  We investigate the long-time properties of a dynamic, out-of-equilibrium
network of individuals holding one of two opinions in a population consisting
of two communities of different sizes. Here, while the agents' opinions are
fixed, they have a preferred degree which leads them to endlessly create and
delete links. Our evolving network is shaped by homophily/heterophily, which is
a form of social interaction by which individuals tend to establish links with
others having similar/dissimilar opinions. Using Monte Carlo simulations and a
detailed mean-field analysis, we study in detail how the sizes of the
communities and the degree of homophily/heterophily affects the network
structure. In particular, we show that when the network is subject to enough
heterophily, an "overwhelming transition" occurs: individuals of the smaller
community are overwhelmed by links from agents of the larger group, and their
mean degree greatly exceeds the preferred degree. This and related phenomena
are characterized by obtaining the network's total and joint degree
distributions, as well as the fraction of links across both communities and
that of agents having fewer edges than the preferred degree. We use our
mean-field theory to discuss the network's polarization when the group sizes
and level of homophily vary.
