2023 Asia Pacific Signal and Information Processing Association Annual Summit and Conference (APSIPA ASC)

Evolutionary Analysis and Cultural Transmission
Models of Color Style Distributions in Painting Arts
Eita Nakamura∗ and Yasuyuki Saito†∗ Kyoto University, Kyoto, Japan

E-mail: eita.nakamura@i.kyoto-u.ac.jp Tel/Fax: +81-75-753-5396† National Institute of Technology, Kisarazu College, Chiba, Japan

E-mail: saito@j.kisarazu.ac.jp Tel/Fax: +81-438-30-4137
Abstract—This paper studies the evolutionary dynamics andcultural transmission of color styles in painting arts. Creative
cultures, such as visual arts, develop through repeated processesof knowledge transmission and modification, and characteristicsof these processes can lead to macroscopic evolutionary patterns.
While recent studies have found intriguing trends and dynamic
patterns in the evolution of visual arts, the relationship between
these trends and cultural transmission processes remains poorly
understood. We first analyze the evolutionary dynamics of color
distributions in oil paintings by applying a clustering method.
The results reveal that some of the frequencies of the color style
clusters exhibit synchronous changes that are related to creators’
biases in producing paintings belonging to specific clusters. We
subsequently construct models of cultural evolution to represent
how these biases are transmitted between creators, based on two
hypothetical transmission modes: one guided by influencer-todescendant connections and another guided by community-todescendant connections. Through statistical inference, we showthat the influencer-guided transmission model better fits the historical data. Furthermore, highly influential creators, as inferredby the model purely from content data, remarkably align withthose commonly mentioned in the literature on art history.

I. I NTRODUCTION
The recent advancement of generative AI [1]–[3] has drawn
attention to the application of image processing in the fieldof arts, such as painting. The automatic creation technology
based on machine learning enables the generation of images indiverse styles. While machine learning relies heavily on a large
accumulation of image data historically created by humans,
the generation of paintings in new creation styles remains a
big challenge [4]. In fact, there is still limited quantitativeand scientific understanding of the process in which humans
develop new creation styles. If we can understand this human’s
creative intelligence, the knowledge can be utilized to realize
automatically generating paintings in a much wider variety ofstyles than current systems.
Creative cultures have evolved through repeated processes oftransmission and modification of creative knowledge. Cultural
evolution including such cultural transmission processes has
been studied based on theoretical analogies with biologicalevolution, and two factors, environmental and dynamical, inthe evolution process are generally considered to shape evolutionary processes [5]. Environmental factors refer to various
factors that generate selective pressure, and determine thedirection and constraints of evolution. In the case of painting

culture, these include people’s preferences, the supply of
painting materials, and social backgrounds such as economics
and politics. On the other hand, dynamical factors refer to theeffects caused by the processes of knowledge transmission,
mutation, and selection, and are generally relevant for understanding the speed and regularities of evolution.
Research on the evolutionary analysis of painting art data,
mainly focusing on Western art, has recently become an active
field [6], thanks to the availability of large-scale digital data
and the development of image analysis techniques including
deep learning. Previous studies have found trends in certain time periods, in the entropy-complexity features used inphysical system analysis [7] and in the features extracted byconvolutional neural networks (CNNs) [8], [9]. It has also
been shown that the characteristic temporal changes found inthese analyses generally agree with the analysis by art experts,
and the creators who represent the style of each era coincidewith those widely mentioned in art history [7], [8], [10].
Another study suggested that there were significant changes
in the average features of paintings, particularly during the15th and 16th centuries, and in the late 19th century, from theanalysis of color contrast, which can be associated with the
popularization of oil paints and the increase in individuality inpainting styles [11]. The existence of trends spanning several
centuries and the presence of revolutionary periods with rapidchanges have also been observed in music data, where similar
evolutionary analysis has been conducted [12]–[15]. These
quantitative results have revealed the relationship between thetemporal changes in painting styles and social/environmental
factors. However, the impact of dynamic factors due to culturaltransmission on art evolution still remains poorly understood.
In this study, focusing on the fact that creators usually
produce artworks in several creation styles, we investigate
the impact of cultural transmission of these creative style
distributions on macroscopic evolution of the painting culture.
First, we analyze the color styles of oil paintings by using
a clustering method and show that there are synchronouschanges in the frequencies of certain color style clusters. We
also confirm, through an analysis of the network of clusters
obtained from creators belonging to multiple style clusters, thatthe creators’ biased distribution of color styles had significant
influence on the aforementioned synchrony.
To understand the process by which creators select theDownsamplingandcolor reduction

