  Since their inception about a decade ago, dynamic networks which adapt to the
state of the nodes have attracted much attention. One simple case of such an
adaptive dynamics is a model of social networks in which individuals are
typically comfortable with a certain number of contacts, i.e., preferred
degrees. This paper is partly a review of earlier work of single homogeneous
systems and ones with two interacting networks, and partly a presentation of
some new results. In general, the dynamics does not obey detailed balance and
the stationary distributions are not known analytically. A particular limit of
the latter is a system of extreme introverts and extroverts - the XIE model.
Remarkably, in this case, the detailed balance condition is satisfied, the
exact distribution and an effective Hamiltonian can be found explicitly.
Further, the model exhibits a phase transition in which the total number of
links in the system - a macroscopically interesting quantity, displays an
extreme Thouless effect. We show that in the limit of large populations and
away from the transition, the model reduces to one with non-interacting agents
of the majority subgroup. We determine the nature of fluctuations near the
transition. We also introduce variants of the model where the agents show
preferential attachment or detachment. There are significant changes to the
degree distributions in the steady state, some of which can be understood by
theoretical arguments and some remain to be explored. Many intriguing questions
are posed, providing some food for thought and avenues for future research.
