Nonequilibrium phase transition due to social group isolation

We introduce a simple model of a growing system with $m$ competing communities. The model corresponds to the phenomenon of defeats suffered by social groups living in isolation. A nonequilibrium phase transition is observed when at critical time $t_c$ the first isolated cluster occurs. In the one-dimensional system the volume of the new phase, i.e. the number of the isolated individuals, increases with time as $Z \sim t^3$. For a large number of possible communities the critical density of filled space equals to $\rho_c = (m/N)^{1/3}$ where $N$ is the system size. A similar transition is observed for Erd\H{o}s-R\'{e}nyi random graphs and Barab\'{a}si-Albert scale-free networks. Analytic results are in agreement with numerical simulations.