This article uses computer simulations as a means of assessing two different models for the competition between two languages from the interdisciplinary perspective of complex systems. These models allow for the analysis of the role of bilingual speakers and they consider two of the basic factors determining the use of each language: their relative prestige and the language loyalty of their speakers. Here we assess the effect of these parameters in different social networks, with the aim of contributing to the understanding of the role of bilingualism and social networks in processes of language contact. We first present a qualitative analysis based on a simulation tool that we have designed, in order to visualize the dynamics of the models. Secondly, we consider a qualitative analysis of the role of complex networks, identifying mechanisms for the rapid extinction of a language and for situations of language segregation. Finally, we present a quantitative analysis which shows, counter-intuitively, a transition from a final scenario of language maintenance to one of language shift as language loyalty increases. Other findings are: (i) that the scenario of language coexistence is reduced in the presence of bilingual agents; and (ii) that networks which allow for local effects reduce further the situations for which language maintenance is possible.