In many graph applications, the structure of the graph changes dynamically over time and may require real time analysis. However, constructing a large graph is expensive, and most studies for large graphs have not focused on a dynamic graph data structure, but rather a static one. To address this issue, we propose DegAwareRHH, a high performance dynamic graph data store designed for scaling out to store large, scale-free graphs by leveraging compact hash tables with high data locality. We extend DegAwareRHH for multiple processes and distributed memory, and perform dynamic graph construction on large scale-free graphs using emerging 'Big Data HPC' systems such as the Catalyst cluster at LLNL. We demonstrate that DegAwareRHH processes a request stream 206.5x faster than a state-of-the-art shared-memory dynamic graph processing framework, when both implementations use 24 threads/processes to construct a graph with 1 billion edge insertion requests and 54 million edge deletion requests. DegAwareRHH also achieves a processing rate of over 2 billion edge insertion requests per second using 128 compute nodes to construct a large-scale web graph, containing 128 billion edges, the largest open-source real graph dataset to our knowledge.