#### Design, Generation, and Validation of Extreme Scale Power-Law Graphs

##### Jeremy Kepner, Siddharth Samsi, William Arcand, David Bestor, Bill Bergeron, Tim Davis, Vijay Gadepally, Michael Houle, Matthew Hubbell, Hayden Jananthan, Michael Jones, Anna Klein, Peter Michaleas, Roger Pearce, Lauren Milechin, Julie Mullen, Andrew Prout, Antonio Rosa, Geoff Sanders, Charles Yee, Albert Reuther

Massive power-law graphs drive many fields: metagenomics, brain mapping, Internet-of-things, cybersecurity, and sparse machine learning. The development of novel algorithms and systems to process these data requires the design, generation, and validation of enormous graphs with exactly known properties. Such graphs accelerate the proper testing of new algorithms and systems and are a prerequisite for success on real applications. Many random graph generators currently exist that require realizing a graph in order to know its exact properties: number of vertices, number of edges, degree distribution, and number of triangles. Designing graphs using these random graph generators is a time-consuming trial-and-error process. This paper presents a novel approach that uses Kronecker products to allow the exact computation of graph properties prior to graph generation. In addition, when a real graph is desired, it can be generated quickly in memory on a parallel computer with no-interprocessor communication. To test this approach, graphs with $10^{12}$ edges are generated on a 40,000+ core supercomputer in 1 second and exactly agree with those predicted by the theory. In addition, to demonstrate the extensibility of this approach, decetta-scale graphs with up to $10^{30}$ edges are simulated in a few minutes on a laptop.

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