Transmission of cultural traits in layered ego-centric networks
Although a number of models have been developed to investigate the emergence of culture and evolutionary phases in social systems, one important aspect has not yet been sufficiently emphasized. This is the structure of the underlaying network of social relations serving as channels in transmitting c...
Gespeichert in:
| Datum: | 2014 |
|---|---|
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут фізики конденсованих систем НАН України
2014
|
| Schriftenreihe: | Condensed Matter Physics |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/153481 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Transmission of cultural traits in layered ego-centric networks / V. Palchykov, K. Kaski, J. Kert\'esz // Condensed Matter Physics. — 2014. — Т. 17, № 3. — С. 33802:1-10. — Бібліогр.: 18 назв.. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-153481 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1534812019-06-15T01:27:17Z Transmission of cultural traits in layered ego-centric networks Palchykov, V. Kaski, K. Kert\'esz, J. Although a number of models have been developed to investigate the emergence of culture and evolutionary phases in social systems, one important aspect has not yet been sufficiently emphasized. This is the structure of the underlaying network of social relations serving as channels in transmitting cultural traits, which is expected to play a crucial role in the evolutionary processes in social systems. In this paper we contribute to the understanding of the role of the network structure by developing a layered ego-centric network structure based model, inspired by the social brain hypothesis, to study transmission of cultural traits and their evolution in social network. For this model we first find analytical results in the spirit of mean-field approximation and then to validate the results we compare them with the results of extensive numerical simulations. Хоча було розвинено ряд моделей для дослiдження появи культури та еволюцiйних фаз у соцiальних системах, один важливий аспект не був достатньо висвiтлений. Цим аспектом є роль структури мереж соцiальних зв’язкiв, якi служать каналами для передачi культурних особливостей i якi вiдiграють важливу роль в еволюцiйних процесах у соцiальних системах. У данiй статтi ми дослiджуємо цю роль у моделi переачi культурних особливостей та їх еволюцiю в соцiальних системах, спираючись на пластоподiбнi структури егоцентричних мереж, надхненi гiпотезою соцiального мозку. Для цiєї моделi ми отримали аналiтичний розв’язок у дусi середнього поля та дослiдили його застосовнiсть, порiвнюючи отриманi передбачення з результатами чисельних симуляцiй. 2014 Article Transmission of cultural traits in layered ego-centric networks / V. Palchykov, K. Kaski, J. Kert\'esz // Condensed Matter Physics. — 2014. — Т. 17, № 3. — С. 33802:1-10. — Бібліогр.: 18 назв.. — англ. 1607-324X PACS: 89.75.Fb, 89.65.Ef, 89.75.Hc DOI:10.5488/CMP.17.33802 arXiv:1405.6009 http://dspace.nbuv.gov.ua/handle/123456789/153481 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
Although a number of models have been developed to investigate the emergence of culture and evolutionary phases in social systems, one important aspect has not yet been sufficiently emphasized. This is the structure of the underlaying network of social relations serving as channels in transmitting cultural traits, which is expected to play a crucial role in the evolutionary processes in social systems. In this paper we contribute to the understanding of the role of the network structure by developing a layered ego-centric network structure based model, inspired by the social brain hypothesis, to study transmission of cultural traits and their evolution in social network. For this model we first find analytical results in the spirit of mean-field approximation and then to validate the results we compare them with the results of extensive numerical simulations. |
| format |
Article |
| author |
Palchykov, V. Kaski, K. Kert\'esz, J. |
| spellingShingle |
Palchykov, V. Kaski, K. Kert\'esz, J. Transmission of cultural traits in layered ego-centric networks Condensed Matter Physics |
| author_facet |
Palchykov, V. Kaski, K. Kert\'esz, J. |
| author_sort |
Palchykov, V. |
| title |
Transmission of cultural traits in layered ego-centric networks |
| title_short |
Transmission of cultural traits in layered ego-centric networks |
| title_full |
Transmission of cultural traits in layered ego-centric networks |
| title_fullStr |
Transmission of cultural traits in layered ego-centric networks |
| title_full_unstemmed |
Transmission of cultural traits in layered ego-centric networks |
| title_sort |
transmission of cultural traits in layered ego-centric networks |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2014 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/153481 |
| citation_txt |
Transmission of cultural traits in layered ego-centric networks / V. Palchykov, K. Kaski, J. Kert\'esz // Condensed Matter Physics. — 2014. — Т. 17, № 3. — С. 33802:1-10. — Бібліогр.: 18 назв.. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT palchykovv transmissionofculturaltraitsinlayeredegocentricnetworks AT kaskik transmissionofculturaltraitsinlayeredegocentricnetworks AT kerteszj transmissionofculturaltraitsinlayeredegocentricnetworks |
