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The determinant of performance in scale-up graph processing on a single system is the speed at which the graph can be fetched from storage: either from disk into memory or from memory into CPU-cache. Algorithms that follow edges perform random accesses to the storage medium for the graph and this can often be the determinant of performance, regardless of the algorithmic complexity or runtime efficiency of the actual algorithm in use. A storage-centric viewpoint would suggest that the solution to this problem lies in recognizing that graphs represent a unique workload and therefore should be treated as such by adopting novel ways to access graph structured data. We approach this problem from two different aspects and this paper details two different efforts in this direction. One approach is specific to graphs stored on SSDs and accelerates random access using a novel prefetcher called RASP. The second approach takes a fresh look at how graphs are accessed and suggests that trading off the low cost of random access for the approach of sequentially streaming a large set of (potentially unrelated) edges can be a winning proposition under certain circumstances: leading to a system for graphs stored on any medium (main-memory, SSD or magnetic disk) called X-stream. RASP and X-stream therefore take - diametrically opposite - storage centric viewpoints of the graph processing problem. After contrasting the approaches and demonstrating the benefit of each, this paper ends with a description of planned future development of an online algorithm that selects between the two approaches, possibly providing the best of both worlds.