In this article we extend the model of language evolution exemplified in Ringe et al. 2002, which recovers phylogenetic trees optimized according to a criterion of weighted maximum compatibility, to include cases in which languages remain in contact and trade linguistic material as they evolve. We describe our analysis of an Indo-European (IE) dataset (originally assembled by Ringe and Taylor) based on this new model. Our study shows that this new model fits the IE family well and suggests that the early evolution of IE involved only limited contact between distinct lineages. Furthermore, the candidate histories we obtain appear to be consistent with archaeological findings, which suggests that this method may be of practical use. The case at hand provides no opportunity to explore the problem of conflict between network optimization criteria; that problem must be left to future research.