The language game approach is widely adopted to study conventionalization of linguistic knowledge. Most of contemporary models concentrate on the dynamics of language games in random or predefined social structures, but neglect the role of communicative constraints. This paper adopts one form of language games, the category game, to discuss whether some simple distance-related communicative constraint may affect the conventionalization of linguistic categories. By comparing the simulation results with those based on another form of language games, the naming game, we point out some essential differences between these two games which cause their distinct performances under the same communicative constraint. This study fills the gap between the dynamics of language games in random structures and that in complex networks, and suggests that internal properties of language games may reversely influence communicative constraints and social structures.