Phase transitions in social networks inspired by the Schelling model

A generalized Schelling social segregation model is applied for the description of evolving social networks in which the number of connections of each agent (individual) remains approximately constant in time, though may be different for different agents. We propose two models. The first two-parametric model describes a "polychromatic" society, where each "color" designates some typical "social category", to which the agent belongs. One control parameter, $\mu$, reflects the condition that three individuals within the same social category have some preference to form a triad. The second parameter, $\nu$, controls the preference for two individuals from different social categories to be connected. The polychromatic model enjoys several regimes separated by phase transitions in the $(\mu,\nu)$ parameter space. The second "colorless" model describes a society in which advantage or disadvantage of forming small fully connected communities (short cycles or cliques in a graph) is controlled by a single "energy" parameter, $\gamma$. We analyze the topological structure of a social network in this model and demonstrate that for some critical threshold, $\gamma^+>0$, the entire network splits into the set of weakly connected clusters, while at another threshold, $\gamma^-<0$, the network acquires a bipartite graph structure. Our results shed light on social segregation and self-organized formation of cliques in social networks.