An infinite stochastic model of social network formation

We consider an infinite interacting particle system in which individuals choose neighbors according to evolving sets of probabilities. If x chooses y at some time, the effect is to increase the probability that y chooses x at later times. We characterize the extremal invariant measures for this process. In an extremal equilibrium, the set of individuals is partitioned into finite sets called stars, each of which includes a “center” that is always chosen by the other individuals in that set.