Journal of Archaeological Method and Theory
Self-Organized Cultural Cycles andtheUncertainty
AleksandrDiachenko1 · IwonaSobkowiak‑Tabaka2
Accepted: 4 January 2022
© The Author(s) 2022
Contributing to the issue of complex relationship between social and cultural evolu-
tion, this paper aims to analyze repetitive patterns, or cycles, in the development of
material culture. Our analysis focuses on culture change associated with sociopo-
litical and economic stasis. The proposed toy model describes the cyclical charac-
ter of the quantitative and qualitative composition of archaeological assemblages,
which include hierarchically organized cultural traits. Cycles sequentially process
the stages of uniﬁcation, diversity, and return to uniﬁcation. This complex dynamic
behavior is caused by the ratio between cultural traits’ replication rate and the pro-
portion of traits of the higher taxonomic order’s related unit. Our approach identiﬁes
a shift from conformist to anti-conformist transmission, corresponding with open
and closed phases in cultural evolution in respect to the introduction of innovations.
The model also describes the dependence of a probability for horizontal transmis-
sion upon orders of taxonomic hierarchy during open phases. The obtained results
are indicative for gradual cultural evolution at the low orders of taxonomic hierarchy
and punctuated evolution at its high orders. The similarity of the model outcomes
to the patters of material culture change reﬂecting societal transformations enables
discussions around the uncertainty of explanation in archaeology and anthropology.
Keywords Cultural evolution· Diversity· Taxonomic hierarchy· Innovation·
* Aleksandr Diachenko
1 Institute ofArchaeology, National Academy ofSciences ofUkraine, Kyiv, Ukraine
2 Faculty ofArchaeology, Adam Mickiewicz University inPoznań, Poznań, Poland
Discussions on the evolutionary development of cultural traits as the result of either
adaption to the environment and socioeconomic transformations or migrationist
approaches to culture change have generally framed the clash of archaeological par-
adigms for over a century. This debate returned to the key topics of world archaeol-
ogy during the ongoing Third Science Revolution in archaeology, as deﬁned by K.
Kristiansen (2014). The aDNA and strontium isotope data signiﬁcantly bias current
understandings of culture change towards migrationist explanations (e.g., Furholt,
2021). At the same time, ﬁne-grained chronologies based on the large series of AMS
dates provide a strong ground for testing the evolutionary models. Studies over the
last two decades empirically identiﬁed repetitive patterns, or cycles, in social and
cultural evolution. Such patterns extend the migrationist-evolutionary debate to the
factors and mechanics of long-term human development considered by Darwinian
archaeology and the archaeology of complex systems (e.g., Bentley & Maschner,
2004, 2009; Boyd & Richerson, 1985, 2005; Cavalli-Sforza & Feldman, 1981;
Lyman & O’Brien, 1998; Mesoudi, 2011; O’Brien & Lyman, 2000; O’Brien &
Shennan, 2010; Shennan, 2002, 2008; Shennan, 2009).
Contributing to our understanding of the complex relationship between social
and cultural evolution, this paper aims to analyze the internal factors which cause
the cyclical behavior of hierarchically organized material culture. Similar to evo-
lutionary biology and paleontology, archaeology traditionally addresses its data
through taxonomic ordering. Numerous discussions on the classiﬁcation of mate-
rial culture demonstrate the lack of agreement on strict rules producing taxonomic
units as measures of similarities in material culture. However, these units’ hierar-
chical organization provides a credible understanding of data structure (O’Brien
& Lyman, 2000; Lyman & O’Brien, 2003; cf. Premo, 2021). This paper aims
to understand how the behavior of cultural traits at diﬀerent taxonomic orders is
interrelated. We use the term “orders” in respect to levels of taxonomic hierarchy,
sequentially numbered from the bottom to top.
Cyclical patterns in the dynamics of diﬀerent phenomena have been considered
since the philosophers of Ancient Greece (e.g., Diachenko etal., 2020; Gronen-
born etal., 2014). This study considers repetitive patterns (cycles) as representing
the transition of archaeological assemblages from more uniﬁed to more diverse
and then back to more uniﬁed. We distinguish between two types of such cul-
tural cycles, which may be exempliﬁed by the following empirical cases. The recent
studies of D. Gronenborn and his co-authors identiﬁed cycles in the uniﬁcation
and diversity of LBK pottery styles, which correlate with cycles of social dynam-
ics and conﬂict among the LBK populations of Central Europe (Gronenborn etal.,
2014, 2017, 2018, 2020; see also Mosionzhnik, 2006; Turchin, 2003; Turchin
& Nefedov, 2009). In contrast with D. Gronenborn and his co-authors’ results,
the work on the Western Tripolye culture in Eastern Europe indicated a similar
cyclical pattern of unification and diversity of pottery forms, which does not
correspond with the transformations in socioeconomic organization or spatial
demography of this population (Diachenko etal., 2020).
Self-Organized Cultural Cycles andtheUncertainty of…
These two types of cultural cycles are conditionally labeled as “reﬂective cycles”
and “self-organized cycles.” Reﬂective cycles (exempliﬁed by the LBK case) char-
acterize the socioeconomic and spatio-demographic transformations mirrored in
material culture. Self-organized cycles (exempliﬁed by the Western Tripolye culture
case), which referred to social stasis, are assumed to result from the neutral (unbi-
ased) cultural evolution or conformist (biased) transmission. The latter is associated
with the disproportionally high probability of acquiring the more common variant
(Boyd & Richerson, 1985; Deﬀner etal., 2021; Denton etal., 2020). Conformist
transmission and its opposite, anti-conformist transmission, are chosen without
considering the cost of success (Boyd & Richerson, 2005). We assume that self-
organized cycles occur during the periods of sociopolitical and economic stasis in
populations’ long-term development, while reﬂective cycles result from deep soci-
etal transformations. Overlap between the two types of cycles at diﬀerent taxonomic
orders should also not be excluded. Developing our understanding of repetitive pat-
terns in cultural dynamics, this study speciﬁcally focuses on self-organized cycles.
