growth
period) to the full regression curve.

###  4.2 Discussion of the Time Scale and Continuous Sequences

Figure 2 shows that the culturally continuous and institutionally continuous
time series result in a similar logistic regression to the full data. Notably,
table 1 shows that the culturally continuous time series have a much longer
duration than the institutionally continuous ones. In particular, in the
Yellow River Valley, Upper Egypt, Kachi Plain and Susiana, the culturally
continuous time series is approximately as long (or even longer) as the
characteristic time scale of SPC1 growth. This is expected for regions that
saw the emergence of large, complex states relatively early in history and
without any precedent from neighbouring societies – the so-called ’pristine’
or ’primary’ states [35, 36] – which these regions all experienced. However,
this is not the case for most other NGAs, indicating that in those NGAs, the
growth from the lower to the higher SPC1 plateau did not take place over the
course of just one culturally continuous era, but rather included developments
across cultural spheres and, in most cases, including developments being
’brought in from the outside’ in the form of direct conquest or more indirect
influence. The institutionally continuous time series are all significantly
shorter than the characteristic growth time, as expected from the criteria
used to generate those sequences. This is notable, as it suggests that in
order to transition from low to high social complexity, major shifts in the
NGA’s governing institutions are necessary to facilitate the increase in
social complexity. In other words, our findings suggest that major transitions
in social complexity are not feasible for a single polity to accomplish, but
require multiple social formations or ’phases’ of rule building successively
(but not monotonically, as the above figures illustrate) on prior
developments. Nevertheless, the general similarity of the three regression
curves in figure 2 shows that our analysis is stable with respect to the exact
selection of time periods and different cutout criteria used.

It is interesting to compare those NGAs that crossed the threshold
$\textmd{SPC}1_{0}$ to those that failed to do so and stayed at lower
complexity values. The latter group had a mean of only 6.4 recorded data
points, i.e. there were only complex social formations coded as part of the
Seshat sample for a period of roughly six centuries. On the other hand, the
NGAs that did reach a high complexity and exceeded the threshold
$\textmd{SPC}1_{0}$ had a mean of 57.3 recorded data points, corresponding to
almost 6 millennia of observed data. Partly this is explained by different
availability of historical and archaeological evidence in different regions,
but it suggests also that cultural developments in the low complexity NGAs
could have followed the same trajectory of logistic growth, if they had been
given enough time. Unfortunately, the necessity to identify an anchor time for
this analysis means that all NGAs from the Seshat world region Oceania-
Australia had to be discarded for this research. The bias introduced by this
has to be kept in mind while interpreting our results.

###  4.3 Interpretation and Comparison to Previous Work

With the shifted time index RelTime, the logistic regression model of the SPC1
time series achieves a high accuracy in capturing the evolution of socio-
political complexity measured by SPC1. Previous work has already demonstrated
a significant amount of cross-cultural generality in the factors contributing
to the evolution of socio-political complexity ([6], supplemented by findings
in [3, 37]). Notably, a previous study has already identified a characteristic
growth pattern of SPC1 and the second principal component SPC2 and found that
a rapid period of scale is first followed by a growth of information
processing and economic complexity and then by further growth in scale [3].
Here, we expand on this prior work by identifying that the time scales
involved in these developments also exhibit a general, characteristic shape.
Nevertheless, the evolution of social complexity is a lengthy and non-
monotonous process; this emerges clearly from our analyses distinguishing the
full regional time-series involved in the transition from low to high
thresholds of SPC1 from sequences of cultural or institutional continuity. We
see no examples of this evolution accomplished during a single
institutionally-continuous sequence. Further, in all NGAs there are noisy
periods during which SPC1 grows but also crises during which socio-political
complexity sharply declines, only to recover later and continue increasing.
These findings highlight both that different parts of the world experienced
similar processes of social complexity growth, involving multiple phases of
cultural and socio-political structures building off of (and occasionally
recovering from) prior developments in each region.
While the sample of past societies explored in this article is certainly not
exhaustive, they comprise a fairly representative sample of regions from
different parts of the world and include societies from different periods,
cultures and different developmental experiences. Our results thus lends novel
empirical support to the idea (from e.g. [14, 5]) that socio-cultural
evolution does indeed occur in similar time scales across different cultures
and geographies. Future research can expand these insights by including
additional societies and exploring alternate thresholds of complexity to
identify anchor times to include more NGAs from the original sample, because
the current thresholding procedure in particular excluded some NGAs from
modern-day Oceania-Australia from our analysis.
In terms of the underlying approach, our study tries to single out the
autocatalytic effect of social complexity growth. To this end, it not only
focused on one NGA at a time, but also compares our regression results to the
culturally and institutionally continuous periods for the respective NGA.
Thus, we uncover an empirical pattern in the temporal evolution of SPC1 that
has not yet been fully discussed by previous work, e.g in causal analyses of
the drivers of social complexity like in [7]. Our methodology differs from
e.g. the regression model in [7] by deliberately choosing a very simple model
to single out the temporal evolution whilst disregarding possible drivers of
the observed dynamics. We believe this approach can be utilised to answer
other questions about long-run cultural evolution, for instance the processes
by which key technologies (e.g. metallurgy, military technology,
communications media etc.) are invented in certain locations and then adopted
in others.
While the autocatalytic growth model provides an elegant interpretation of our
findings (the current level of complexity facilitates further growth until the
presence of an upper boundary of complexity is approached), it has to be
regarded with caution: We sought as far as possible to disentangle culturally
and institutionally endogenous developments from those driven by interactions
with other polities, though even the internal developments are not free from
external influence. Previous work, for instance, shows the strong effect of
military conflicts with other states on the growth of sociopolitical
complexity [7, 28]. Hence, the autocatalytic model might be a useful low-
dimensional description of the data, but not an exhaustive explanation. In
short, our findings exposes a cross-cultural temporal pattern whose causes
need to be fleshed out in future work.
Finally, the findings of the present article can be used as a benchmark for
future additions to the Seshat data: if a new NGA is added to the databank and
shows a clear divergence from the logistic curve, it may be prudent to either
check, if there are any mistakes in the data generation and interpolation, or
if the divergences can be explained by historical developments. Such a
benchmark may thus be useful for further expansion of the Seshat databank.