| first_indexed |
2025-07-14T04:37:36Z |
| last_indexed |
2025-07-14T04:37:36Z |
| _version_ |
1837595735906844672 |
| fulltext |
Condensed Matter Physics, 2014, Vol. 17, No 3, 33802: 1–10
DOI: 10.5488/CMP.17.33802
http://www.icmp.lviv.ua/journal
Transmission of cultural traits in layered ego-centric
networks
V. Palchykov1,2,3, K. Kaski1,4,5, J. Kertész6,1
1 Department of Biomedical Engineering and Computational Science, Aalto University,
00076 Aalto, Finland
2 Institute for Condensed Matter Physics, National Academy of Sciences of Ukraine, 79011 Lviv, Ukraine
3 Lorentz Institute for Theoretical Physics, Leiden University, 2300 RA Leiden, The Netherlands
4 CABDyN Complexity Center, Said Business School, University of Oxford, Oxford OX1 1HP, UK
5 CCNR and Physics Department, Northeastern University, Boston, MA 02115, USA
6 Center for Network Science, Central European University, Nador 9, H–1051, Budapest, Hungary
Received May 22, 2014, in final form June 12, 2014
Although a number of models have been developed to investigate the emergence of culture and evolutionary
phases in social systems, one important aspect has not yet been sufficiently emphasized. This is the structure
of the underlaying network of social relations serving as channels in transmitting cultural traits, which is ex-
pected to play a crucial role in the evolutionary processes in social systems. In this paper we contribute to the
understanding of the role of the network structure by developing a layered ego-centric network structure based
model, inspired by the social brain hypothesis, to study transmission of cultural traits and their evolution in
social network. For this model we first find analytical results in the spirit of mean-field approximation and then
to validate the results we compare them with the results of extensive numerical simulations.
Key words: ego-centric network, evolution of culture, mean-field approximation
PACS: 89.75.Fb, 89.65.Ef, 89.75.Hc
1. Introduction
Emergent natural phenomena like ferromagnetism in materials, and social phenomena like forma-
tion of collective opinion or evolution of culture in a society have clearly different origins. However,
during the last two decades it has become increasingly clear that there are important common features,
namely they are all complex systems, in which a large number of interacting units contribute to entirely
new cooperative qualities. Formulating this way, it is not anymore surprising that concepts, methods, and
modelling techniques of statistical physics have increasingly been applied to a number of problems be-
yond the traditional realm of physics, giving rise to such new discipline names as “econophysics” [1] and
“sociophysics” [2, 3].
In physics sense, modelling means conceptual simplification, in which one aims to find a small num-
ber of assumptions to capture the essence of the phenomenon investigated. This is exemplified by amodel
of two possible spin states for each particle, like the Ising model for magnet, which in spite of its simplic-
ity is capable of describing spontaneous symmetry breaking and the related emergence of magnetization.
Similarly, simple models of disease spreading with two possible states (either “susceptible” or “infected”)
for the members of the society permit to describe the emergence of a pandemic state in the system. A
major difference between modelling natural and social phenomena is, however, that in the former case
general principles like variational laws or microscopic equations of motion are usually known, while in
the latter case of social systems, the situation is more complicated. In this case, one uses plausible as-
sumptions about the processes leading to stochastic rules, which are then tested by comparing the results
with reality - usually in a qualitative manner.
© V. Palchykov, K. Kaski, J. Kertész, 2014 33802-1
http://dx.doi.org/10.5488/CMP.17.33802
http://www.icmp.lviv.ua/journal
V. Palchykov, K. Kaski, J. Kertész
Here we will discuss, from the sociophysics perspective, the problem of competition and coexistence
of two cultural traits in a society. The situation is in a sense similar to binary alloys, in which for some
parameters the components mix, while for the others a phase separation takes place. This analogy raises
the question about a kind of cultural phase diagram, which is a known problem in cultural anthropology
[4, 5].Whilemuch of research has been done from themean field point of view, only recently the topology
of the interactions has been taken into account [6].
The natural way to describe the processes of transmission of values in a social system is to assume
transition rates for the evolution of the local properties by which Markov chain equations can be for-
mulated for the behavior of the entire system. While in this context the Markovian approximation has
its obvious limitations, it is still worth studying for several reasons. First, the Markovian approximation
should hold for a large time scale; second, the solutions can serve as references and third, this is the
technique we are most familiar with.