distribution of color styles, we construct cultural evolution
models incorporating a transmission process of this distribution
between creators and compare them with a random selection
model. In cultural transmission, various modes of learning
are generally possible, such as learning from a specific influencer, as in a master-descendant relationship, or learningfrom numerous creators or artworks. We here consider two
extreme possibilities: influencer-guided transmission, where
one influencer is mainly involved in the transmission, andcommunity-guided transmission, where the average featuresfrom a community of creators are transmitted. We formulate
the mathematical models to represent these hypothetical cases.
Through statistical inference based on these models, we showthat the influencer-guided transmission model better explains
the historical data. Furthermore, we validate the potential ofthis model, which allows for the estimation of the creator’sfitness (influence) from image data by unsupervised learning,by measuring the degree of match between creators with highestimated fitness and those mentioned in an influential textbook
on art history.

A

Color statistics(40 dim)All artworks
V. van GoghV. van Gogh
1

1

2

2

K
Creator’s artworks

3

・・・

3

・・・

・・・

B

K

Style clusters

Creator’s style distribution

Fig. 1. Processes for analyzing color styles. A: Preprocessing for extracting
color statistics. B: Color style clusters and creator’s style distribution.
100
10-1

Frequency

10-2
10-3
10-4
10-5
10-6

1500

1600

1700

Year

1800

1900

2000

Fig. 2. Evolution of color frequencies. The plotted lines are colored in theindicating base colors (while line is shaded). Arrows indicate notable trends.

where L is an auxiliary parameter corresponding to a “sample
size” to compute the probability. The clustering algorithm canbe derived based on the expectation-maximization (EM) algorithm. In the E-step, the posterior probability is computed as
P (k|n) ∝ πk P (θn |θ̄k ), where L is set as L = 10 t (t denotes
the iteration time) and plays a similar role as the inverse
temperature in the simulated annealing method.
P In the M-step,
the mixture probability is updatedasπ←kn P (k|n) and thePmean statistics as θ̄ki ← n P (k|n)θni . Note that we apply
the clustering without using the temporal information tn . Afterthe convergence of the EM algorithm, we assign each artwork
n to the most probable cluster kn = argmaxk P (k|n), forsubsequent analyses.
To get an overview of the temporal changes of the distribution of data samples, we also project the color statistics on a
two-dimensional visualization space. For this purpose, we use
the t-SNE method [16] with the Jensen-Shannon divergenceused as the distance measure. This dimensional reduction
process is conducted independently of the clustering and isused only for visualization.

k=1

where K is the number of clusters, πk denote the probabilities
of clusters (called the mixture probabilities), and θ̄k is themean statistics of cluster k. The k-th component discrete
distribution model P (θn |θ̄k ) is defined as
P (θn |θ̄k ) =Reduced image(100 x 100)
40 base colors

II. C LUSTER ANALYSIS OF COLOR STYLE
A. Materials and method
We used image data of mostly Western art paintings obtained from a public website WikiArt.org. Some meta data areassociated with the obtained image data, including the yearof production, creator’s name, and media (materials). For ouranalysis, we extracted artworks with media labeled as “oil”,
resulting in 32 401 artworks. We then extracted color frequency
statistics from each of the images as follows. First, all images
were converted to 100 × 100 pixels by downsampling (Fig. 1).
Second, we applied color reduction to I = 40 representative
colors, which form a set of equally spaced points in thecone-shaped HSV color space. Third, the relative frequencies
θn = (θni )Ii=1 of the representative colors i for each image
n were computed. The tuple (an , tn , θn ) of creator an , yearof production tn , and color statistics θn was used for thefollowing analysis. The years of production in the data ranged
from 1270 to 2022. The number of distinct creators was 1128.
To quantitatively study the evolution of creation styles,
we analyze the temporal changes in the distribution of color
statistics. To deal with the high-dimensional space of thestatistics, we first apply clustering in this space to identify
several concentrated regions, which we call color style clusters.
Specifically, we use for clustering the discrete distribution
mixture model defined as
KX
P (θn ) =πk P (θn |θ̄k ),IYOriginal image