#### Author Contributions

TW performed all analyses and drafted the manuscript; DH assisted in
conceptual development and drafting the manuscript.

#### Acknowledgements

Initial ideas behind this paper were developed at a workshop held by the
Complexity & Collapse Research Group of the Complexity Science Hub, Vienna.
The authors thank all the participants at that event, particularly Mateusz
Iskrzyński for valuable contributions at early stages of this project.
Financial support for this work was provided by the “Complexity Science”
research initiative supported by the Austrian Research Promotion Agency FFG
under grant #873927 and by the German Academic Scholarship Foundation
(Studienstiftung des deutschen Volkes).

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##  Appendix A Methods and Technical Details

###  A.1 Logistic Regression Curve

Logistic regression is used to model time series data which is mostly
distributed at two plateaus with a transitory area between them [38]. It is
based on the characteristic sigmoid curve of the logistic growth model
described in [34], which models an exponential growth process constrained by a
carrying capacity. The logistic curve has the functional form $f$ with an
asymptotic behaviour

$f(x)=\frac{a}{1+\exp(-c(x-d))}+b,\quad f(-\infty)=b\quad\textmd{ and }\quad
f(\infty)=a+b.$ (4)

Often, data is scaled such that $b=0$ and $a=1$, i.e. an asymptotic behaviour
between two binary plateaus at height $0$ and $1$.

####  A.1.1 Reversing the Direction

Estimating the coefficients $(a,b,c,d)$ can lead to numerical instabilities
because it is possible to transform a logistic curve with $c>0$ to an
equivalent equation $\hat{f}$ with $c<0$. Consider e.g. $a=1,b=0,c=1$ and
$d=0$, then

$f(x)=\frac{1}{1+\exp(-x)}=\frac{\exp(x)}{\exp(x)+1}=\frac{\exp(x)+1-1}{\exp(x)+1}=1+\frac{-1}{1+\exp(x)}.$
(5)

The last reformulation of $f$ can now be parametrised via
$\hat{a}=-1,\hat{b}=1,\hat{c}=-1$ and $\hat{d}=0$. This ambiguity can lead to
the regression algorithm yielding positive and negative results for $c$ during
multiple runs. This can be prevented by setting an initial parameter guess
with $c>0$, which locks the algorithm into positive values for $c$.

###  A.2 Kernel Density Estimation (KDE)

A KDE tries to reconstruct a probability density function based on a sample
$x_{1},\dots,x_{n}$ of measurement data by smoothing the histogram of the data
[39, 40]. The estimated density $\hat{\rho}(x)$ is modelled as a weighted sum
of probability densities (kernels) centred around the measured $x_{i}$. In
this article, the Gaussian density is used as the kernel via
scipy.stats.gaussian_kde [33].

###  A.3 Residuals and Root Mean Squared Error

For an algorithm $f$ which estimates values $\hat{y}$ from data $X$ with true
values $y$, there are several methods to evaluate the accuracy of $f$. One of
them is the root mean squared error $RMSE$. It is defined as

$RMSE=\sqrt{\frac{1}{n}\sum_{i=1}^{n}r_{i}^{2}}$ (6)

via the residuals $r_{i}=\hat{y}_{i}-y_{i}$. An $RMSE$ much smaller than the
range of measured values $y_{i}$ means that the model shows only little
deviation from the data. A roughly symmetric distribution of the residuals
around $0$ indicates that the model does not have a bias towards particular
values.

