Phase transitions in social networks inspired by the Schelling model

We propose two models of social segregation inspired by the Schelling model. Agents in our models are nodes of evolving social networks. The total number of social connections of each node remains constant in time, though may vary from one node to the other. The first model describes a "polychromatic" society, in which colors designate different social categories of agents. The parameter $\mu$ favors/disfavors connected "monochromatic triads", i.e. connected groups of three individuals \emph{within the same social category}, while the parameter $\nu$ controls the preference of interactions between two individuals \emph{from different social categories}. The polychromatic model has several distinct regimes in $(\mu,\nu)$-parameter space. In $\nu$-dominated region, the phase diagram is characterized by the plateau in the number of the inter-color connections, where the network is bipartite, while in $\mu$-dominated region, the network looks as two weakly connected unicolor clusters. At $\mu>\mu{crit}$ and $\nu >\nu{crit}$ two phases are separated by a critical line, while at small values of $\mu$ and $\nu$, a gradual crossover between the two phases occurs. The second "colorless" model describes a society in which the advantage/disadvantage of forming small fully connected communities (short cycles or cliques in a graph) is controlled by a parameter $\gamma$. We analyze the topological structure of a social network in this model and demonstrate that above a critical threshold, $\gamma^+>0$, the entire network splits into a set of weakly connected clusters, while below another threshold, $\gamma^-<0$, the network acquires a bipartite graph structure. Our results propose mechanisms of formation of self-organized communities in international communication between countries, as well as in crime clans and prehistoric societies.