Fast Computation for the Forest Matrix of an Evolving Graph

Haoxin Sun, Xiaotian Zhou, Zhongzhi Zhang
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Abstract

The forest matrix plays a crucial role in network science, opinion dynamics, and machine learning, offering deep insights into the structure of and dynamics on networks. In this paper, we study the problem of querying entries of the forest matrix in evolving graphs, which more accurately represent the dynamic nature of real-world networks compared to static graphs. To address the unique challenges posed by evolving graphs, we first introduce two approximation algorithms, \textsc{SFQ} and \textsc{SFQPlus}, for static graphs. \textsc{SFQ} employs a probabilistic interpretation of the forest matrix, while \textsc{SFQPlus} incorporates a novel variance reduction technique and is theoretically proven to offer enhanced accuracy. Based on these two algorithms, we further devise two dynamic algorithms centered around efficiently maintaining a list of spanning converging forests. This approach ensures $O(1)$ runtime complexity for updates, including edge additions and deletions, as well as for querying matrix elements, and provides an unbiased estimation of forest matrix entries. Finally, through extensive experiments on various real-world networks, we demonstrate the efficiency and effectiveness of our algorithms. Particularly, our algorithms are scalable to massive graphs with more than forty million nodes.
快速计算演化图的森林矩阵
森林矩阵在网络科学、舆论动力学和机器学习中发挥着至关重要的作用,能深入揭示网络的结构和动态。在本文中,我们研究了在演化图中查询森林矩阵条目的问题,与静态图相比,演化图更准确地代表了真实世界网络的动态性质。为了解决演化图带来的独特挑战,我们首先介绍了两种针对静态图的近似计算算法--textsc{SFQ}和textsc{SFQPlus}。\textsc{SFQ}采用了对森林矩阵的概率解释,而textsc{SFQPlus}则采用了一种新颖的方差缩小技术,并在理论上被证明可以提供更高的精度。在这两种算法的基础上,我们进一步设计了两种动态算法,其核心是有效地维护跨度收敛森林列表。这种方法确保了更新(包括边的添加和删除)以及查询矩阵元素的运行时间复杂度为 $O(1)$,并提供了对森林矩阵条目的无偏估计。最后,通过在各种真实世界网络上的广泛实验,我们证明了我们算法的效率和有效性,特别是我们的算法可以扩展到拥有超过 4000 万个节点的大规模图。
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