B. Results
We used images whose year of production is 1450 or laterbecause there were only a small number of images produced

Lθni
θ̄ki
,

i=1

2

1400s

1500s

1600s

1700s

1800s

1900s

2000s

Clusters

B

Style clusterA

Mean statistics
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
1500

1600

1700
1800
Year

1900

2000

Fig. 3. A: Distribution of artworks in the visualization space. Red dots indicate artworks in each period and gray dots represent all the artworks. In the ‘cluster’panel, artworks are illustrated in the colors of the clusters they belong to (cluster colors are arbitrary). B: Evolution of style cluster frequencies.

before the year. Fig. 2 shows the temporal evolution of therelative frequencies of colors, where the frequencies were
calculated for each 25-years window. The most frequent colors
have low brightness and low saturation, and bright and highsaturation colors are generally rare. Several trends can beobserved in this figure, as indicated by arrows. First, from the15th century to the 17th century, some of the most frequentcolors increased their frequencies in the data, whereas somerelatively frequent bright colors decreased their frequencies.
The frequencies of these colors changed in opposite directions from the mid-18th century to the 20th century. Second,
several rare bright colors had an overall gradual increase of
the frequency from the 16th century to the 20th century.
Last, the frequencies of vivid colors that were rarely used inearly centuries increased rapidly from the mid-19th century
onwards. While these general trends provide some insights,
the periodwise average frequencies cannot represent variation
and biases of color combinations in individual artworks. The
analysis method based on style clusters solves this problem,
as presented in the results below, thus providing a usefulframework for studying underlying evolutionary processes.

some style clusters with synchronous rises and declines. For
example, clusters 1 and 3, and clusters 17 and 18 respectively
have synchronous rises in different centuries. These results
indicate the presence of regularities in the evolution of color
styles. In the next section, we analyze this synchrony in moredetail and study its relationships with creators’ behavior.
III. S YNCHRONOUS CHANGES OF STYLE CLUSTERFREQUENCIES AND CREATORS ’ STYLE DISTRIBUTIONS
A. Analysis method
The relative frequencies of style clusters, as depicted inFig. 3, can be expressed as a time-dependent mixture probability πk (t). The local synchrony Skk0 (t) between the rises
and declines of clusters k and k 0 can be formulated as
Skk0 (t) = √dπk (t) dπk0 (t)1,dt
Ck Ck0 dt

R
where Ck = dt (dπk (t)/dt)2 is a normalization
R constant.
The (overall) synchrony can be defined as Skk0 = dt Skk0 (t).
In the numerical analysis, the mixture probabilities are observed at discrete time steps, and the differentials and integration in these equations should be replaced with differences
and summation w.r.t. time. It is noted that since the relative
frequencies must sum up to unity, a rise of a cluster naturally
induces a decline of another, and Skk0 generally has a negativevalue if the relative frequencies of clusters k and k 0 fluctuate
independently. Therefore, a positive value of Skk0 indicatessynchronous changes between the two clusters.
There are at least two possible factors that cause synchronous changes of cluster frequencies. First, some style
clusters with similar mean color statistics, such as those locatedclosely in the visualization space in Fig. 3, can represent
separate parts of the same color style. In such a case, achange in the frequency of a color style can result in coherentchanges in the frequencies of the relevant clusters. Second, thebehavior of creators who usually produce artworks in several
creation styles can induce the synchrony between clusters
that do not necessarily represent similar color statistics. For

The results of clustering and visualization of color statistics
are shown in Fig. 3, where we used K = 20 clusters toapproximate the distribution. In the visualization space, mostartworks in early centuries are located in the lower half region,
which represents color combinations containing mostly darkcolors. Several important observations can be made in theevolution of the relative frequencies of color style clusters.
First, we can observe concurrent and transient cluster structure[13], [17], that is, several clusters are active in each time
period and active clusters gradually change over time. That
there always exist several active clusters reflects the fact that
there are usually multiple art styles and genres in any society.
Second, there are two notable periods where multiple style
clusters arose: from the 15th to the 16th centuries, and fromthe mid-19th to the mid-20th centuries. These time periods
coincide with the previous result [11] showing periods withsignificant changes in color contrast. Third, we can observe
3

1

1

3

0.75

5

0.5

Cluster

7

0.25

9

0

11

13

–0.25

15

–0.5

17

Network A

Derived synchrony S(A)Uniform
Feature similarity
Creator-mediated association
Frequency-based association−0.042−0.0070.033−0.034

–0.75

19
1

Fig. 4.

