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Fast Computation for the Forest Matrix of an Evolving Graph

Haoxin Sun, Xiaotian Zhou, Zhongzhi Zhang

TL;DR

This work tackles the problem of efficiently querying entries of the forest matrix $\boldsymbol{\Omega}=(\mathbf{I}+\mathbf{L})^{-1}$ on evolving graphs. It introduces SFQ and SFQPlus for static graphs, leveraging a probabilistic interpretation of $\omega_{ij}$ and variance-reduction strategies to obtain unbiased, low-variance estimators via extensions of Wilson's algorithm. For dynamic graphs, it develops Insert-Update and Delete-Update procedures that maintain a uniformly sampled list of spanning converging forests with a prune mechanism to cap growth, achieving $O(1)$-time query and update performance in practice. Extensive experiments on large real-world networks demonstrate that SFQPlus consistently outperforms SFQ in accuracy and that the approach scales to graphs with tens of millions of nodes, where exact solvers fail. The techniques enable fast, scalable forest-matrix analyses essential for network science, opinion dynamics, and related domains.

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.

Fast Computation for the Forest Matrix of an Evolving Graph

TL;DR

This work tackles the problem of efficiently querying entries of the forest matrix on evolving graphs. It introduces SFQ and SFQPlus for static graphs, leveraging a probabilistic interpretation of and variance-reduction strategies to obtain unbiased, low-variance estimators via extensions of Wilson's algorithm. For dynamic graphs, it develops Insert-Update and Delete-Update procedures that maintain a uniformly sampled list of spanning converging forests with a prune mechanism to cap growth, achieving -time query and update performance in practice. Extensive experiments on large real-world networks demonstrate that SFQPlus consistently outperforms SFQ in accuracy and that the approach scales to graphs with tens of millions of nodes, where exact solvers fail. The techniques enable fast, scalable forest-matrix analyses essential for network science, opinion dynamics, and related domains.

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 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.
Paper Structure (19 sections, 8 theorems, 3 equations, 4 figures, 2 tables, 4 algorithms)

This paper contains 19 sections, 8 theorems, 3 equations, 4 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Graph and Laplacian Matrix
  5. Spanning Converging Forests and Forest Matrix
  6. Fast Query of Entries in Forest Matrix for Digraphs
  7. Extension of Wilson's Algorithm
  8. Probabilistic Interpretation and Unbiased Estimators of Entries in Forest Matrix
  9. Enhanced Estimators with Reduced Variance
  10. Fast Query of Entries of Forest Matrix for Evolving Graphs
  11. Problem Statement
  12. Edge Insertion
  13. Edge Deletion
  14. Prune Technique and Algorithm Details
  15. Experiments
  16. ...and 4 more sections

Key Result

Lemma 4.1

For two different nodes $i$ and $j$ in graph $\mathcal{G}$ and a uniformly chosen spanning converging forest $\phi \in \mathcal{F}$, $\widetilde{\omega}_{ij}(\phi) = \frac{1}{1+d_j}\sum_{k \in N^-(j)}\widehat{\omega}_{ik}(\phi)$ is an unbiased estimator of $\omega_{ij}$. The variance of this estimat

Figures (4)

  • Figure 1: A toy digraph $\mathcal{G}_0$ and updated graph $\mathcal{G}_1$ with their spanning converging forests. Blue nodes are roots.
  • Figure 2: A toy digraph $\mathcal{G}_0$ and updated graph $\mathcal{G}_1$ with their spanning converging forests.
  • Figure 3: Comparison of average relative errors of the diagonals for algorithms SFQ and SFQPlus on four graphs: web-Web-Stanford (a), Delicious (b), web-Google (c), Youtube(d), where suffix -S indicates the results on static graphs and -D denotes results on the updated graphs.
  • Figure 4: Comparison of average relative errors of the forest closeness centrality measures for algorithms SFQ and SFQPlus on four graphs: web-Web-Stanford (a), Delicious (b), web-Google (c), Livejournal(d), where suffix -S indicates the results on static graphs and -D denotes results on the updated graphs.

Theorems & Definitions (8)

  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • Theorem 4.5
  • Lemma 5.1
  • Theorem 5.2
  • Theorem 5.3