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Efficient Algorithms for Computing Random Walk Centrality

Changan Liu, Zixuan Xie, Ahad N. Zehmakan, Zhongzhi Zhang

TL;DR

This paper tackles the impractical cubic-time computation of Random Walk Centrality (RWC) on large graphs by introducing a pivot-based reformulation that reduces the core work to a single Laplacian solve plus lightweight diagonal estimation. It delivers two near-linear-time algorithms: FastChol, which uses incomplete Cholesky factorization and sparse inverse estimation, and FastWalk, which employs Wilson’s loop-erased random walks to sample rooted spanning trees. The authors prove theoretical guarantees, demonstrate substantial speedups and memory savings over the state-of-the-art, and show scalability to graphs with millions of nodes and edges while maintaining high approximation quality. The methods significantly broaden the practical applicability of global centrality analysis on real-world networks and offer a foundation for extensions to dynamic or grouped centrality computations.

Abstract

Random walk centrality is a fundamental metric in graph mining for quantifying node importance and influence, defined as the weighted average of hitting times to a node from all other nodes. Despite its ability to capture rich graph structural information and its wide range of applications, computing this measure for large networks remains impractical due to the computational demands of existing methods. In this paper, we present a novel formulation of random walk centrality, underpinning two scalable algorithms: one leveraging approximate Cholesky factorization and sparse inverse estimation, while the other sampling rooted spanning trees. Both algorithms operate in near-linear time and provide strong approximation guarantees. Extensive experiments on large real-world networks, including one with over 10 million nodes, demonstrate the efficiency and approximation quality of the proposed algorithms.

Efficient Algorithms for Computing Random Walk Centrality

TL;DR

This paper tackles the impractical cubic-time computation of Random Walk Centrality (RWC) on large graphs by introducing a pivot-based reformulation that reduces the core work to a single Laplacian solve plus lightweight diagonal estimation. It delivers two near-linear-time algorithms: FastChol, which uses incomplete Cholesky factorization and sparse inverse estimation, and FastWalk, which employs Wilson’s loop-erased random walks to sample rooted spanning trees. The authors prove theoretical guarantees, demonstrate substantial speedups and memory savings over the state-of-the-art, and show scalability to graphs with millions of nodes and edges while maintaining high approximation quality. The methods significantly broaden the practical applicability of global centrality analysis on real-world networks and offer a foundation for extensions to dynamic or grouped centrality computations.

Abstract

Random walk centrality is a fundamental metric in graph mining for quantifying node importance and influence, defined as the weighted average of hitting times to a node from all other nodes. Despite its ability to capture rich graph structural information and its wide range of applications, computing this measure for large networks remains impractical due to the computational demands of existing methods. In this paper, we present a novel formulation of random walk centrality, underpinning two scalable algorithms: one leveraging approximate Cholesky factorization and sparse inverse estimation, while the other sampling rooted spanning trees. Both algorithms operate in near-linear time and provide strong approximation guarantees. Extensive experiments on large real-world networks, including one with over 10 million nodes, demonstrate the efficiency and approximation quality of the proposed algorithms.
Paper Structure (22 sections, 12 theorems, 41 equations, 5 figures, 2 tables, 3 algorithms)

This paper contains 22 sections, 12 theorems, 41 equations, 5 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Graph and Corresponding Matrices
  5. Random Walk Centrality
  6. Existing Methods
  7. New formulation of RWC
  8. $\textsc{FastChol}$ Algorithm: Matrix Factorization
  9. Theoretical Foundation
  10. Algorithm Design
  11. Performance Analysis
  12. $\textsc{FastWalk}$ Algorithm: Sampling Spanning Trees
  13. The Sampling Algorithm
  14. Experimental Evaluation
  15. Experimental Setting
  16. ...and 7 more sections

Key Result

Theorem 3.1

Let $v\in V$ be a designated pivot node, $\mathcal{L}_v$ be the submatrix of the normalized Laplacian matrix $\mathcal{L}$ obtained by deleting the $v$-th row and the $v$-th column of $\mathcal{L}$, then we have the following reformulation of RWC for all nodes $u\in V$:

Figures (5)

  • Figure 1: Running time comparison for different algorithms on several networks: (a) Facebook, (b) DBLP, (c) Youtube, (d) Orkut, (e) soc-Livejournal and (f) hetero-Livejournal.
  • Figure 2: Mean relative error for different algorithms on several networks: (a) Facebook, (b) DBLP, (c) Youtube, (d) Orkut, (e) soc-Livejournal and (f) hetero-Livejournal.
  • Figure 3: Kendall tau distance for different algorithms.
  • Figure 4: Ablation study for $\textsc{FastChol}$ on the effects of key components. (a) time comparison and (b) error comparison.
  • Figure 5: Sensitivity of hyperparameters: (a) base window size and (b) drop tolerance.

Theorems & Definitions (20)

  • Theorem 3.1
  • proof
  • Remark 3.2: RWC versus effective resistance Te91
  • Lemma 3.3
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Theorem 4.3
  • proof
  • ...and 10 more