Graph sampling frequently compresses a large graph into a limited screen space. This paper proposes a hierarchical structure model that partitions scale-free graphs into three blocks: the core, which captures the underlying community structure, the vertical graph, which represents minority structures that are important in visual analysis, and the periphery, which describes the connection structure between low-degree nodes. A new algorithm named hierarchical structure sampling (HSS) was then designed to preserve the characteristics of the three blocks, including complete replication of the connection relationship between high-degree nodes in the core, joint node/degree distribution between high- and low-degree nodes in the vertical graph, and proportional replication of the connection relationship between low-degree nodes in the periphery. Finally, the importance of some global statistical properties in visualization was analyzed. Both the global statistical properties and local visual features were used to evaluate the proposed algorithm, which verify that the algorithm can be applied to sample scale-free graphs with hundreds to one million nodes from a visualization perspective.