As an alternative to the majority of other models of cultural evolution, the math-
ematical component of our approach simultaneously considers the behavior of
cultural traits at diﬀerent taxonomic orders and does not include innovation rate
as one of the model input parameters. Therefore, the development of this alterna-
tive approach suggests the introduction of the toy model of cultural evolution. As
already discussed in archaeological studies (and beyond), the label “toy model”
does not denigrate the model’s relevance. Instead, application of toy models
aims to understand the relationship between the key parameters, evaluate the
eﬀect of these parameters on model behavior, and avoid turning models into
“black boxes” (Drost & Vander Linden, 2018). Following the logic of modeling
in physics (Feynman, 1965), we develop the model to explain speciﬁc patterns
and processes in the archaeological record and then test its utility to explain the
results of other models. More speciﬁcally, the repetitive patterns in hierarchi-
cally organized data are addressed through the replacement of cultural traits. We
then explore the utility of the toy model to explain a shift from conformist to anti-
conformist transmission, open and closed phases in cultural evolution, probabilities
for horizontal transmission at diﬀerent orders of taxonomical hierarchy during open
phases, and the impact of various factors on assemblages’ diversity. Finally, we dis-
cuss the uncertainty of explanation in archaeology. However, let us begin by analyz-
ing uncertainty (informational entropy) as the methodological tool underlying the
identiﬁcation of cultural cycles.
Entropy andCultural Cycles
Current understandings of prehistoric cultural cycles are strongly related to the
concept of informational entropy as a quantitative approach to the analysis of the
archaeological record. Entropy is a measure of the diversity and uncertainty of infor-
mation, and the probability of introducing a new element to the system (Shannon
& Weaver, 1963). While developed in information theory, the concept of entropy
is widely applied in various natural and social sciences to analyze systems’ internal
diversity. Archaeological applications mainly consider Shannon’s (1948) entropy,
also known as Shannon’s diversity index (Bevan et al., 2013; Bobrowsky & Ball,
1989; Crema, 2015; Dickens Jr. & Fraser, 1984; Drost & Vander Linden, 2018;
Fedorov-Davydov, 1987; Furholt, 2012; Gjeseld etal., 2020a, b; Gronenborn etal.,
2014, 2017, 2018, 2020; Justeson, 1973; Kandler & Crema, 2019; Neiman, 1995;
Nolan, 2020; Premo & Kuhn, 2010; Shott, 2010). According to this approach, “uni-
ﬁcation” is not opposed to “diversity”; instead, both categories are considered to
gradually replace each other. Notions of “more uniﬁed” or “more diverse” assem-
blages are replaced by the values of Shannon’s entropy (H). The latter is estimated
where pi is the proportion of elements belonging to the ith type (∑pi = 1), and K is
the normalizing coeﬃcient (Shannon, 1948).
Archaeological literature often intuitively considers assemblage diversity by
accounting for only the number of artifact types or stylistic variation. For example,
an increase in the number of pottery types intuitively means an increase in pottery
diversity. In terms of analytical approaches, in this case, diversity equals the cate-
gory of “richness” or maximal entropy discussed below (Lyman, 2008; Shott, 2010).
Application of Shannon’s entropy simultaneously considers qualitative and quan-
titative characteristics of the analyzed sets. Let us illustrate this statement using the
following example. Assume an assemblage of artifacts is composed of three types,
A, B, and C. Let artifacts of type A comprise 0.8 of this set, while artifacts of types
B and C account for 0.1 each (Fig.1a: 1). According to formula 1 (K = 1), the
entropy of the assemblage is estimated at c. 0.28 (Fig.1b: 1). Now let us add arti-
facts of type D to this assemblage, setting the distribution of types C and D at 0.05
each (Fig.1a: 2). Then, the proportion of artifacts of type A is decreased to 0.6,
while the proportion of type B and C artifacts is increased to 0.2 each (Fig.1a: 3).
Remarkably, the change in artifact proportions results in higher internal assemblage
diversity than the change in its qualitative composition. The obtained values of H
are c. 0.41 (change in proportions, Fig.1b: 3) and c. 0.32, respectively (change in
composition, Fig.1b: 2).
The maximal entropy, also known as Hartley’s entropy (Hartley, 1928), corre-
sponds with the uniform distribution of cultural traits across all types. This is illus-
trated by the increase of entropy in our example to c. 0.48, which corresponds with
an equal proportion of cultural traits of the three types (Fig.1a: 4 and 1b: 4). Hart-
ley’s entropy (Hmax) is expressed as follows:
where S is the number of system states.
Both Shannon’s entropy and Hartley’s entropy can be used to analyze artifact
types belonging to a single order of taxonomic hierarchy. However, as recently
explored by Ray J. Rivers and his co-authors (Rivers, in preparation), hierarchi-
cally organized archaeological data requires the application of Rényi’s entropy.
Hmax =log S
Self-Organized Cultural Cycles andtheUncertainty of…
Rényi’s entropy also generalizes Shannon’s and Hartley’s entropy, and is esti-
mated as follows:
where Hλ is Rényi’s entropy of order λ (Rényi, 1961).
Fig. 1 Hypothetical example of the composition of an artifact set (a) and its diversity estimated as Shan-
non’s entropy (b)
Formula 3, as the value of Hλ approaches zero, has properties of Hartley’s
entropy (formula 2). If λ in formula 3 approaches one, then formula 3 has properties
of Shannon’s entropy (formula 1).