TABLE ID ERIVED SYNCHRONIES FOR DIFFERENT CLUSTER NETWORKS .

3

5

7

9 11 13 15 17 19
Cluster–1

Pairwise synchrony Skk0 between clusters.

example, there are many creators who produced both portraits
and landscapes, each of which are usually painted in different
color combinations, and a change of artistic taste shared in acommunity can result in a synchronous change of color stylesof both genres. To study this aspect, we calculate the color
style distribution πa = (πak )K
k=1 of creator a, by counting thenumber of artworks by the creator belonging to cluster k andnormalize the frequencies.
In both cases, some relations, feature similarity or association through creators, induce connections between clusters
that lead to synchronous changes in the cluster frequencies. As
in the case of association through creators, such connections
can temporally change in general. These connections can be
expressed as a weighted network A(t) = {Akk0 (t)} of clusters, where edges Akk0 (t) represent the degree of connection
between clusters k and k 0 at time t. The contribution ofaR networkon the synchrony can be quantified as S(A) =P
dt kk0 Akk0 (t)Skk0 (t). To compare different networks, we
impose conditions that self-connections
P are zero (Akk (t) = 0)
and that a normalization condition kk0 Akk0 (t) = 1 holds.
To examine possible factors of the synchrony of style cluster
frequencies, we consider the following four cluster networks.
(i) Uniform network: Akk0 = 1/(K −1) (k 6= k 0 ). This
P is a null
hypothesis. (ii) Feature similarity network: Akk0 ∝ i θ̄ki θ̄k0 i .
This represents the cosine similarity between the mean statistics of the clusters.
(iii) Creator-mediated association network:P
Akk0 (t) ∝ a D(t, a)πak0 πak0 , where D(t, a) = 1 if creator
a is relevant at time t and 0 otherwise. This represents theco-occurrence of style clusters in the artworks produced byindividual creators. (iv) Frequency-based association network:
Akk0 (t) ∝ πk (t)πk0 (t). This represents a hypothetical situation
where creators choose the style clusters of their artworks byrandomly sampling from the cluster frequencies of the time,and consequently their style distributions πak coincide withπk (t) of time t when creators a are active. We compare these
networks in terms of the derived synchrony S(A); if a network
Akk0 (t) is relevant to the macroscopic evolution of the cluster
frequencies, then S(A) is expected to be large.

1450–1550

1550–1650

1650–1750

1750–1850

1850–1950

1950–2050

Fig. 5. Creator-mediated cluster networks in different centuries. The nodes
are located similarly as the cluster centers in the visualization space in Fig. 3.

while a majority of the elements have negative values. Table I
lists the derived synchronies for the four cluster networks. Only
the creator-mediated association network yielded a positivesynchrony, indicating that the creators’ style distributions had
an impact on the dynamics of style cluster frequencies andinduced the synchronous changes. The result also suggeststhat the feature similarity of clusters also induced the synchrony. On the other hand, the small derived synchrony by
the frequency-based network indicates that creators’ style
distributions deviated from the random selection case, in abiased and coherent (i.e. shared by many creators of the sametime) way.
The creator-mediated cluster networks visualized in Fig. 5clearly show that the association of clusters changed considerably over centuries. The connection weights between someclusters are much larger than others, and we can observe someedges between distant clusters with distinct styles also havelarge weights. Overall, these results show that creators’ color
style distributions are biased in a coherent way that cannot be
simply explained by the feature similarity or random selectionof style clusters, and as they vary over time these biased style
distributions markedly affected the macroscopic evolution ofcolor styles in painting arts.
IV. C ULTURAL TRANSMISSION OF COLOR STYLE
DISTRIBUTIONS

A. Cultural evolution models for creators’ style distributions
The biased and coherent color style distributions of creators
discussed in the previous section pose an important question:
How did the creators’ style distributions determine? Whilecertain social factors such as the demand from the church or
patrons are conceivable, we here focus on a simple possibility

