Social network structure has been argued to shape the structure of languages, as well as affect the spread of innovations and the formation of conventions in the community. Specifically, theoretical and computational models of language change predict that sparsely connected communities develop more systematic languages, while tightly knit communities can maintain high levels of linguistic complexity and variability. However, the role of social network structure in the cultural evolution of languages has never been tested experimentally. Here, we present results from a behavioral group communication study, in which we examined the formation of new languages created in the lab by micro-societies that varied in their network structure. We contrasted three types of social networks: fully connected, small-world, and scale-free. We examined the artificial languages created by these different networks with respect to their linguistic structure, communicative success, stability, and convergence. Results did not reveal any effect of network structure for any measure, with all languages becoming similarly more systematic, more accurate, more stable, and more shared over time. At the same time, small-world networks showed the greatest variation in their convergence, stabilization, and emerging structure patterns, indicating that network structure can influence the community's susceptibility to random linguistic changes (i.e., drift).