We study an infinite horizon game in which pairs of players connected in a network are randomly matched to bargain. Players who reach agreement are replaced by new players at the same positions in the network. We show that all equilibria are payoff equivalent. The payoffs and the set of agreement links converge as players become patient. Several new concepts—mutually estranged sets, partners, and shortage ratios—provide insights into the relative strengths of the positions in the network. We develop a procedure to determine the limit equilibrium payoffs for all players. Characterizations of equitable and nondiscriminatory networks are also obtained.