###  A.4 Coefficient of Prediction $\rho^{2}$

Another method to evaluate the quality of an estimated function $f$ is the
coefficient of prediction $\rho^{2}$ used in [6]. It takes the value of
$\rho^{2}=1$, if the prediction is always exactly true, and $\rho^{2}=0$, if
the prediction is only as accurate as always using the mean $\bar{y}$. It is
defined by

$\rho^{2}=1-\frac{\sum_{i=1}^{n}(\hat{y}_{i}-y_{i})^{2}}{\sum_{i=1}^{n}(\bar{y}-y_{i})^{2}}.$
(7)

###  A.5 Bootstrapping

Bootstrapping is used to estimate standard deviations and confidence intervals
in a model-free approach. A sample $z_{1},\dots,z_{n}$ is re-sampled with
replacement, i.e. a new sample $\tilde{Z}=z_{i_{1}},\dots,z_{i_{n}}$ is
created that for some $j\neq k$ fulfils $i_{j}=i_{k}$. This procedure is
repeated $N$ times so that there are $\tilde{Z}_{1},\dots,\tilde{Z}_{N}$
bootstrapped samples. If $N$ is large enough, then e.g. the mean
$\tilde{\mu}(z)$ of the re-sampled data will converge to the true mean of the
original sample, but the empirical distribution of the re-sampled means
$\tilde{\mu}_{1}(z),\dots,\tilde{\mu}_{N}(z)$ enables the calculation of the
confidence interval of the empirical mean [41]. This approach can be adapted
to make inference on the standard deviation and CIs of any statistical
property of the original sample.

##  Appendix B Example of the Data Preprocessing

We illustrate the preprocessing of the raw $\textmd{SPC}1$ time series using
the NGA ’Latium’ (modern day central Italy) as an example. Figure 3 shows the
shift between the original time series to the RelTime time frame relative to
the anchor time $T^{(Latium)}_{a}=500\textmd{ BC}$. Additionally, this figure
depicts how the $\textmd{SPC}1$ time series of Latium is dissected into
culturally or institutionally continuous time series intervals of SPC values
for the NGA. Note that the time index is given in $RelTime$ for the shifted
data and real-world time for the original data.

Figure 3: Illustration of how the $\textmd{SPC}1$ time series of the Latium
NGA is shifted by its anchor time to the new relative time frame $RelTime$.
Hence, at $RelTime=0$, the shifted time series is equal to the threshold value
$\textmd{SPC}1_{0}$. The two recorded discontinuities for the Latium NGA (red)
are used to divide its $\textmd{SPC}1$ time series into different intervals.
The interval containing the threshold value $\textmd{SPC}1_{0}$ (the "Relevant
Snippet" with the solid line) is used for further analysis whereas the rest
(dotted) is discarded. Note that the times of the discontinuities do not line
up with the 100-year time intervals of the $\textmd{SPC}1$ measurement which
is why the transition between the black dashed/solid lines does not perfectly
align with the red vertical lines.

##  Appendix C Detailed Data

The SPC1 data from figure 1 is shown in figure 4, but spread out onto several
subplots to help the reader identify different NGAs.

Figure 4: Spreading the various time series from figure 1 onto multiple plots
to make it easier to distinguish the NGAs from each other.

###  C.1 Data Collection and Availability

As described in the main text, the data used in this paper is derived from
[26], supplemented with information provided by the authors. The original data
is described in [42]. It was gathered, cleaned, reviewed, and managed by
members of the Seshat Databank project following standard project methods, as
described in the references cited in the main text. For more details on the
social complexity data utilised here and development of the SPC1 values, see
especially [6] and [7].
The full dataset used in the analyses presented here is available on [31].
This shows:

  * •

NGA: Name of the NGA

  * •

PolID: Unique identifier for each polity in the sample

  * •

AbsTime: The ’absolute’ or calendar time-point for each SPC1 value. Note that
we sample at 100-year intervals, for as many intervals as there are polities
in the sample occupying each NGA

  * •

RelTime: The shifted time-series, such that RelTime = 0 in the century
interval in each NGA during which SPC1 values crossed the calculated
threshold, as described in the main text. Other rows in the NGA are express
their relation before or after this threshold, in century intervals. Note that
rows that fall outside of the central cultural or institutional sequence are
not expressed

  * •

SPC1: Raw SPC1 values for each polity at each century interval, calculated as
described in the main text (but not yet min-max scaled)

  * •

Culture.Sequence: Label indicating the central time-series identified as part
of the culturally-continuous sequence that surrounds each NGA’s RelTime=0
threshold (labelled ’cultural.continuity’); all other century intervals in the
NGA are labelled ’outside.central’ to indicate they fall outside of this
central interval sequence

  * •

Institutions.Sequence: Same as above, but indicating the institutionally
continuous sequence.