成对和逐列的通勤时间与Katz得分的快速矩阵计算
Fast matrix computations for pair-wise and column-wise commute times and Katz scores
AI总结：
本文利用矩阵、矩和正交方法，通过Lanczos过程逼近节点对之间的通勤时间和Katz得分，并给出上下界；同时采用共轭梯度法变体计算通勤时间，利用Katz矩阵的局部化性质计算逐列Katz得分，在17个真实图上验证了方法的理论性质和性能。
摘要：
我们首先探索了逼近节点对之间通勤时间和Katz得分的方法。这些方法基于数值线性代数社区开发的矩阵、矩和正交方法。它们依赖于Lanczos过程，并提供了成对得分估计的上下界。我们还探索了从图中一个节点到所有其他节点的通勤时间和Katz得分的逼近方法。这里，我们的通勤时间方法基于共轭梯度算法的一个变体，它提供了矩阵逆的所有对角线的估计。我们的Katz得分技术利用了Katz矩阵的经验局部化性质。我们将用于个性化PageRank计算的算法应用于这些Katz得分，并从理论上证明了该方法的收敛性。我们在17个真实世界图上评估了这些方法，图的大小从1000到1,000,000个节点不等。我们的结果表明，我们的成对通勤时间方法和逐列Katz算法都具有良好的理论性质和实证性能。
英文摘要：
We first explore methods for approximating the commute time and Katz score between a pair of nodes. These methods are based on the approach of matrices, moments, and quadrature developed in the numerical linear algebra community. They rely on the Lanczos process and provide upper and lower bounds on an estimate of the pair-wise scores. We also explore methods to approximate the commute times and Katz scores from a node to all other nodes in the graph. Here, our approach for the commute times is based on a variation of the conjugate gradient algorithm, and it provides an estimate of all the diagonals of the inverse of a matrix. Our technique for the Katz scores is based on exploiting an empirical localization property of the Katz matrix. We adopt algorithms used for personalized PageRank computing to these Katz scores and theoretically show that this approach is convergent. We evaluate these methods on 17 real world graphs ranging in size from 1000 to 1,000,000 nodes. Our results show that our pair-wise commute time method and column-wise Katz algorithm both have attractive theoretical properties and empirical performance.
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