In our present study we focus on modelling a biased transmission of cultural traits, developed to de-
scribe the evolution of culture in a society [7]. Among different types of biased transmission there is direct
bias with one cultural trait being more attractive than the others or indirect bias with an individual using
a characteristic not immediately related to the cultural trait (e.g., individual success). Here we focus on
the former one, namely on direct bias. The assumption that one cultural trait is more attractive than the
other, will cause its frequency to increase in time. However, the effect of this behavior depends on the
network of social interactions that allow the cultural traits to be transmitted between the members of a
social system. Here we adopt the view of ego-centric social networks to describe the emotional closeness
of the social brain hypothesis [8], which besides the restriction on the number of active social network
members for each individual (the Dunbar number of about 150) suggests the entire ego-centric network
consisting of the layers that correspond to different levels of intensities of interactions or emotional close-
ness [9–11].
In this paper we first describe the biased transmission model in section 2; develop a mean-field ap-
proximation for the evolution of a cultural trait in the entire system with the given ego-centric network
in section 3; construct two distinct cases of entire networks with a fixed ego-centric structure in section 4;
and investigate the applicability of the derived mean field approximation. In section 5 we draw conclu-
sions.
2. Transmission models with direct bias
Assuming that there are two possible cultural traits xi (t) = {0,1} for each individual i = 1, . . . , N in
a system, transmission models allow each individual to change his or her cultural trait in time t as a
result of adopting it from available sources. The model of direct bias suggests some bias in choosing
and adopting cultural trait, which means that one of the traits is more attractive. Then we introduce a
quenched quantity b to determine the strength of this bias, the value of which varies within the range
b ∈ (−1,1). For b > 0, individuals are predisposed towards choosing trait 1 while for b < 0 cultural trait 0
is favoured. The possible sources of cultural trait for each ego i correspond to the “friends” of individual
i ego-centric social network and to the ego i him- or herself.
Let us now assume that the ego-centric social network of individual i consists of Ni friends and that
the strength of relation between the ego i and source j is ωi j . It reflects the influence of each of the
source j on the ego i . The values of ωi j are normalized
∑
〈 j 〉i
ωi j = 1, (1)
where 〈 j 〉i means that j runs over the sources of the cultural trait for ego i .
For the direct bias transmission model [7], each ego i decides to choose one of his or her source of
cultural trait and adopts the trait from that source. The decision which source to choose depends both on
ωi j and on the combination of the bias b and the cultural trait x j (t) of the source j . The probability qi j
that the ego i adopts cultural trait from the source j is determined [7] as
qi j =
ωi j (1+β j )
∑
〈k〉i
ωik (1+βk )
, (2)
33802-2
Transmission of cultural traits in layered ego-centric networks
where
β j =
{
b, for x j (t) = 1,
−b, for x j (t) = 0.
(3)
In equation (2) k runs over all available sources of a cultural trait for ego i (including the ego him- or
herself, i.e., k = i ). Note that the value of qi j in equation (2) depends on the whole configuration {x j (t)}i
of cultural traits x j (t) of all the sources for ego i . Then the probability pi (t +1|{x j (t)}i ) that an individual
i will adopt cultural trait xi (t + 1) = 1 at time t + 1 given the configuration {x j (t)}i at time t may be
expressed as follows:
pi (t +1|{x j (t)}i ) =
∑
〈 j 〉i
qi j x j (t). (4)
Equation (4) provides a general rule for the evolution of individual cultural trait. The fraction of individ-
uals p(t), as characterized by cultural trait 1 at time t for the large enough population follows
p(t) =
1
N
N
∑
i=1
pi (t), (5)
where pi (t) is the probability that an individual i has cultural trait xi (t) = 1 at time t . To investigate the
evolution of a cultural trait in the system one has to specify both the structure of the network of social
interactions and the initial conditions. Below we consider two types of ego-centric social networks and
apply a mean-field approximation that allows us to analytically estimate the evolution of a cultural trait
in the entire system.
3. Mean-field approximation
Here, we investigate two cases of social networks that reflect possible sources of a cultural trait. First,
we consider a simple ego-centric network [7], where each individual has only one external source of cul-
tural trait at each time t , but this source changes over time and may be considered as a randomly chosen
individual in a system. Second, we consider layered ego-centric network, assuming that the number of
sources within each layer is large enough. The mean field approximations are obtained assuming that
the probability pi (t) of having a cultural trait xi (t) = 1 is independent of i , and thus pi (t) = p(t). Below
we consider the preference towards cultural trait 1, i.e., b > 0.
3.1. Single external source of a cultural trait
Let us assume that each individual i has only two possible sources of a cultural trait labelled with j :
(i) himself or herself ( j = i ) or (ii) an external source ( j , i ), so that the strength ωi j of social influence of
the source j reads as follows:
ωi j =
{
α0, if j = i ,
1−α0, if j , i .