B. Results
Fig. 4 shows the pairwise synchrony Skk0 between style
clusters, where we have used discrete time steps of 25-year
width. We see some off-diagonal elements with positive values,
4

s

s

of cultural transmission. Since complex knowledge about theart of paining is usually learned from other creators or theirartworks, it is possible that the color style distribution istransmitted in this learning process.
We construct two models to examine what particular modeof cultural transmission is plausible to explain the data. In
cultural phenomena, unlike in genetic evolution, the number ofcultural parents is not limited to one (asexual) or two (sexual),
and transmission from a varying number of cultural parents
are possible. We thus consider two hypothetical situations that
represent extreme cases within a broad range of possibilities. In
the first influencer-guided transmission model, we suppose thata creator learns his/her creation style mainly from a specificinfluencer’s artworks, as in a master-descendant relationship.
In the second community-guided transmission model, we suppose that a creator learns the creation style from numerous
creators’ artworks belonging to some stylistic community. In
the following, we mathematically formulate these models.
We formulate the cultural evolution models based on astochastic description of the transmission-mutation-selection
process [5]. This is a formulation similar to the models of
biological evolution used in population genetics [18], exceptthat the potential of a past creator to produce cultural offspring
can in principle continue for long after his/her death. We
denote by tsa the year that creator a started creation, which
is defined as the earliest year of the creator’s artworks inthe data. Let π̃a = (π̃ak )K
k=1 denote the frequency countsof style clusters in artworks of creator a. By normalizingthese frequencies, we obtain the style distribution πa . We also
denote by πa<t the style distribution obtained from the artworksof creator a produced before year t. Our cultural evolution
models are formulated as explicit realizations of the probability
distribution P (π̃a ) for all creators a.

represented by a weighting factor e−(ta −ta0 )/τc , where the time
constant τc represents the time scale for the bias. The second
bias is the (intrinsic) fitness of an influencer, which represents
a time-independent likelihood of being learned by successors
due to possible factors such as prestige and visibility. To
represent this bias, we introduce a fitness parameter va0 foreach creator. We also require that a potential influencer a0 musthave produced at least Nbound artworks (specifically, we set
Nbound = 7). The selection probability can then be given as

B. Influencer-guided transmission model

where the summation is taken over all artworks n produced
before t, kn denotes the cluster assigned to n, and τp denotes
the time constant for artworks. The parameters η and  represent the strength of the second and third terms and are called
the oblique transmission coefficient and the random mutation
rate, respectively. It is noted that the time-decaying factors
in Eqs. (2) and (3) reduce the influence of past creators andartworks as time proceeds. Additionally, the restrictions ontime in these equations ensure the causal constraint that future
information is not used in the definition of probability P (π̃a ).

ts

<ts

s

s

Psela (a0 ) ∝ I(tsa0 < tsa ) I(Na0 a ≥ Nbound ) e−(ta −ta0 )/τc +va0 ,(2)where I(C) = 1 if condition C is true and otherwise I(C) = 0,
and Na<t
0 denotes the number of artworks produced by creatora0 before year t.
The transmitted style distribution represents the influencer’s
style distribution with modifications including mutations. Here,
we include the influence of creators other than the influencer
since it is certainly possible that a learner creator observes
many others’ artworks and is influenced by the average style
distribution of the time. This situation can be formulated as<tss

πa0 →a = (1 − η − )πa0 a + η π <ta +  1/K,
swhere π <ta represents the average style distribution at (justbefore) time tsa , 1/K = (1/K, . . . , 1/K) denotes a uniform
distribution, and η and  are nonnegative constants. The first,
second, and third terms on the right-hand side represent
the transmission from the influencer, transmission from theaverage style distribution of the time, and random mutation,
respectively. The average style distribution π <t also incorporates the recency effect and is defined asX
πk<t ∝
I(tn < t)I(kn = k) e−(t−tn )/τp ,(3)
n:tn <tIn the influencer-guided transmission model, each learnercreator selects an influencer creator among previous creatorsand learns the influencer’s style distribution. This model canbe formulated asXts
P (π̃a ) =Psela (a0 )Pdis (π̃a |πa0 →a ),a0 :tsa0 <tsawhere the summation sis taken over all potential creatorst
a0 with tsa0 < tsa , Psela (a0 ) is the selection probability of0
the influencer a of creator a, πa0 →a represents the style
distribution transmitted from a0 to a, and the last factor isgiven as a discrete probability distribution
Y
Pdis (π̃a |πa0 →a ) =[π(a0 →a)k ]π̃ak .
(1)

