 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 beyond 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 number of assumptions to capture the essence of the phenomenon investigated. This is exemplified by a model of two possible spin states for each particle, like the Ising model for magnet, which in spite of its simplicity 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 assumptions about the processes leading to stochastic rules, which are then tested by comparing the results with reality -usually in a qualitative manner.
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]. While much of research has been done from the mean 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 formulated 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 describe 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 closeness [9][10][11].
In this paper we first describe the biased transmission model in section 2; develop a mean-field approximation 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 conclusions.

Transmission models with direct bias
Assuming that there are two possible cultural traits x i (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 N i friends and that the strength of relation between the ego i and source j is ω i j. It reflects the influence of each of the source j on the ego i. The values of ω i j are normalized 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 ω i j and on the combination of the bias b and the cultural trait x j (t ) of the source j. The probability q i j that the ego i adopts cultural trait from the source j is determined [7] as 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 q i 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 p i (t +1|{x j (t )} i ) that an individual i will adopt cultural trait x i (t + 1) = 1 at time t + 1 given the configuration {x j (t )} i at time t may be expressed as follows: where p i (t ) is the probability that an individual i has cultural trait x i (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.

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 cultural 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 p i (t ) of having a cultural trait x i (t ) = 1 is independent of i, and thus p i (t ) = p(t ). Below we consider the preference towards cultural trait 1, i.e., b > 0.

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 ω i j of social influence of the source j reads as follows: In order to reproduce a temporal evolution of an individual cultural trait [equation (4)] the entire configuration {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 realization of each particular configuration {x j (t )} i. Then equation (4) may be expanded as follows: 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:

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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, ω i j, into account [equation (6)], one obtains an estimation for the evolution of individual cultural trait p i (t +1), which in the mean-field approximation is equal to the fraction of population p(t +1) [equation (5)] characterised by a favoured cultural trait [7]: The result (9) was obtained in 1980-s and it shows that the system starting from arbitrary small number 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 simulations 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 relations 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 considered by the ego.

Layered ego-centric network
Here we develop a mean-