(6)
In order to reproduce a temporal evolution of an individual cultural trait [equation (4)] the entire config-
uration {x j (t)}i of cultural traits x j (t) of all the sources for ego i must be specified. In the approximation
used, the exact configuration {x j (t)}i is not available and we deal with the probability P ({x j (t)}i ) of real-
ization of each particular configuration {x j (t)}i . Then equation (4) may be expanded as follows:
pi (t +1) =
∑
{x j (t )}i
[
P ({x j (t)}i )
∑
〈 j 〉i
qi j x j (t)
]
, (7)
where the first sum runs over four possible configurations of {x j (t)}i . Recalling that x j (t) for each source
j of cultural trait at time t is as follows:
x j (t)=
{
1, with probability p(t),
0, with probability 1−p(t),
(8)
33802-3
V. Palchykov, K. Kaski, J. Kertész
one may simply calculate the values of P ({x j (t)}i ) for each of the four possible configurations {x j (t)}i .
Substituting these values into equation (7) and taking the strength of social influence of each source
j on ego i , ωi j , into account [equation (6)], one obtains an estimation for the evolution of individual
cultural trait pi (t+1), which in themean-field approximation is equal to the fraction of population p(t+1)
[equation (5)] characterised by a favoured cultural trait [7]:
p(t +1) = p(t)+p(t)
[
1−p(t)
] 4bα0(1−α0)
1−b2(2α0 −1)2
. (9)
The result (9) was obtained in 1980-s and it shows that the system starting from arbitrary small num-
ber of individuals with cultural trait 1 reaches a final state in which all individuals are characterized by
the favored trait 1 for an arbitrarily small value of bias b > 0. A more complicated structure of a social
network (decision among a number of sources) makes the calculations harder, and numerical simula-
tions were then used to investigate the system behavior. R.I.M. Dunbar [12] considered the behavior of
the system where each ego has two layers of sources of a cultural trait, and where the strength of re-
lations decays with the layer number. Numerical calculations demonstrated a qualitatively similar but
sufficiently faster spreading of the cultural trait in a system when more sources are simultaneously con-
sidered by the ego.
3.2. Layered ego-centric network
Here we develop a mean-field approximation for the evolution of a cultural trait in a system with a
layered ego-centric social network. Let us assume that there are two layers of friends that correspond
to different levels of emotional closeness: Let each individual have n1 first- and n2 second layer friends.
Then the strength of social influence can be written as follows:
ωi j =
α0, if i = j ,
α1, if j is the first-layer friend for ego i ,
α2, if j is the second-layer friend for ego i ,
(10)
and satisfy the normalization condition (1). If the number of sources within each layer is large enough,
then nl p(t) and nl [1−p(t)] approximate the numbers of l -th layer friends with the cultural trait equal to
1 and 0, respectively. Assuming that nl p(t) and nl [1−p(t)] match the corresponding numbers, the influ-
ence of the first- and second-layer friends on the ego i may be considered as an effective eternal field; and
this will allow us to avoid considering all possible configurations {x j (t)}i of cultural traits, as it is shown
below. Within this assumption, qi j among the whole configuration {x j (t)}i of cultural traits depends only
on two of them, namely on xi (t) and x j (t) being implemented into βi and β j , correspondingly:
qi j =
ωi j (1+β j )
α0(1+βi )+
∑2
l=1
{αl nl p(t)(1+b)+αl nl [1−p(t)](1−b)}
. (11)
The normalization condition (1) being substituted into equation (11) allows us to exclude both α1n1 and
α2n2 from the equation (11):
qi j =
ωi j (1+β j )
α0(1+βi )+ [1−α0]{p(t)(1+b)+ [1−p(t)](1−b)}
. (12)
To investigate the temporal evolution of a cultural trait, equation (7) is used again. Within the approx-
imation used here, the nested sum of this equation becomes dependent only on the cultural trait xi (t) of
ego i :
∑
〈 j 〉i
qi j x j (t)=
α0(1+βi )xi (t)+ [1−α0]p(t)(1+b)
α0(1+βi )+ [1−α0]{p(t)(1+b)+ [1−p(t)](1−b)}
. (13)
Now to estimate pi (t +1), it is enough to consider only two possible values of xi (t) among the whole set
of configurations {x j (t)}i in equation (7). If xi (t) = 1, then the probability that an individual i conserves
the cultural trait at time t +1 follows
∑
〈 j 〉i
qi j x j (t)
∣
∣
∣
xi (t )=1
=
α0(1+b)+ [1−α0]p(t)(1+b)
α0(1+b)+ [1−α0]p(t)(1+b)+ [1−α0][1−p(t)](1−b)
. (14)
33802-4
Transmission of cultural traits in layered ego-centric networks
Instead, if xi (t) = 0 the probability that the individual i will adopt a cultural trait xi (t +1) = 1 at time t +1
follows
∑
〈 j 〉i
qi j x j (t)
∣
∣
∣
xi (t )=0
=
[1−α0]p(t)(1+b)
α0(1−b)+ [1−α0]p(t)(1+b)+ [1−α0][1−p(t)](1−b)
. (15)
According to equation (7), the probability pi (t +1) that an individual i is characterized by a cultural
