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 bothlink prediction and recommendation.