Analysis of large graphs is critical to the ongoing growth of search engines and social networks. One class of queries centers around node affinity, often quantified by random-walk distances between node pairs, including hitting time, commute time, and personalized PageRank (PPR). Despite the potential of these "metrics," they are rarely, if ever, used in practice, largely due to extremely high computational costs.In this paper, we investigate methods to scalably and efficiently compute random-walk distances, by "embedding" graphs and distances into points and distances in geometric coordinate spaces. We show that while existing graph coordinate systems (GCS) can accurately estimate shortest path distances, they produce significant errors when embedding random-walk distances. Based on our observations, we propose a new graph embedding system that explicitly accounts for per-node graph properties that affect random walk. Extensive experiments on a range of graphs show that our new approach can accurately estimate both symmetric and asymmetric random-walk distances. Once a graph is embedded, our system can answer queries between any two nodes in 8 microseconds, orders of magnitude faster than existing methods. Finally, we show that our system produces estimates that can replace ground truth in applications with minimal impact on application output.

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