Graphs are a natural fit for modeling concepts used in solving diverse problems in science, commerce, engineering, and governance. Responding to the variety of graph data and algorithms, many parallel and distributed graph-processing systems exist. However, until now these platforms use a static model of deployment: they only run on a pre-defined set of machines. This raises many conceptual and pragmatic issues, including misfit with the highly dynamic nature of graph processing, and could lead to resource waste and high operational costs. In contrast, in this work we explore a dynamic, elastic model of deployment. To conduct an in-depth elasticity study of distributed graph processing, we build a prototype, JoyGraph, which is the first such system that implements complex, policy-based, and finegrained elasticity. Using the state-of-the-art LDBC Graphalytics benchmark and the SPEC Cloud Group's elasticity metrics, we show the benefits of elasticity in graph processing: improved resource utilization, and aligned operation-workload dynamicity. Furthermore, we explore the cost of elasticity in graph processing. We identify a key drawback: although elasticity does not degrade application throughput, graph-processing workloads are sensitive to data movement while leasing or releasing resources.