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Scaling Up Graph Propagation Computation on Large Graphs: A Local Chebyshev Approximation Approach

Yichun Yang, Rong-Hua Li, Meihao Liao, Longlong Lin, Guoren Wang

TL;DR

The paper tackles scalable graph propagation (GP) for SSPPR and HKPR on large graphs, where Taylor-based PwMethod and Push-based approaches converge slowly. It introduces a novel Chebyshev expansion of GP functions, deriving explicit coefficients for SSPPR and HKPR and using them to build two algorithms: ChebyPower (a Chebyshev-accelerated power method) and ChebyPush (a local push-style method via a subset Chebyshev recurrence). Theoretical results show faster convergence (roughly O(√N) over Taylor) and provable error guarantees, with ChebyPush offering graph-size-independent runtime under suitable thresholds; the paper also generalizes to broader GP operators and bidirectional schemes. Extensive experiments on five real-world graphs demonstrate substantial speedups (often 3–8×) over state-of-the-art baselines for both SSPPR and HKPR, validating the practicality and scalability of the approach for large-scale graph analysis and graph neural network workflows.

Abstract

Graph propagation (GP) computation plays a crucial role in graph data analysis, supporting various applications such as graph node similarity queries, graph node ranking, graph clustering, and graph neural networks. Existing methods, mainly relying on power iteration or push computation frameworks, often face challenges with slow convergence rates when applied to large-scale graphs. To address this issue, we propose a novel and powerful approach that accelerates power iteration and push methods using Chebyshev polynomials. Specifically, we first present a novel Chebyshev expansion formula for general GP functions, offering a new perspective on GP computation and achieving accelerated convergence. Building on these theoretical insights, we develop a novel Chebyshev power iteration method (\ltwocheb) and a novel Chebyshev push method (\chebpush). Our \ltwocheb method demonstrates an approximate acceleration of $O(\sqrt{N})$ compared to existing power iteration techniques for both personalized PageRank and heat kernel PageRank computations, which are well-studied GP problems. For \chebpush, we propose an innovative subset Chebyshev recurrence technique, enabling the design of a push-style local algorithm with provable error guarantee and reduced time complexity compared to existing push methods. We conduct extensive experiments using 5 large real-world datasets to evaluate our proposed algorithms, demonstrating their superior efficiency compared to state-of-the-art approaches.

Scaling Up Graph Propagation Computation on Large Graphs: A Local Chebyshev Approximation Approach

TL;DR

The paper tackles scalable graph propagation (GP) for SSPPR and HKPR on large graphs, where Taylor-based PwMethod and Push-based approaches converge slowly. It introduces a novel Chebyshev expansion of GP functions, deriving explicit coefficients for SSPPR and HKPR and using them to build two algorithms: ChebyPower (a Chebyshev-accelerated power method) and ChebyPush (a local push-style method via a subset Chebyshev recurrence). Theoretical results show faster convergence (roughly O(√N) over Taylor) and provable error guarantees, with ChebyPush offering graph-size-independent runtime under suitable thresholds; the paper also generalizes to broader GP operators and bidirectional schemes. Extensive experiments on five real-world graphs demonstrate substantial speedups (often 3–8×) over state-of-the-art baselines for both SSPPR and HKPR, validating the practicality and scalability of the approach for large-scale graph analysis and graph neural network workflows.

Abstract

Graph propagation (GP) computation plays a crucial role in graph data analysis, supporting various applications such as graph node similarity queries, graph node ranking, graph clustering, and graph neural networks. Existing methods, mainly relying on power iteration or push computation frameworks, often face challenges with slow convergence rates when applied to large-scale graphs. To address this issue, we propose a novel and powerful approach that accelerates power iteration and push methods using Chebyshev polynomials. Specifically, we first present a novel Chebyshev expansion formula for general GP functions, offering a new perspective on GP computation and achieving accelerated convergence. Building on these theoretical insights, we develop a novel Chebyshev power iteration method (\ltwocheb) and a novel Chebyshev push method (\chebpush). Our \ltwocheb method demonstrates an approximate acceleration of compared to existing power iteration techniques for both personalized PageRank and heat kernel PageRank computations, which are well-studied GP problems. For \chebpush, we propose an innovative subset Chebyshev recurrence technique, enabling the design of a push-style local algorithm with provable error guarantee and reduced time complexity compared to existing push methods. We conduct extensive experiments using 5 large real-world datasets to evaluate our proposed algorithms, demonstrating their superior efficiency compared to state-of-the-art approaches.

Paper Structure

This paper contains 21 sections, 13 theorems, 45 equations, 9 figures, 1 table, 5 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Existing Solutions and Their Limitations
  5. A Chebyshev Power Method
  6. Chebyshev Expansion of GP Function
  7. Why Chebyshev Expansion
  8. The Proposed $\mathsf{ChebyPower}$ Algorithm
  9. A Novel Chebyshev Push Method
  10. Subset Chebyshev Recurrence
  11. $\mathsf{ChebyPush}$: Chebyshev Push Method
  12. Theoretical Analysis of $\mathsf{ChebyPush}$
  13. Generalizations
  14. Computing More General GP Matrix
  15. Generalizing to a Bidirectional Method
  16. ...and 6 more sections

Key Result

lemma 1

The following Chebyshev expansion holds for any graph propagation function $f$: where $c_k=\langle f,T_k\rangle=\frac{2}{\pi}\int_{-1}^{1}{\frac{1}{\sqrt{1-x^2}}f(x)T_k(x)dx}$ if $k\geq 1$; $c_0=\langle f,T_0\rangle =\frac{1}{\pi}\int_{-1}^{1}{\frac{1}{\sqrt{1-x^2}}f(x)T_0(x)dx}$ .

Figures (9)

  • Figure 1: Illustration of the deviation propagation
  • Figure 2: Illustration of $\mathsf{ChebyPush}$ to compute $T_3(\mathbf{P})\mathbf{e}_{v_1}$
  • Figure 3: The difference between $\mathsf{Push}$ v.s. $\mathsf{ChebyPush}$
  • Figure 4: Query time of different $\mathsf{SSPPR}$ algorithms under $l_1$-error. The lower (resp., upper) bar of each figure represents the query time of each algorithm to reach low-precision (resp., high-precision) $l_1$-error.
  • Figure 5: Query time of different $\mathsf{SSPPR}$ algorithms under Normalized RelErr. The lower (resp., upper) bar of each figure represents the query time of each algorithm to reach low-precision (resp., high-precision) Normalized RelErr.
  • ...and 4 more figures

Theorems & Definitions (34)

  • definition 1: Matrix Function DBLP:books/daglib/0086372
  • definition 2: Graph Propagation (GP) Function
  • definition 3: Single-source Personalized PageRank ($\mathsf{SSPPR}$) vector andersen2006local
  • definition 4: Heat Kernel PageRank ($\mathsf{HKPR}$) vector chung2007heat
  • definition 5
  • definition 6: Chebyshev polynomials
  • lemma 1: chebyshev expansion for general GP function
  • proof
  • lemma 2: chebyshev expansion for $\mathsf{SSPPR}$
  • proof
  • ...and 24 more