C. Community-guided transmission model
In the community-guided transmission model, we consider
that a learner creator first analyzes previous artworks and finds
communities of creators sharing similar style distributions. The
learner creator then selects a community and learns the meanstyle distribution of the community. This model is described
as
C
X
<ts
P (π̃a ) =ψc a Pdis (π̃a |πc→a ),(4)k

We incorporate two biases in the selection probability torepresent potential tendencies of learner creators, which aresimilar to those used for modeling citation dynamics ofscientific papers [19]. The first is the recency bias, which
represents the creators’ tendency to more likely choose areference influencer that lived more recently. This bias can bec=1

where c indexes communities of creators, C is the numberof communities, ψc<t represents the selection probability of5

TABLE IIP ERPLEXITIES AND ESTIMATED PARAMETERS OF THE EVOLUTIONMODELS . R ESULT FOR THE MODEL USED FOR ANALYZING CREATOR
FITNESS ( SEE TEXT ) IS ALSO SHOWN AT THE BOTTOM .

communities at time t, and πc→a represents the style distribution transmitted from community c to creator a. The
discrete probability distribution Pdis (π̃a |πc→a ) is the sameas in Eq. (1). The creator communities are formalized as adynamic mixture model with the following log-likelihood:
XX−(t−tsa )/τc
<t
<t <t
e
L(t) =ln
ψc Pdis (π̃a |πc ) ,a:tsa <t,
Na<t ≥Nboundc(5)
where τc is the creator time constant, ψc<t is the mixture probability, and πc<t is the mean style distribution of community c.
Assuming that how creators analyze the community structure istime-dependent and recent creators are given more attention insthis process, the time-decaying factor e−(t−ta )/τc is introduced
to weigh the creators’ contributions on the likelihood. We use
in Eq. (4) the parameter values of ψc<t and πc<t that maximize
Eq. (5). See Sec. IV-D for the inference method.
The transmitted style distribution in Eq. (4) is formulated
similarly as in the influencer-guided transmission model: we
suppose that the mean style distribution of the selected community has the dominant influence, but the mean average style
distribution of the time also has some influence and there isa small chance of random mutation. This situation can beformalized as<ts

Perplexityτc (yrs)

η

19.1
15.6
16.4
16.2
16.1

NA
117
160
110
220

NA
0.37
0.38
0.52
0.58

Influencer-guided (for analysis)

16.2

50

0.1

optimized the other parameters are fixed. For creators’ fitness
va , we use the expectation-maximization (EM) algorithm with
the influencer variable a0 treated as a latent variable for eachlearner creator a, which can be derived in a manner similar to
the Gaussian mixture model. For parameters τc and η, we use
the grid search method.
In the community-guided model, the model parameters to be
estimated are {ψc<t , πc<t }, τc , and η. We again use an iterative
method for optimizing these parameters, and the parameters τcand η are optimized by grid search. We use the EM algorithm
for estimating ψc<t and πc<t with the likelihood function in
Eq. (5). To ensure the temporal consistency of the community
estimation, we first perform community estimated on a coarse
time scale of 25-years width and use this result to initialize
the parameters ψc<t and πc<t for individual years.

sπc→a = (1 − η − )πc a + η π <ta +  1/K,
where η represents the oblique transmission coefficient also inthis model.

E. Results

D. Inference method

Table II shows the model comparison result. As a reference,
we also compared the random model where a creator generates
artworks with cluster frequencies π̃a according to the temporalssaverage style distribution π <ta : P (π̃a ) = Pdis (π̃a |π <ta ). The
creator time constant τc was estimated in the resolution of approx. 1 year for the influencer-guided model, and 10 years forthe community-guided model, due to the large computational
cost of the latter model. We tested C = 10, 20, and 30 for thecommunity-guided model. Compared to the random model,
the influencer-guided and community-guided models both hadsignificantly lower perplexities, indicating a clear effect of
cultural transmission. We see that the number of communities
C does not have much impact on the perplexity, while large
values of the oblique transmission coefficient were estimatedfor C = 20 and 30, suggesting that these models overfitted
the community structure. By comparison, the influencer-guided
model had a smaller perplexity, which shows that influencerguided transmission better fits our data. On the other hand,the oblique transmission coefficient was also relatively large
for this model, implying that the color style distribution islearned not only from one influencer but also from others.
The estimated value τc = 117 yrs suggests that the influenceof past creators sustains for a relatively long time.
An interesting possibility of the influencer-guided modelis that it can estimate the fitness (influence) of creators in
an unsupervised manner. To examine the model’s potential,