One of the general properties of entropy is that it decreases with reduction
(Gromov, 2013). In our case, reduction means a combination of lower order tax-
onomic units into higher order taxonomic units. Therefore, the number of units
(system states) is reduced with the increasing order of taxonomic hierarchy,
and Rényi’s entropy increases as the order of a unit in archaeological taxonomy
decreases. This statement can be illustrated by the following example: consider
two artifact classes, with proportions of 0.6 and 0.4. Each class includes two types
distributed with proportions of 0.6 and 0.4. Each of these types is divided into two
subtypes, which also comprise 0.6 and 0.4. The entropy of the subtypes (the lowest
“archaeological order,” λ → 0) is estimated at c. 0.9. The entropy of types (the sec-
ond “archaeological order,” λ → 1) is estimated at c. 0.58, while the entropy of class
(the third “archaeological order,” λ = 2) is estimated at c. 0.28 (Fig.2). Since Rényi’s
entropy decreases as the order of taxonomic hierarchy increases, human culture in
its widest sense is associated with Min-entropy, Hmin, which is expressed as:
Let us consider artifact diversity at a single order of taxonomic hierarchy by con-
sidering the other hypothetical example. Diversity is estimated by applying Shan-
non’s entropy to simplify explanations for cultural cycles observed in the archae-
ological record. According to the selected diversity measure (formula 1), more
uniﬁed artifact sets are associated with the quantitative predominance of artifacts
belonging to one or few types (Fig.1: 1). Therefore, in the case of simple replace-
ment of cultural traits by each other, cultural cycles of “uniﬁcation–diversity–uni-
ﬁcation” are represented by the quantitative predominance of either one or several
types shifted towards a more even distribution of diﬀerent types, and then towards
Fig. 2 Hypothetical example of the increase in Rényi’s entropy corresponding to the decrease of the
order in archaeological taxonomy. A total of 0.6 and 0.4 are the proportions of artifacts in taxonomic
units and H is the Rényi’s entropy
Self-Organized Cultural Cycles andtheUncertainty of…
the predominance of other type(s). One may also consider a stable relative number
of the most frequent type(s); this is contrasted with a sequential change among less
frequent cultural traits. Both scenarios illustrate the vital role of types characterized
by higher proportions (further labeled as primary types) in shaping the initial and
ﬁnal stages of cultural cycles. Therefore, self-organized cultural cycles are consid-
ered for the assessment of the frequency seriation, which is graphically represented
as “battleship curves” (Lyman & O’Brien, 2006; O’Brien & Lyman, 2000, 2002), in
the framework of informational entropy. Let us now consider the model of dynamic
behavior which results in “battleship curves.”
Self‑Organized Cultural Dynamics: aToy Model
Having framed modeling cycles in the context of the “culture as information”
approach (Aunger, 2009; Boyd & Richerson, 1985; Cavalli-Sforza & Feldman,
1981; Krakauer et al., 2020; Nolan, 2020), we should begin by introducing the
parameter of “informational capacity.” This is deﬁned as the sum of the total num-
ber of cultural traits which can be maintained by the system at all taxonomic scales
and cultural memory. Cultural memory may be deﬁned as mostly incomplete infor-
mation on cultural traits previously available in the system, but not currently main-
tained. Informational capacity is further framed by interaction and communication,
which are greatly impacted by both population size and density (Bentley etal., 2011;
Chabai etal., 2020; Deﬀner etal., 2021; Fletcher, 1995; Henrich, 2004; Knappett,
2011; Lyman etal., 2000; Powell etal., 2010; Shennan, 2018; Sterelny, 2021).
Social knowledge is stored in collective mind. Individual human beings can
remember c. 1.5 billion bytes of information (Landauer, 1986). However, expand-
ing or reducing informational capacity does not simply mean adding or subtracting
a person to or from a group. The clue to the change in informational capacity is
the degree of social complexity reﬂected in the extent of similarity and diﬀerence
between information shared by individuals in a group (Bentley etal., 2011). There-
fore, the values of informational capacity may increase or decrease over time as a
result of economic or sociopolitical transformations, which impact their interaction-
communication networks. In this case, system behavior leads to reﬂection cycles
(e.g., Gronenborn etal., 2020; Hilbert, 2015; Kohler etal., 2009; Turchin, 2003).
Since self-organized cultural cycles are unassociated with societal transformations,
they are characterized instead by their stable informational capacity which remains a
constant value over a period of time.
Our toy model distinguishes between more frequent primary traits and less
frequent “secondary” cultural traits at the same order of taxonomic hierarchy;
while conditionally dubbed “secondary” traits for the convenience of presenta-
tion, they also include tertiary types. The model is based on the assumption that
system behavior results from primary components at diﬀerent taxonomic hier-
archical orders. Therefore, the behavior of secondary components has a limited
freedom, as they are framed by the behavior of primary components. Assuming
a limited number of cultural traits whose behavior deﬁnes a culture in its narrow
sense (e.g., archaeological culture: Roberts & Vander Linden, 2011), the model
describes the subsequent replacement of these cultural traits by each other over
time. In the case of conformist (biased) transmission, the extremely rapid spread
of new “fashions” may be considered in the ﬁeld of memetics, which originates
from R. Dawkins’ (1976/2016) Selﬁsh Gene, ﬁrst published in 1976. The inten-
sive spread and “mutation” of cultural traits and their replacement by subsequent
cultural traits over time lie at the core of archaeological thought and, probably,
speciﬁcally distinguish cultural and biological evolution (e.g., Aunger, 2009;
Laland etal., 2001; Shennan, 2002).
