The language game approach has recently been adopted to explore the conventionalization of linguistic knowledge in a social environment. Most contemporary studies focus on the dynamics of language games in random or predefined social networks, but neglect the reverse roles of communicative constraints in language evolution and social structures. This article, based on two forms of language games (the naming game and the category game), examines whether a simple, distance-based communicative constraint can affect the conventionalization of linguistic knowledge. The study bridges the gap between random networks and complex social structures, and illustrates that the internal properties of language games can influence the effects of communicative constraints and social structures.