Graph databases have become an increasingly common infrastructure component. Yet existing systems either operate on offline snapshots, provide weak consistency guarantees, or use expensive concurrency control techniques that limit performance. In this paper, we introduce a new distributed graph database, called Weaver, which enables efficient, transactional graph analyses as well as strictly serializable ACID transactions on dynamic graphs. The key insight that allows Weaver to combine strict serializability with horizontal scalability and high performance is a novel request ordering mechanism called refinable timestamps. This technique couples coarse-grained vector timestamps with a fine-grained timeline oracle to pay the overhead of strong consistency only when needed. Experiments show that Weaver enables a Bitcoin blockchain explorer that is 8x faster than Blockchain.info, and achieves 12x higher throughput than the Titan graph database on social network workloads and 4x lower latency than GraphLab on offline graph traversal workloads.