Structured sequences are popularly used to describe graph data with time-evolving node features and edges. A typical real-world scenario of structured sequences is that unknown class labels continuously arrive and thus the training and testing often across different class spaces. This scenario is also referred to as the open-world learning problem on structured sequences. In this paper, we present a new Dense Open-world Structured Sequence Learning model (DOSSL for short) to learn graph streams in the open-world learning setting. To capture both structural and temporal information, DOSSL uses a GNN-based stochastic recurrent neural network for learning node representation in graph streams, then a truncated Laplacian distribution to describe the latent distribution of graph nodes, and a sampling function is used to generate node representations. Further, DOSSL learns dense target embeddings for the known classes to improve the compactness of known class distribution and reserve enough space for open-world unknown classes. The ultimate open-world classifier is optimized to detect the samples from unknown classes under the constraints of DVAE loss, label loss, class uncertainty loss, and dense target loss. Through empirical analysis conducted on real-world datasets, it has been demonstrated that the advanced technique known as DOSSL exhibits the ability to acquire precise node classifiers by harnessing the power of graph streams.