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Graph processing is becoming commonplace in many applications to analyze huge datasets. Much of the prior work in this area has assumed I/O devices with considerable latencies, especially for random accesses, using large amount of DRAM to trade-off additional computation for I/O accesses. However, emerging storage devices, including currently popular SSDs, provide fairly comparable sequential and random accesses, making these prior solutions inefficient. In this paper, we point out this inefficiency, and propose a new graph partitioning and processing framework to leverage these new device capabilities. We show experimentally on an actual platform that our proposal can give 2X better performance than a state-of-the-art solution.