Data of many problems in real-world systems such as link prediction and one-class recommendation share common characteristics. First, data are in the form of positive-unlabeled (PU) measurements ( e.g. Twitter “following”, Facebook “like”, etc.) that do not provide negative information, which can be naturally represented as networks. Second, in the era of big data, such data are generated temporally-ordered, continuously and rapidly, which determines its streaming nature. These common characteristics allow us to unify many problems into a novel framework - PU learning in streaming networks. In this paper, a principled probabilistic approach SPU is proposed to leverage the characteristics of the streaming PU inputs. In particular, SPU captures temporal dynamics and provides real-time adaptations and predictions by identifying the potential negative signals concealed in unlabeled data. Our empirical results on various real-world datasets demonstrate the effectiveness of the proposed framework over other state-of-the-art methods in both link prediction and recommendation.