Evolution of a social network: The role of cultural diversity

We present a simple deterministic and based on local rules model of evolving social network, which leads to a network with the properties of a real social system, e.g., small-world topology and assortative mixing. The state of an individual $S_{\mathrm{i}}$ is characterized by the values of $Q$ cultural features, drawn from Gaussian distribution with variance $\sigma$. The other control parameter is sociability $T_{\mathrm{i}}$, which describes the maximal number of connections of an individual. The state of individuals and connections between them evolve in time. As results from numerical computations, an initial diversity of cultural features in a community has an essential influence on an evolution of social network. It was found that for a critical value of control parameter ${\sigma }_{\mathrm{c}}\left(Q\right)$ there is a structural transition and a hierarchical network with small-world topology of connections and a high clustering coefficient emerges. The emergence of small-world properties can be related to the creation of subculture groups in a community. The power-law relation between the clustering coefficient of a node and its connectivity $C\left(k\right)\sim k^{-\beta }$ was observed in the case of a scale-free distribution of sociability $T_{\mathrm{i}}$ and a high enough cultural diversity in a population.