Fast Computation for the Forest Matrix of an Evolving Graph

TL;DR

This paper introduces fast, scalable algorithms for efficiently approximating and maintaining the forest matrix in evolving graphs, enabling real-time queries and updates in massive dynamic networks.

Contribution

The paper presents novel approximation and dynamic algorithms for the forest matrix in evolving graphs, achieving constant-time updates and queries with proven accuracy.

Findings

Algorithms are scalable to graphs with over 40 million nodes.

Proposed methods outperform existing approaches in efficiency and accuracy.

Dynamic algorithms maintain unbiased estimates with O(1) update time.

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.