Since behavior of primary types is assumed to be independent and is approxi-
mated using the frequency seriation (see above), primary types’ quantitative
dynamics may be approached by applying the discrete version of the logistic
equation. Widely known from chaos theory, the discrete version of the logistic
equation describes diﬀerent processes, such as turbulence and population dynam-
ics (Feigenbaum, 1978, 1979; May, 1976). A number of studies applied this equa-
tion to archaeological data as well (Barceló & Del Castillo, 2016; also see Kohler
etal., 2009; Turchin & Korotaev, 2006). The discrete version of the logistic equa-
tion is expressed as follows:
where Mt and Mt + 1 are the proportions of a component M in a system at the time
periods t and t+1 respectively, and x is the growth rate of component M (x ≤ 4).
The iterations of formula 5 lead to the transition from simple periodic behavior
to a regime with complex aperiodic growth with period-doublings. System behav-
ior is caused by the values of the growth rate x. If x exceeds 1 but it is less than 2,
then Mt + 1 stabilizes near the following value:
If x belongs to the range between 2 and 3, the values of Mt + 1 oscillate around
the same value (formula 6), and then stabilize near this value. If x is greater than
3 but is less than 3.45, Mt + 1 oscillates between two values; if x exceeds 3.45 but
is less than 3.54, Mt + 1 oscillates between four values etc. (Feigenbaum, 1978,
1979; May, 1976).
We have made two modiﬁcations to the logistic equation. The ﬁrst modiﬁca-
tion introduces thresholds to the proportions in formula 5. The second considers
the decline in the proportion of cultural traits belonging to a primary type.
Thresholds are applied to consider the heterogonous character of archaeologi-
cal data (and, therefore, culture in its wide sense) and its taxonomic structure
(the number of cultural traits belonging to lower hierarchical orders is limited
by the number of traits at higher orders’ related units). Given that a primary trait
α belongs to the set of an order λ composed of m primary traits and the related
number of secondary traits, all taxonomically related to m primary traits of an
order λ + 1, the threshold
is introduced into the logistic equation as its ﬁrst
Self-Organized Cultural Cycles andtheUncertainty of…
are the proportions of cultural traits of the primary type α of
an order λ at time t and t+1 respectively, and
is the replication rate of cultural
traits of the primary typeα.
Formula 7 preserves the properties of formula 6 discussed above.
According to the second modiﬁcation to the logistic equation, after the pro-
portion of primary type starts to decrease, the secondary types become primary,
while the primary type shifts to a secondary type. This decrease in proportion of
traits may be slow or rapid. Slow decreases are associated with the relatively low
. Rapid declines are associated with relatively high values of
Let us now consider the behavior of secondary types framed by the changes in
primary type proportions. For the sake of explanation, we will begin with a hypo-
thetical data set composed of two types of artifacts. These are primary typeα and
secondary type β. Both types have a common threshold
equal to 1. Since the
proportion of artifacts of type β depends on the change of proportion of artifacts of
type α and replication rate of type β,
are the proportion of cultural traits of secondary type β at the
time t and t+1 respectively, and rβ is the replication rate of secondary typeβ.
Basing on formulas 7 and 8, the total change in the proportions of artifacts
referred to the primary type α and secondary type β may be expressed as follows:
Therefore, secondary type β artifact replication rate is estimated as:
In order to consider the replication rates of multiple secondary types at diﬀerent
orders of taxonomic hierarchy, formula 10 may be generalized as follows:
is the replication rate of nth secondary type cultural traits and
is the proportion of nth secondary type cultural traits having a common threshold
with primary type
Considering multiple primary types at diﬀerent taxonomic orders, the overall
change in the proportions of cultural traits belonging to the secondary and primary
types may be presented as:
Let us now return to the properties of the ratio
. At this ratio’s increas-
ing high values, the proportion of cultural traits belonging to primary type α rapidly
increases and then signiﬁcantly drops. At the given limits of
7 and 10), the respective increase in the proportion of cultural traits belonging to
secondary types may be lower than the decrease of traits belonging to primary types.
Therefore, there are cases when formula 12 is not satisﬁed.
Fig.3 exempliﬁes the identiﬁed singularity in the model equations, i.e., system
behavior reaching a point when the model equations break down. Assume a change
in the proportion of artifacts belonging to two types, α and β. The primary type rep-
lication rate equals 3.645. The secondary type replication rate is estimated according
to formula 10. The proportion of primary type α artifacts increases from 0.6 in the
ﬁrst unit of time to 0.875 in the second unit of time, while the proportion of artifacts
belonging to secondary type β decreases from 0.4 to 0.125. In the third unit of time,
the proportion of artifacts belonging to type α drops to 0.399. However, under the
highest possible value of rβ given by formula 10, artifacts belonging to type β can
reach only 0.439. Therefore, the sum of the proportion of artifacts of both types is
equal to 0.838 (Fig.3).
Fig. 3 Hypothetical example representing the singularity in equation12 (the replication rate in primary
type is equal to 3.645)
Self-Organized Cultural Cycles andtheUncertainty of…
This singularity may be interpreted as the open phase of the system, when
innovations are introduced to the existing set of cultural traits. This interpreta-
tion is similar to the understanding of a singularity in artiﬁcial intelligence and
technology, as an unpredictable societal transformation subsequently replacing
a rapid increase in technology (Shanahan, 2005; also see Bentley etal., 2011).