trait 1 at time t +1 as follows:
pi (t +1) = p(t)
∑
〈 j 〉i
qi j x j (t)
∣
∣
∣
xi (t )=1
+ [1−p(t)]
∑
〈 j 〉i
qi j x j (t)
∣
∣
∣
xi (t )=0
. (16)
Substituting equations (14) and (15) into equation (16) and taking into account that pi (t) = p(t), one
obtains the mean-field approximation for the fraction of the population with the cultural trait favored by
bias as follows:
p(t +1) = p(t)+p(t)(1−p(t))(1−α0)
[ 1+b
Σ(t)−α0b
−
1−b
Σ(t)+α0b
]
, (17)
where
Σ(t) = 1− (1−α0)b[1−2p(t)]. (18)
Under the assumption that the numbers n1 and n2 of first- and second layer friends are large enough
to apply the mean-field approximation and due to the normalization condition (1), the result (17) ob-
tained does not depend either on α1 or on α2. Only the value of α0 is important. For similar reasons, the
existence of additional layers does not change the result (17) provided that the number of sources within
each layer is large enough. To verify the validity of the mean-field approximation (17), in the next section
we compare it with numerical simulations.
4. Numerical simulations
Finally, we perform numerical simulations of the biased transmission model on layered ego-centric
networks in order to test the results of the mean-field approximation (17). According to the social brain
hypothesis [8], the capacity of a human brain limits the number of individuals with whom one can main-
tain real social relationships. After reaching that limit of about 150 acquaintances, the quality of rela-
tionships changes significantly. Naturally, within this set of acquaintances there are large differences as
regards emotional closeness of the acquaintances to an ego. R.A. Hill and R.I.M. Dunbar (see [13] and ref-
erences therein) state that different layers of emotional closeness exist, creating a set of inclusive layers,
the sizes of which increase in factors of about 3 with each layer, starting from a support clique of ∼ 5, to
a sympathy group of ∼ 12, to a band of ∼ 35 and to a cognitive group of ∼ 150) individuals. Moreover,
these layers of emotional closeness are correlated with the frequency of contacts between individuals
[14], so that an ego contacts with a member of a support clique on a daily basis, with a member of a
sympathy group on a weekly basis, with a member of a band on monthly basis and with a member of
a cognitive group on a yearly basis. For our numerical simulations we will implement this ego-centric
network structure to investigate the transmission of a cultural trait in social networks, as we describe
herein below.
4.1. Network generation
We generate several networks of social interactions that satisfy the following conditions: i) each indi-
vidual has a number of ego-centric friendship layers up to the third, fourth or fifth levels; and ii) the sizes
of these inclusive layers increase with the scaling ratio 3, thus consisting of 5, 15, 45, 135 etc. members.
An example of such an ego-centric network is shown in figure 1 (a). Since the entire networks with the
given ego-centric picture may be designed in a number of ways, we generate two types of networks that
satisfy described the ego-centric properties.
Initially, the corresponding random networks were generated. For this reason, we generate a set of
nodes the number of which is given in table 1. Then, each individual i is connected to randomly chosen
five other nodes in the system so that each ego arrives at 5 first layer neighbors connected by the strongest
33802-5
V. Palchykov, K. Kaski, J. Kertész
(a) (b)
Figure 1. (Color online) Example of an ego-centric social network (a) with two layers of friends, and an
example of the corresponding entire hierarchical social network of a minimal size (b). Dark links (red
online) connect the first layer friends and light-grey links (yellow online) correspond to the second layer
friends.
Table 1. Basic network characteristics: the number of nodes N and the total number of links L for the
3-, 4-, and 5-layer networks. The given number of nodes N is the minimal number of nodes that are
necessary to reproduce the network with the described ego-centric picture.
N L
3 layers 54 1251
4 layers 162 10935
5 layers 486 98415
links of weight α1. Then, we repeat this procedure for each consecutive layer l (link weight αl ) avoiding
having multiple links between any pair of nodes. Thus, each an individual is expected to have 10 links of
weight α2 to the second layer friends, 30 links of weight α3 to the third layer friends, etc. The expected
number of links L in a network is shown in table 1. Due to this construction procedure, a few nodes
may have a smaller number of the weakest links in the network. However, we expect that this does not
affect the final result since this number is small enough and concerns only the weakest links in a random
network.
