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Finding similar user pairs is a fundamental task in social networks, withnumerous applications in ranking and personalization tasks such as linkprediction and tie strength detection. A common manifestation of usersimilarity is based upon network structure: each user is represented by avector that represents the user's network connections, where pairwise cosinesimilarity among these vectors defines user similarity. The predominant taskfor user similarity applications is to discover all similar pairs that have apairwise cosine similarity value larger than a given threshold $\tau$. Incontrast to previous work where $\tau$ is assumed to be quite close to 1, wefocus on recommendation applications where $\tau$ is small, but stillmeaningful. The all pairs cosine similarity problem is computationallychallenging on networks with billions of edges, and especially so for settingswith small $\tau$. To the best of our knowledge, there is no practical solutionfor computing all user pairs with, say $\tau = 0.2$ on large social networks,even using the power of distributed algorithms.  Our work directly addresses this challenge by introducing a new algorithm ---WHIMP --- that solves this problem efficiently in the MapReduce model. The keyinsight in WHIMP is to combine the "wedge-sampling" approach of Cohen-Lewis forapproximate matrix multiplication with the SimHash random projection techniquesof Charikar. We provide a theoretical analysis of WHIMP, proving that it hasnear optimal communication costs while maintaining computation cost comparablewith the state of the art. We also empirically demonstrate WHIMP's scalabilityby computing all highly similar pairs on four massive data sets, and show thatit accurately finds high similarity pairs. In particular, we note that WHIMPsuccessfully processes the entire Twitter network, which has tens of billionsof edges.

When Hashes Met Wedges: A Distributed Algorithm for Finding High Similarity Vectors