The key point in this similarity is the rapid growth preceding signiﬁcant system’s
reorganization. At ﬁrst glance, such an approximation for the rapid decline in cul-
tural trait proportion might seem an oversimpliﬁcation. However, this may be an
example of cultural mutation or a form of selection bias. To illustrate this, con-
sider an individual’s music playlist. They may choose to play some tracks increas-
ingly often for a given period of time, before the frequency with which a particu-
lar track is played suddenly drops. While the track is still listened to, it is played
less frequently, while other songs increasingly take its place, and additional new
tracks are added to the playlist. Examples of the rapid increase followed by signif-
icant drops may be also found in names, buzzwords, scientiﬁc terms etc. (Bentley
Modeling the selection process and the introduction of innovations requires
developing new equations or modifying previously existing approaches for fur-
ther integration into our toy model. We currently suggest the following correction
to the model equations, which allows us to avoid the identiﬁed singularity by add-
ing the parameter describing the introduction of innovations to a set. Following
our assumption on open phases in the system behavior, the sum of the proportions
of innovative traits introduced into a set (
) is framed by the proportions of
cultural traits belonging to the primary and secondary types in the set in the pre-
ceding time unit:
While the ratio
does not reach substantively high values to move a set
to a singularity in the subsequent time unit, the
equals zero. When this ratio
reaches substantively high values, a set is moved to a singularity in a subsequent
time unit. During the time unit associated with the singularity, the proportion of pri-
mary trait available in the set drops, and the values of
Integrating diversity at diﬀerent scales, the dynamics of cultural traits replac-
ing each other and the introduction of innovations are represented by:
Formula 14 describes cultural dynamics at the two nearest orders of taxonomic
hierarchy, which result in repetitive patterns sequentially processing the stages
of uniﬁcation, diversity, and return to uniﬁcation of archaeological assemblages.
Iterating this equation at the other taxonomic orders, the one obtains assem-
blage diversity composed of hierarchically organized cultural traits, which are
selectively neutral or passed through the conformist transmission. Let us illustrate
the model behavior and its outcomes.
One more hypothetical set of cultural traits exempliﬁes the behavior of our toy
model (Fig.4). Consider a class of artifacts composed of two types, “A” and “B,”
distributed with proportions 0.7 and 0.3. Each type is composed of two variants.
These are the variants “A1” (70% of type “A”), “A2” (30% of type “A”), “B1” (70%
of type “B”), and “B2” (30% of type “B”). The threshold of variants (
the proportions of types they compose at each unit of time. The values of
estimated according to formula 12. The replication rate
of primary variants is
identical to the replication of the related types.
The replication rate of primary type “A” and variant “A1” is initially set for 2.5.
Primary type “A” decreases with this replication rate and becomes a secondary type.
Respectively, secondary type “B” increases with the replication rate of c. 2.26 and
becomes primary. Type “B” further grows supported by the replication rate, adding
0.2 for each time unit. The model represents a gradual replacement of type “A” by
type “B” (Fig.4a). The corresponding variants’ behavior is characterized by ﬂuctua-
tions in variants of type “B” (Fig.4b).
Variant “A1” decreases during time unit 2, subsequently increases during time
units 3 to 7, and decreases again during time units 8 and 9. Variant “A2” rapidly
increases in time unit 2, then decreases in time units 3–7, and increases again in
time units 8 and 9. Variant “B1” oscillates in frequency during time units 1–5. The
decline in variant “B2” in time unit 5, when formula 12reaches a singularity, causes
the introduction of innovating variant “B3” (formula 14). After its introduction, the
replication rate in one of the secondary variants,
, is set to an arbitrary taken
value of 8 in time units 5–9. Notably, despite setting the replication rate of vari-
ant “B3” to a constant value, its proportion neither remains constant nor gradually
increases or decreases over time as one would expect. Instead, the proportion of var-
iant “B3” increases in time unit 7 and decreases in time units 8 and 9, in the result
of changing the proportions of type “B,” i.e., the related unit of the higher order of
hierarchy. Variants “B1” and “B2” ﬂuctuate in relatively narrow ranges during time
units 7–9 (Fig.4b).
Diﬀerent behavior at diﬀerent taxonomic hierarchal orders results in diﬀerent val-
ues of entropy Hλ (Fig.4c). This makes four important conclusions possible. First,
as suggested by the properties of Rényi’s entropy incorporated into the model, the
diversity increases from the top down in archaeological taxonomies.
Fig. 4 Model behavior illustrated by the hypothetical example of the cultural set. The hypothetical arti-
fact set is divided into two types, each composed of two variants. Artifact types replace each other over
time (a). The resulting behavior in variants leads to the introduction of innovation, “Variant B3” (b). The
resulted values of Rényi’s entropy indicate diﬀerent intensiveness of cultural dynamics at diﬀerent orders
of taxonomic hierarchy (c)
Self-Organized Cultural Cycles andtheUncertainty of…
Second, changes in the cultural trait proportions result in cyclical trends, which
represent the transition of an assemblage from uniﬁcation to diversity and then
back to uniﬁcation (increase and subsequent decrease in entropy are represented in
Fig.4c: types increase in time unit 1–2; variants increase in time units 1–3 and 5–8).
More uniﬁed assemblages exhibit the quantitative predominance of one or a few cul-
tural traits. More diverse sets are characterized by a more even distribution of traits
belonging to the same taxonomic order. Therefore, the highest values of entropy in
our hypothetical example (Fig.4c, types–time unit 2, variants–time units 3, 7, and 8)
correspond to the lowest diﬀerence in trait frequency (Fig.4a: time unit 2, and 4b:
time unit 3), also accounting an increase in the proportion of innovative trait occur-
ring in the system (Fig.4b: time units 7 and 8).
Third, the intensity of cultural dynamics increases from the top down in archaeo-
logical hierarchies. Type behavior in our example is shaped into a single cycle rep-
resenting the increase and subsequent gradual decrease in diversity. Two smaller
cycles represent the behavior of variants (Fig.4c: time units 1–5 and 5–9).