Then, as another alternative for a network with ego-centric property, we construct a hierarchical
network. Initially, the entire set of nodes (see table 1) is divided into groups of six individuals each, and
each pair of the nodes within each group are connected by the links of the strongest weight α1. Thus,
each ego has got 5 links to the first layer friends. Next, the set of these groups are grouped into clusters,
each consisting of 3 groups and thus containing altogether 18 nodes. In order to build the second layer
of ego-centric network, each node of the initial group is connected to 5 out of 6 individuals of each of the
selected groups in the cluster with the link weight α2. The choice of the five nodes is done in the way to
obtain a maximally regular network. The example of such network is shown in figure 1 (b). In order to
obtain a hierarchical network of higher order, we repeat the procedure by merging the three clusters,
etc. The constructed hierarchical network has a strong relation between the number of nodes N and the
number of links L in the network, see table 1.
The distribution of link weights between an ego and the members of each layer l of his or her ego-
33802-6
Transmission of cultural traits in layered ego-centric networks
centric network shows a proportional decrease of the link weight αl with the layer number l with a
decreasing rate:
r =αl /αl+1 . (19)
Having generated these two different networks of social interactions, we will next perform numerical
simulations of the biased transmission model on them with different values of r .
4.2. Biased transmission model
Now let us consider the model of biased transmission equation (4) on the above-discussed network
structures. In the simulations, we use the same value α0 = 0.5 for each individual i = 1, . . . , N in the
system, which gives the probability that an individual i considers him- or herself as a source of cultural
trait in unbiased case b = 0. The bias parameter is set fixed to b = 0.1 and we use three different values of
parameter r (19): r = {2,5,10}. For the initial (t = 0) conditions, only a single randomly chosen individual
k is characterized by a favored cultural trait xk (0) = 1 and all the others m , k are characterized by
xm (0) = 0. The results of numerical simulations are averaged over 1000 runs of the model and a random
network has been regenerated for each run. Since initially only a single node k is characterized by amore
attractive cultural trait xk (0) = 1, the favored trait has disappeared in more than 60% of runs at the early
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 20 40 60 80 100 120 140
a
r = 2
r = 5
r = 10
Eq. (3.4)
Eq. (3.12)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 20 40 60 80 100 120 140
b
r = 2
r = 5
r = 10
Eq. (3.4)
Eq. (3.12)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 20 40 60 80 100 120 140
c
r = 2
r = 5
r = 10
Eq. (3.4)
Eq. (3.12)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 20 40 60 80 100 120 140
d
r = 2
r = 5
r = 10
Eq. (3.4)
Eq. (3.12)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 20 40 60 80 100 120 140
e
r = 2
r = 5
r = 10
Eq. (3.4)
Eq. (3.12)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 20 40 60 80 100 120 140
f
time (t)
p
(t
)
r = 2
r = 5
r = 10
Eq. (3.4)
Eq. (3.12)
Figure 2. (Color online) Fraction p(t) of trait favored by bias as a function of time t . Top panels [(a), (b)]
correspond to three layer ego-centric network of social interactions, panels (c) and (d) correspond to the
four layer network, and the bottom panels (e) and (f) to the five layer networks. The left hand side panels
demonstrate evolution of cultural trait in random networks while the right hand side panels correspond
to the hierarchical networks. The evolution of cultural trait on networks with different values of link
weight rate r = {2,5,10} are represented by balls, squares and triangles, respectively. Dot-dashed line
corresponds to the results of the single external source mean-field approximation (9) and the dotted line
represents the results of the mean field approximation (17) for layered ego-centric networks.
33802-7
V. Palchykov, K. Kaski, J. Kertész
stage of simulations. Thus, below we consider only those runs at time t for which the system contains at
least a single node with the trait 1 at t . The results of our investigation are shown below for networks
with the different number of layers.
Let us start with 3-layer egocentric network, where each individual has 5 first-, 10 second- and 30
third-layer friends, so that inclusive layer sizes follow 5, 15 and 45. The fraction p(t) of individuals with
a favored cultural trait on a random network is shown in figure 2 (a) as a function of time t . The same de-
pendency on a hierarchical network with 3-layer ego-centric networks is shown in figure 2 (b). The mean
field approximations (9) and (17) are shown by dot-dashed and dotted lines, respectively. The results
show that the approximation (17) describes the evolution of a cultural trait for a random network much
better than the single source approximation [7]. Moreover, the distribution of the link weights among
the layers, described by parameter r , does not affect the evolution of a cultural trait in random networks
as predicted by the mean-field approximation. However, the value of r affects the process on hierarchi-
cal networks, so that for large values of r , the evolution of the favored cultural trait in the hierarchical
network becomes slower than expected by mean field approximation.
