一个面向大型真实图的几何距离预言机
A Geometric Distance Oracle for Large Real-World Graphs
摘要：
许多图处理算法需要确定任意数量的节点对之间的最短路径距离。由于大型图的所有节点对之间的精确距离的计算，例如，10M以上的节点在计算时间和存储空间上都是非常昂贵的，通常使用距离近似来代替精确计算。在本文中，我们提出了一种新的和可扩展的距离预言机，利用现实世界的大型图的双曲核心快速和可扩展的距离近似。我们的经验表明，建议的预言机显着优于以前的预言机上的一组随机测试用例从公共领域的图形库。有两组先前的工作，我们基准我们的方法。第一组，这往往优于其他预言，采用嵌入到低维欧氏空间的图形精心构造的双曲距离，但不提供任何保证的距离估计误差。第二组利用图的Gromov型树收缩，保证加性误差不超过$2δ\log{n}$，其中$δ$是图的双曲常数。我们表明，我们提出的预言机1）明显快于那些使用双曲嵌入（第一组）的预言机，具有类似的近似误差，并且，也许令人惊讶的是，2）与Gromov类树收缩（第二组）相比，表现出明显更低的平均估计误差。我们通过对来自多个域的十几个真实世界网络和合成测试用例的集合进行数值计算来证实我们的主张，这些网络和测试用例的大小从数千到数百万个节点不等。
英文摘要：
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 $2δ\log{n}$, where $δ$ 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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