Fourth, as demonstrated by the behavior of “Type B” and “Variant B3” which
replication rate was set to a constant value, the proportion of the lower order’s
taxonomic unit is framed by the proportion of the higher order’s taxonomic unit
(Fig.4b). Therefore, these results are indicative for interrelated behavior at diﬀerent
Let us now turn to the utility of our approach to generalize the outcomes of other
models dealing with patterns and processes in cultural evolution.
The presented model describes the dynamics of cultural systems incorporating hier-
archically organized traits, which results in self-organized cultural cycles. These pat-
terns do not correspond to bias in assemblages following economic or sociopoliti-
cal changes. The increase and rapid decline in primary types and their subsequent
replacement are controlled by the ratio
, or the ratio between the replica-
tion rate of cultural traits and the proportion of traits of the related unit of the higher
taxonomic order. The relationship between cultural traits’ behavior at diﬀerent taxo-
nomic orders is controlled by the parameter
The replication rate of a cultural trait is also a key parameter in a number of neu-
tral models of cultural evolution. For example, F. Neiman’s (1995) model produces
similar behavior depending on two factors: the rate of innovation and the random
process of cultural drift. E. Gjeseld and his co-authors’ model (2020a) considers
cultural development in the context of replication rates and the extension of traits.
The factor of cultural transmission may be added to these two factors (Gjeseld
The replication rate represents a change in the proportion of a cultural trait
over time. Its values may be aﬀected by demography, the analyzed phenomena’s
duration, and artifacts’ use life. In this way, the proposed approach generalizes
the outcomes of other models.
Self-Organized Cultural Cycles andtheUncertainty of…
A trait’s replication rate may be increased or decreased depending on the
growth or decline of the number of “consumers” and “producers.” This agrees
with the outcomes of our model and approaches which link cultural diversity
to population size (Bentley etal., 2011; Creanza et al., 2017; Crema & Lake,
2015; Henrich, 2004; Lycett & Norton, 2010; Neiman, 1995; Richerson et al.,
2009). The increase in population size and density is considered as one of the
factors which inﬂuence the diﬀerent spread rates between modern and prehistoric
cultural traits (e.g., Bauman, 2001; Baumeister, 2005). Of course, when dealing
with cultural evolution, we should consider the eﬀective population size instead
of the total population number, as was ﬁrst suggested in the analysis of genetic
analogies decades ago (Wright, 1931). In genetics, eﬀective population size is
a measure of the random genetic drift on a population genetic feature of inter-
est in a real population (Premo, 2016). In respect to culturally evolving traits,
the complex relationship between census population size and eﬀective popula-
tion size comprises innovation ﬂow, diﬀerent modes of cultural transmission, and
social network structure (Deﬀner etal., 2021; Premo, 2015, 2016, 2021; Premo &
Other factors which impact replication rates, and which therefore inﬂuence
increases or decreases in cultural diversity, are the duration of occupations or ana-
lyzed periods, and diﬀerent artifact type life use (Mesoudi & O’Brien, 2008; Perreault,
2019; Premo, 2014; Schiﬀer, 1987; Shott, 1989, 2010; Surovell, 2009). For instance,
variability in Paleolithic and Mesolithic arrowheads is generally higher than the diver-
sity of scrapers (Tomasso & Rots, 2021), because the use life of an arrowhead is much
shorter than the use life of a scraper. Of course, both guided and non-guided variations
in these categories also impact their internal diversities (Deﬀner & Kandler, 2019;
Shennan, 2009). However, the major diﬀerence in these diversities may be explained
exclusively by diﬀerent replication rates. The proposed toy model produces cyclical
cultural dynamics even when population size, artifact life use, and duration of the ana-
lyzed data are set as constant.
The model equations were initially focused on replacing cultural traits and the
cycling nature of culture. However, the model also describes several empirically
known patterns of cultural evolution: ﬁrst, the cumulative nature of culture; and sec-
ond, its punctuated, dynamic development, as reﬂected in the “open” and “closed”
phases associated with diﬀerent types of cultural transmission. The other important
model outcomes are the appearance of innovations as large complex packages and
the mechanics of cultural memory.
According to formula 11, the decrease in cultural traits among secondary types
does not necessarily reach zero. For example, their number may take the values of
10−5 or lower orders, and as such is too small to be associated with absolute num-
bers greater than one. We tend to interpret such signiﬁcantly small values as a rep-
resentation of cultural memory (see above). This way the model “preserves” types
(or the “ideas of a style”) in a cultural system, even if they are not produced for a
period of time. Even when they have passed out of use, these styles are preserved in
cultural memory, and may be used again with the increase of values of the related
. Such borrowing of “new” styles from the past is well-known from
experiments and empirical evidence (Mesoudi, 2010; Premo & Scholnick, 2011).
Singularity in the model equations, which is associated with a signiﬁcant decline
in cultural traits among primary types, is interpreted as a state of openness to inno-
vations. As an alternative to other models which assume a continuous innova-
tion rate (Crema & Lake, 2015; Henrich, 2004; Neiman, 1995; also see: Kandler
& Crema, 2019), our approach suggests a discrete character of new cultural trait
acceptance. This corresponds with the empirical observations and outcomes of other
models, which consider the appearance of innovations as large complex packages
(e.g., O’Brien & Shennan, 2010; Shennan, 2002).