This behavior is even more pronounced for the systems having four and five layers in ego-centric
social networks. The results for the four-layer ego-centric networks, where each individual has 5 first-, 10
second-, 30 third- and 90 fourth layer friends are depicted in figure 2 (c) and (d). Note that in this case the
last inclusive layer contains 135 external sources corresponding to a rough upper limit of active human
relations an individual can maintain.
For the sake of completeness, we also consider network structures that contain five layers where the
last layer contains 270 friends. Here, the inclusive fifth layer with the total of 405 friends goes beyond
the Dunbar number limit, thus including individuals who are not actively related to the central ego. The
results of the spreading simulations for the hierarchical and random networks are shown in figure 2 (e)
and 2 (f) for comparison. These results confirm the previously described behavior of the 3-layer ego-
centric networks, so that the slowing down effect is even further enhanced in hierarchical networks for
large values of r .
5. Conclusions
To summarise, we have here developed a model of transmission of cultural traits in layered and
weighted egocentric networks. The network structure was generated either randomly or to have regular
hierarchical structure in accordance with the ego-centric friendship layering showing decreasing emo-
tional closeness with an increasing distance from the ego, as suggested by Robin Dunbar. We studied this
model first analytically with two types of mean field approximations, namely one based on a single ex-
ternal source of cultural trait and the other on a layered ego-centric network. In order to validate these
mean field results, we compared them with the results obtained by extensive numerical simulations, and
found them behaviorally similar, but especially the approximation of the layered ego-centric network
seemed to compare well with the simulation results.
In the simulation studies, we observed a distinct slowing down effect in the evolutionary process of
a cultural trait. This may sound somewhat surprising against the fact that social networks show a small
world topology [15], reflecting that any individual is only a few steps away from any other. However, a
similar behavior is seen in a spreading process following susceptible - infected dynamics, which, on the
one hand, was caused by the inner community structure of the social network and, on the other hand,
by the bursty nature of social interactions between a pair of individuals [16]. Similar mechanisms were
found responsible in maintaining long-term cultural diversity beside the emergent short-term collective
behavior [17]. The origins of these kinds of systemic behavior bear close resemblance with the critical
slowing down round the critical phase transition points of materials where regardless of the long range
correlations, the existence of ordered domains slow down the relaxation of the whole system towards the
stable state.
The slowing down in our study is related to the appearance of a well defined community structure
or network clusters. The clustered structure for random networks does not exist by definition and the
obtained mean-field approximation is capable of properly describing the evolution of the cultural trait
in a social system. However, the appearance of a clustered structure in hierarchical networks for large
33802-8
Transmission of cultural traits in layered ego-centric networks
values of the control parameter r results in slowing down that cannot be captured by the mean-field
approximation. On the one hand, this limits the validity of the derived approximation (17) and, on the
other hand, underlines the importance of community structure for the spreading and evolution processes
in social systems. Besides the importance of the structure of individual ego-centric social networks in
creating channels to transmit cultural trials, the composition of these ego-centric networks together with
the whole system play an important role in the functionality of the system.
Acknowledgements
This paper is dedicated to the memory of A. Olemskoi, a physicist whose research was closely related
to synergy and investigation of complex systems. VP acknowledges the good memory of writing the first
review paper [18] on complex networks in Ukrainian together with A. Olemskoi.
This work was supported by the FiDiPro program (TEKES, Finland). JK acknowledges support from
FuturICT.hu (grant no.: TÁMOP–4.2.2.C–11/1/KONV–2012–0013).
References
1. Mantegna R.N., Stanley H.E., An Introduction to Econophysics: Correlations and Complexity in Finance, Cam-
bridge University Press, Cambridge, 2000.
2. Sen P., Chakrabarti B.K., Sociophysics: An Introduction, Oxford University Press, Oxford, 2013.
3. Castellano C., Fortunato S., Loreto V., Rev. Mod. Phys., 2009, 81, 591; doi:10.1103/RevModPhys.81.591.
4. Axelrod R., J. Conflict Resolut., 1997, 41, 203; doi:10.1177/0022002797041002001.
5. Centola D., González-Avella J.C., Eguíluz V.M., San Miguel M., J. Conflict Resolut., 2007, 51, 905;
doi:10.1177/0022002707307632.
6. Guerra B., Poncela J., Gómez-Garden̈es J., Latora V., Moreno Y., Phys. Rev. E, 2010, 81, 056105;
doi:10.1103/PhysRevE.81.056105.
