We study how payoffs and network structure affect reaching the payoff-dominant equilibrium in a 2×22×2 coordination game that actors play with their neighbors in a network. Using an extensive simulation analysis of over 100,000 networks with 2–25 actors, we show that the importance of network characteristics is restricted to a limited part of the payoff space. In this part, we conclude that the payoff-dominant equilibrium is chosen more often if network density is larger, the network is more centralized, and segmentation of the network is smaller. Moreover, it is more likely that heterogeneity in behavior persists if the network is more segmented and less centralized. Persistence of heterogeneous behavior is not related to network density.