Let us consider how discrete character of innovation occurrence is derived from
the model. According to formula 14, closed phases result in suﬃciently high values
of primary cultural traits’ replication rate. Given a constant probability for copy-
ing errors, the number of copying errors increases proportionally to the increase in
replication rate. The increasing number of copying errors increases the number of
inventions as “creation of new ideas and objects.” Therefore, a rapid increase in rep-
lication rate simultaneously moves an assemblage to the transition from closed to
open phase with respect to the occurrence of innovations and increases the num-
ber of inventions which may be adopted (i.e., may become innovations) during open
phase. Innovation is “an adoption of an idea or object by other agents” (deﬁnitions
for the “invention” and “innovation” are derived from O’Brien and Bentley, 2021).
As shown by Eerkens and Lipo (2005), the increasing number of copying errors
increases the variability of continuous cultural traits. By reassessing the variability
of continuous traits as the increase in the number of discrete taxonomic units in a
hierarchically considered trait, we ﬁnd an agreement between their model and out-
comes of our approach related to neutral evolution. In this way (taking into account
the role of cultural memory), cultural systems incorporate an option for introduc-
ing not only external traits as innovations, but also internally developed inventions.
The latter statement may be illustrated by the previously mentioned example in the
Western Tripolye culture pottery assemblages, where sphero-conical vessels were
replaced by biconical vessels (Diachenko etal., 2020; Ryzhov, 1993, 2000).
With respect to the biased cultural transmission, closed phases in cultural dynam-
ics are associated with conformist transmission, which is replaced by anti-con-
formist transmission and the adoption of innovations during open phases (formulas
13 and 14). Anti-conformist transmission, or negative frequency-dependent bias,
describes the deliberate exclusion of previously “fashionable” traits in the copying
process (Boyd & Richerson, 1985, 2005; Deﬀner et al., 2020; Kohler etal., 2004;
Shennan & Wilkison, 2001).
It is also likely that the probabilities for innovation resulted from horizontal cul-
tural transmission during open phases vary at diﬀerent scales of taxonomic hier-
archy (formula 14). Vertical transmission occurs between generations in the same
population, while horizontal transmission describes the innovation ﬂow between dif-
ferent populations (Boyd & Richerson, 1985; Shennan, 2002).
For the sake of explanation, let us assume the following. First, the probability for
borrowing an idea from “neighbor” assemblages is proportional to the number of
units at the related taxonomic order in “neighbor” sets. Second, the probability for
borrowing an idea from available inventions in a set is proportional to the number of
inventions in this set. Third, there is an equal probability for adopting an available
Self-Organized Cultural Cycles andtheUncertainty of…
invention and a trait from a neighboring group’s assemblage. The increase in the
number of system states from the top to down in archaeological taxonomy gener-
ally decreases the values of
and, therefore, increases the values of the ratio
even with decreasing values of
top-down the taxonomic hierar-
chy (formula 14). Therefore, open phases occur more frequently in lower orders of
archaeological taxonomy (as exempliﬁed by the model results: Fig.4c). The number
of copying errors decreases with the decrease in replication rate top-down the hier-
archy during closed phases. This decrease in inventions reduces the probabilities for
“oﬀering” an idea during an open phase with a decreasing taxonomic order. There-
fore, the probability for horizontal transmission increases top-down in archaeologi-
The increase in the values of
with a growing order of taxonomic hierarchy
during closed phases also requires an increase in the replication rate
to achieve the values of
which support the stable or growing numbers
of primary cultural traits. The increase in
causes an increase in copying errors,
which in turn increases the number of inventions, and thus the number of possible
innovations further occurring in open phase. Therefore, the probabilities for hori-
zontal transmission decrease with the increase of unit’s taxonomic order. For exam-
ple, consider pottery decoration from a taxonomic perspective. Some “elements” of
ornamentation (“lowest” taxonomic order) are horizontally transmitted more fre-
quently than their combinations (“medium” order), while ornamentation “motives”
(comprising combinations of “elements”) are relatively rare. For instance, numerous
examples for this statement are presented in the volume “Import and Imitation in
Archaeology” (Biehl & Rassamakin, 2008), labeling cultural traits passed through
horizontal transmission as “imitations” and “inﬂuences.” Generally, this conclusion
also agrees with the ethnographic observations of Shennan and Steele (1999). The
selection process and constraints of introducing complex traits into diﬀerent cultural
systems should be considered with further development of the model. Also, these
model outcomes limit the explanatory potential of memetics to the lower orders of
archaeological taxonomies resulting in the values of the
ratio, which are
suﬃciently high to support rapid spread, replacement, and mutation of memes.
From the perspective of the model outcomes related to cultural hierarchy and
heterogeneity, diﬀerent concepts of cultural evolution are not considered here as
contradictory. The qualitative and quantitative dynamics of cultural traits of low
orders of taxonomic hierarchy and artifacts characterized by a short-term use life
are better explained by gradual evolution. The evolution of artifacts characterized
by medium- and long-term use life and cultural traits belonging to higher orders
of taxonomic hierarchy ﬁts the concept of punctuated evolution. The latter assumes
long periods of relative stasis, interrupted by periods of rapid bursts of innovation
(Bak, 1996; Bentley, 2003; Dow & Reed, 2011; Gould & Eldredge, 2003; Lyman
& O’Brien, 1998). Closed and open phases of cultural evolution, as presented in the
toy model, are respectively associated with periods of stasis and bursts of change. It
is important to note that the concept of punctuated equilibrium is often misunder-
stood. One of the key issues is the misinterpretation of species branching resulting
from the stratigraphic position of fossils in sediments, which are diﬀerent in age, but
have parallel layers (for the extended discussion see O’Brien & Lyman, 2000). The
crucial agreement between punctuated equilibrium and the outcomes of our model
is the increasing relative stasis (duration of closed phases) with an increasing taxo-
Basing the presented model on entropy as a measure of the diversity and uncer-
tainty of information, it would be reasonable to conclude with uncertainty of
archaeological thought. To a great extent, this uncertainty is caused by the qual-
ity of archaeological record (Perreault, 2019). However, singularities in the cur-
rent version of model equations may lead to a somewhat disappointing conclusion
regarding our limited ability to predict cultural evolution at its widest chronologi-
cal ranges, at least with respect to unbiased and conformist cultural transmission.