The main purpose of the evolution models presented in theprevious sections is to compare the two hypotheses on thecultural transmission process. Both models are formulated asrealizations of the same probability distribution P (π̃a ), and we
can compare them by their total log-likelihoods normalized by
the total number of artworks
X
 X

Ltot =
ln P (π̃a )
π̃ak ,aModel
RandomInfluencer-guided (optimal)
Community-guided (C = 10)
Community-guided (C = 20)
Community-guided (C = 30)

a,kor the exponentiated quantities κ = exp(Ltot ) called the
perplexity. The perplexity can be interpreted as the effective
number of style clusters to choose. The smaller this quantity is,
the more predictive a model is. The log-likelihood value Ltot
depends on the model parameters in both evolution models.
Following the maximum likelihood principle, we estimate themodel parameters to maximize Ltot , that is, we compare thetwo models in their optimal conditions. Since the values ofthe mutation rate  and artwork time constant τp did notaffect the likelihood so much, as they are effectively smoothing
constants, we fixed them as  = 10−6 and τp = 20 yrs. In thefollowing, due to the lack of space, we briefly explain specificmethods for the inference of model parameters.
In the influencer-guided model, the model parameters to be
estimated are {va }, τc , and η. We use an iterative method
for optimizing these parameters; when one parameter (set) is6

40

F-score (%)

TABLE IIIH IGHEST FITNESS CREATORS ESTIMATED BY THE INFLUENCER - GUIDEDMODEL (τc = 50 YRS , η = 0.1). NAMES IN BOLD FONTS INDICATE THOSE
APPEARING IN G OMBRICH ’ S BOOK [20].

τc = 50 yrs, η = 0.1

35
30
25
20

τc = 117 yrs, η = 0.37

15
10

Random

5
0

0

50

100
150
200
List size of top creators

250

300

Fig. 6.
Comparison of F-score measuring the accuracy of estimating
influential creators in comparison with Gombrich’s book [20].

we compared the list of creators with high estimated fitness
with the list of creators appearing in Gombrich’s classicaltextbook on art history [20]. Among the creators listed in[20], we removed a few who did not have major activitiesof painting and obtained a list of 101 creators. Using themodel estimation, we made lists of varying sizes containing
creators with the highest fitness, and we calculated the F-score
to measure how much the results align with the Gombrich’slist. We also conducted a similar calculation for a randomly
sorted list of creators. The result in Fig. 6 shows that the Fscore curve by the influencer-guided model with maximum
likelihood parameters had significantly high values than thatfor the random list. While this result is already remarkable,
we found that with alternative parameterization with a shortertime constant τc = 50 yrs and a smaller oblique transmission
constant η = 0.1, the F-score curve had even higher values(Fig. 6), even though the perplexity becomes higher (Table II).
We discuss this mismatch between the optimized models forthe likelihood measure and the accuracy of estimating “wellknown” creators in the discussion section.
The 25 creators with the highest estimated fitness by the
influencer-guided model with τc = 50 and η = 0.1 are shownin Table III. We can see that the list includes creators active invarious centuries. Importantly, the list contains creators with arelatively small number of artworks in the dataset, including
such well-known creators as S. de Vlieger, R. Lichtenstein,and H. Hoffman. This result shows that the model predicts thefitness not merely from the number of artworks in the data,
which although highly correlates with the creator’s prestige.
That this result was obtained purely from the image data andthe meta data about the year of production for ensuring causal
constraints demonstrates the potential of the present model for
quantitatively and objectively analyzing painting art data andcultural evolution.