7. Boyd R., Richerson P.J., Culture and the Evolutionary Process, University of Chicago Press, Chicago, 1985.
8. Dunbar R.I.M., Evol. Anthropology, 1998, 6, 178.
9. Dunbar R.I.M., Spoors M., Human Nature, 1995, 6, 273; doi:10.1007/BF02734142.
10. Kudo H., Dunbar R.I.M., Anim. Behav., 2001, 62, 711; doi:10.1006/anbe.2001.1808.
11. Zhou W.-X., Sornette D., Hill R.A., Dunbar R.I.M., Proc. R. Soc. B. 2005, 272, 439; doi:10.1098/rspb.2004.2970.
12. Dunbar R.I.M., J. Inst. Econ., 2011, 7, 345; doi:10.1017/S1744137410000366.
13. Hill R.A., Dunbar R.I.M., Hum. Nature-Int. Bios., 2003, 14, 53; doi:10.1007/s12110-003-1016-y.
14. Roberts S.B.G., Dunbar R.I.M., Pers. Relationships, 2011, 18, 439; doi:10.1111/j.1475-6811.2010.01310.x.
15. Barabási A.-L., Linked: How Everything is Connected to Everything Else andWhat It Means for Business, Science,
and Everyday Life, Plume, New York, 2003.
16. Karsai M., Kivelä M., Pan R.K., Kaski K., Kertész J., Barabási A.-L., Saramäki J., Phys. Rev. E, 2011, 83, 025102(R);
doi:10.1103/PhysRevE.83.025102.
17. Valori L., Picciolo F., Allansdottir A., Garlaschelli D., Proc. Natl. Acad. Sci. USA, 2012, 109, 1068;
doi:10.1073/pnas.1109514109.
18. Holovatch Yu., von Ferber C., Olemskoi A., Holovatch T., Mryglod O., Olemskoi I., Palchykov V., J. Phys. Stud.,
2006, 10, 247.
33802-9
http://dx.doi.org/10.1103/RevModPhys.81.591
http://dx.doi.org/10.1177/0022002797041002001
http://dx.doi.org/10.1177/0022002707307632
http://dx.doi.org/10.1103/PhysRevE.81.056105
http://dx.doi.org/10.1007/BF02734142
http://dx.doi.org/10.1006/anbe.2001.1808
http://dx.doi.org/10.1098/rspb.2004.2970
http://dx.doi.org/10.1017/S1744137410000366
http://dx.doi.org/10.1007/s12110-003-1016-y
http://dx.doi.org/10.1111/j.1475-6811.2010.01310.x
http://dx.doi.org/10.1103/PhysRevE.83.025102
http://dx.doi.org/10.1073/pnas.1109514109
V. Palchykov, K. Kaski, J. Kertész
Перенесення культурних особливостей у пластоподiбних
егоцентричних мережах
В. Пальчиков1,2,3, К. Каскi1,4,5, Я. Кертес6,1
1 Вiддiл бiомедичних технологiй та комп’ютерних наук, Унiверситет Аалто, Фiнляндiя
2 Iнститут фiзики конденсованих систем НАН України, вул. Свєнцiцького, 1, Львiв, Україна
3 Iнститут теоретичної фiзики iм. Лоренца, Лейденський унiверситет, Лейден, Нiдерланди
4 CABDyN центр, Бiзнес школа Саїда, Оксфордський унiверситет, Оксфорд, Великобританiя
5 Центр дослiдження складних систем та вiддiл фiзики, Пiвнiчно-схiдний унiверситет, Бостон, США
6 Центр наук про мережi, Центрально-європейський унiверситет, Будапешт, Угорщина
Хоча було розвинено ряд моделей для дослiдження появи культури та еволюцiйних фаз у соцiальних
системах, один важливий аспект не був достатньо висвiтлений. Цим аспектом є роль структури мереж
соцiальних зв’язкiв, якi служать каналами для передачi культурних особливостей i якi вiдiграють важливу
роль в еволюцiйних процесах у соцiальних системах. У данiй статтi ми дослiджуємо цю роль у моделi
переачi культурних особливостей та їх еволюцiю в соцiальних системах, спираючись на пластоподiбнi
структури егоцентричних мереж, надхненi гiпотезою соцiального мозку. Для цiєї моделi ми отримали
аналiтичний розв’язок у дусi середнього поля та дослiдили його застосовнiсть, порiвнюючи отриманi
передбачення з результатами чисельних симуляцiй.
Ключовi слова: егоцентричнi мережi, еволюцiя культури, наближення середнього поля
33802-10
Introduction
Transmission models with direct bias
Mean-field approximation
Single external source of a cultural trait
Layered ego-centric network
Numerical simulations
Network generation
Biased transmission model
Conclusions
|