If Shanahan’s (2005) understanding of singularity as unpredictable transformation
also applies to our model, then precise predictions are only possible between sin-
gularities. Such predictions of the assemblages’ composition during a shift from
conformist to anti-conformist transmission will remain unreached. Since our model
preserves the chaotic properties of the logistic equation, it is sensitively dependent
on the initial conditions, and long-term evolution will remain a space for surprises
and uncertainties. On one hand, this possibility somehow reﬂects the current state
of the art—otherwise archaeologists’ and anthropologists’ recent work on detailed
models would apply to materials from the evolution of Paleolithic geometric orna-
ments on mammoth bones to Impressionism. On the other hand, the ability to pre-
dict the occurrence of singularities from the values of model parameters is already a
Besides indicating possible limitations in approaching cultural evolution, compar-
ing our model with the other models exempliﬁes two signiﬁcant issues in modeling
components of archaeological explanation. These issues are the diﬀerent cultural
and social processes behind similar data distributions and diﬀerent input parameters
causing similar model behavior.
Self-organized and reﬂective cultural cycles represent, respectively, social sta-
bility and social change. However, despite the diﬀerent cultural development
mechanics, both types of cycles may be represented by similar artifact distribution
patterns (cf. Kandler & Crema, 2019; O’Dwyer & Kandler, 2017). This similar-
ity emphasizes that the relationship between societal transformations and cultural
development is far more complex than generally assumed. A signiﬁcant number of
models and empirical observations indicate the reﬂection of adaptation to ﬂuctu-
ating environment or sociopolitical and economic development in material culture
change, including the correspondence between cultural and social cycles (Deﬀner
& Kandler, 2019; Gronenborn etal., 2020; Kohler etal., 2020; Roux, 2010, 2014;
Roux etal., 2017). A number of our model outcomes show that similar patterns in
prehistoric assemblage distributions may strictly result from unbiased or conformist
transmission. This ﬁnds an agreement with the conclusions of Kandler and Crema
(2019), which emphasize the underlying need to carefully interpret the consistency
between data and neutral models, because other transmission models may be equally
consistent. Hence, the identiﬁcation of a cultural cycle does not necessarily mean
the identiﬁcation of social transformations or social stasis.
Our approach ﬁnds an agreement with the results of other models in the possibil-
ity of slowing or accelerating cultural dynamics by changes in population size, the
Self-Organized Cultural Cycles andtheUncertainty of…
life use of artifacts, and/or the duration of the analyzed data. In other words, expla-
nation in particular studies of some artifact categories can be uncertain because of
the possibility of simultaneous eﬀect of these factors on data distribution, regardless
of whether inductive or deductive approaches are applied to data analysis. As we
have shown in the “Entropy and Cultural Cycles” section which assessed “battleship
curves” in frames of informational entropy, one obtains cycles in uniﬁcation–diver-
sity–uniﬁcation of assemblages’ diversity. However, “battleship curves” result from
either the neutral models including innovation rate as an input parameter (Kandler &
Crema, 2019; Neiman, 1995), or, as in our approach in relation to neutral evolution,
without including this parameter.
How can future research ﬁnd a path forward to overcome these issues? The
answer lies in contextualizing analyzed materials in a wider framework based on
their taxonomic hierarchical scale, and the simultaneous behavior of other catego-
ries of interdisciplinary data. At ﬁrst glance, this answer sounds somewhat exag-
gerated. However, it is grounded in the important property of entropy, that is, that
the entropy (uncertainty) of non-interacting systems generally exceeds the entropy
of interacting systems (Gromov, 2013). Therefore, data contextualization as a con-
sideration of diﬀerent interacting categories at diﬀerent scales reduces a myriad of
multiple possible explanations to a much shorter list. Extending this conclusion to
modeling in archaeology, we consider it reasonable to test the utility of models with
the aim of describing speciﬁc patterns and processes to explain other patterns and
Further development of the presented toy model should consider the spatial dis-
tribution of cultural traits. More speciﬁcally, the model should be integrated with
approaches to analyze cultural incubators (Crema & Lake, 2015), cultural distances
(Nakoinz, 2014), isolation-by-distance principle (Shennan etal., 2015), and other
Acknowledgements We are sincerely grateful to Ray J. Rivers for discussing his ideas on cultural
dynamics and approaches to its understanding with us, commenting on the draft of this paper, and for
permission to cite his ongoing, but not yet published research. We are inﬁnitely grateful to Michael J.
O’Brien, who chose to not remain anonymous, and two anonymous reviewers for their valuable critical
comments and suggestions helping to improve this paper. Many thanks are addressed to Erika Ruhl for
editing the English language and her valuable comments on the draft of this manuscript.
Funding This paper was made possible by a grant from the Science Centre of Poland, awarded to Iwona
Sobkowiak-Tabaka (2018/29/B/HS3/01201). Open access to this article was made possible by the pro-
gram “Excellence Initiative – Research University” ﬁnanced by Adam Mickiewicz University in Poznań
(decision no. 011/08/POB5/0053).
Availability of Data and Materials Not applicable
Code Availability Not applicable
Ethics Approval The authors are responsible for correctness of the statements provided in the manuscript.
Consent to Participate Not applicable
Consent for Publication Both authors gave explicit consent to submit an article.
Conﬂict of Interest The authors declare no competing interests.
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