Stable marriage and roommates problems are classic problems with numerous applications in computer science and economics. There is a number of players that strive to be matched in pairs, and the goal is to obtain a stable matching from which no pair of players has an incentive to deviate. In many applications, a player is not aware of all other players and must explore the population before finding a good match. We incorporate this aspect by studying stable matching under dynamic locality constraints in social networks. Our interest is to understand local improvement dynamics and their convergence to matchings that are stable with respect to their imposed information structure in the network.