Fitness eva

Creator aYear tsa

No. of works

1.81
1.77
1.75
1.70
1.63
1.62
1.60
1.58
1.57
1.54
1.53
1.48
1.45
1.44
1.44
1.43
1.43
1.42
1.42
1.41
1.40
1.38
1.38
1.38
1.37

T. van Rysselberghe
I. Aivazovsky
S. de Vlieger
TitianC. Lorrain
C. Hernandez
D. Teniers II
G. Bellini
M. Hafftka
P.-A. Renoir
H. Hofmann
F. Boucher
K. Malevich
P. Picasso
D. Velazquez
T. GainsboroughJ. van Eyck
R. Lichtenstein
C. Monet
H. Matisse
V. van Gogh
C. Corot
P. Gauguin
A. Altdorfer
R. Bujnowski

1880
1835
1629
1503
1630
2001
1633
1455
1986
1858
1902
1734
1904
1893
1616
1745
1426
1961
1861
1890
1881
1825
1873
1507
2001

125
238
29
200
62
17
54
56
8
913
16
51
113
499
127
125
63
26
191
188
747
409
460
25
9

this macroscopic phenomenon. We then constructed cultural
evolution models incorporating transmission and selection processes of creators’ color style distributions and found that the
influencer-guided transmission model better fits the data thanthe community-guided transmission model. We also showedthat the model can estimate the creators’ fitness by unsupervised learning and automatically discover important figuresin the history of painting art only from the image data. All
together, these results indicate that in the cultural transmissionof knowledge about painting styles, not only the average color
combinations but also the distribution of different sets of color
combinations is transmitted from creators to creators, and thismicroscopic process of individual creators can lead to themacroscopic patterns in the evolution of painting style.
It should be emphasized that the present models represent
a simplified description of the real evolutionary process. For
example, the model can be extended to include more than one
cultural parent, or dynamic changes in the parameters such as
the creator time constant and the oblique transmission coefficient. The mismatch of the preferred parameterizations by the
maximum likelihood estimation and the optimal accuracy of
reproducing the human expert’s account, found in Sec. IV-E,
may be an indication of such incompleteness of the current
model. It may also be possible that the present model, which“analyzes” a much larger amount of data than a human can,can find objectively correct results that humans may not findeasily. Therefore, future work should study more elaborated
models and examine how these models will (or will not) alter
the results, and additionally, conduct a benchmark test, for

V. C ONCLUSIONS AND DISCUSSION
We have analyzed the evolution of color styles in painting
arts using statistical machine learning and evolutionary modeling methods. By analyzing the dynamics of color style clusters,
we have found synchronous changes in the cluster frequencies.
We also found that creators’ color style distributions are biased
in a coherent way and this property is a major factor of7example, using synthetic data, to quantitatively measure theability of the present model. It is also important to examine
the influence of the number and values of representative colors,
which were somewhat arbitrarily chosen in this study.
Our result has prospective applications for enhancing thegenerative AI technology. Using the evolution model with
influencer-guided transmission, we can estimate the transmission paths of knowledge about creation style. In scientometrics,
for example, such knowledge reference data (i.e. citation data)
are used as fruitful sources for studying the dynamics ofcreative activities including the quantification of the scientificcontribution [21], the formation of communities [22], and thenature of creativity [23], [24]. Since complete knowledge transmission paths are rarely provided in self-reports or historians’
accounts especially in art fields [25], a data-driven method
would provide useful complemental data for revealing themechanisms by which new creation styles are developed. The
found mechanisms can be integrated with machine learningtechniques to realize “creative” generative AIs, for example,
using the framework of evolutionary computation.
Finally, the present framework of applying a model integrating a probabilistic generative process and cultural evolution
process can be applied to a wide domain of data. While we
focused on the color style in this study, there are other important stylistic features such as spatial composition, brushstroke
pattern, and painted objects. We can apply DNNs for analyzingthese various features [8], [9] and with a modification of thedata generation process, we can conduct a similar analysis
on more general features. Other possible cultural domainsof application include music [26]–[28] and literary art [29],and culinary art [30], where cultural transmission of complex
knowledge is considered equally important to understand thecreative aspect of human intelligence.

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ACKNOWLEDGMENT
We thank Takayuki Itoh, Daichi Mochihashi, Hitomi
Kaneko, and Gumpu Nakamura for useful discussions. This
study was supported by JSPS KAKENHI Grant Numbers
21K12187, 21K02846, and 22H03661, and JST FOREST
Program Grant Number JPMJPR226X.
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