  This paper studies the long-run properties of a stochastic best-response
network formation model. Players stochastically meet and make decisions about
changing the state of their relationship. I give sufficient conditions under
which this game induces a reversible Markov process in the space of networks,
and show that in this case the stationary distribution is given by a Gibbs
measure with an associated aggregating function that depends on players'
utilities. Reversibility also implies the existence of a potential for a
deterministic version of the network formation game. Using the properties of
the Gibbs measure I show that, in simple settings, the long-run behavior of the
model for increasingly large networks is driven by a trade-off between player
incentives and the exponentially increasing number of feasible networks. I use
this framework to obtain network statistics for simple models of social media
following and international trade.
