A massive amount of data generated today on platforms such as social networks, telecommunication networks, and the internet in general can be represented as graph streams. Activity in a network's underlying graph generates a sequence of edges in the form of a stream; for example, a social network may generate a graph stream based on the interactions (edges) between different users (nodes) over time. While many graph mining algorithms have already been developed for analyzing relatively small graphs, graphs that begin to approach the size of real-world networks stress the limitations of such methods due to their dynamic nature and the substantial number of nodes and connections involved. In this paper we present GraphZip, a scalable method for mining interesting patterns in graph streams. GraphZip is inspired by the Lempel-Ziv (LZ) class of compression algorithms, and uses a novel dictionary-based compression approach in conjunction with the minimum description length principle to discover maximally-compressing patterns in a graph stream. We experimentally show that GraphZip is able to retrieve complex and insightful patterns from large real-world graphs and artificially-generated graphs with ground truth patterns. Additionally, our results demonstrate that GraphZip is both highly efficient and highly effective compared to existing state-of-the-art methods for mining graph streams.