#### A Geometric Distance Oracle for Large Real-World Graphs

##### Deepak Ajwani, W. Sean Kennedy, Alessandra Sala, Iraj Saniee

Many graph processing algorithms require determination of shortest-path distances between arbitrary numbers of node pairs. Since computation of exact distances between all node-pairs of a large graph, e.g., 10M nodes and up, is prohibitively expensive both in computational time and storage space, distance approximation is often used in place of exact computation. In this paper, we present a novel and scalable distance oracle that leverages the hyperbolic core of real-world large graphs for fast and scalable distance approximation. We show empirically that the proposed oracle significantly outperforms prior oracles on a random set of test cases drawn from public domain graph libraries. There are two sets of prior work against which we benchmark our approach. The first set, which often outperforms other oracles, employs embedding of the graph into low dimensional Euclidean spaces with carefully constructed hyperbolic distances, but provides no guarantees on the distance estimation error. The second set leverages Gromov-type tree contraction of the graph with the additive error guaranteed not to exceed $2\delta\log{n}$, where $\delta$ is the hyperbolic constant of the graph. We show that our proposed oracle 1) is significantly faster than those oracles that use hyperbolic embedding (first set) with similar approximation error and, perhaps surprisingly, 2) exhibits substantially lower average estimation error compared to Gromov-like tree contractions (second set). We substantiate our claims through numerical computations on a collection of a dozen real world networks and synthetic test cases from multiple domains, ranging in size from 10s of thousand to 10s of millions of nodes.

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