We consider a model for decentralized collaborative networks that is based on stable matching theory. This model is applied to systems with a global ranking utility function, which admits a unique stable configuration. We study the speed of convergence and analyze the stratification properties of the stable configuration, both when all collaborations are possible and for random possible collaborations. As a practical example, we consider the BitTorrent Titfor-Tat policy. For this system, our model provides an interesting insight into peer download rates and a good understanding of upload settings strategy.