Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Accelerated Evolving Set Processes for Local PageRank Computation

Binbin Huang, Luo Luo, Yanghua Xiao, Deqing Yang, Baojian Zhou

TL;DR

This work proposes a novel framework based on nested evolving set processes to accelerate Personalized PageRank (PPR) computation, and shows that the time complexity of such localized methods is upper bounded by $\min\tilde{O}}(R^2/\epsilon^2), \tilde{\mathcal{O}}(m)\}$ to obtain an $\epsilon$-approximation of the PPR vector.

Abstract

This work proposes a novel framework based on nested evolving set processes to accelerate Personalized PageRank (PPR) computation. At each stage of the process, we employ a localized inexact proximal point iteration to solve a simplified linear system. We show that the time complexity of such localized methods is upper bounded by $\min\{\tilde{\mathcal{O}}(R^2/ε^2), \tilde{\mathcal{O}}(m)\}$ to obtain an $ε$-approximation of the PPR vector, where $m$ denotes the number of edges in the graph and $R$ is a constant defined via nested evolving set processes. Furthermore, the algorithms induced by our framework require solving only $\tilde{\mathcal{O}}(1/\sqrtα)$ such linear systems, where $α$ is the damping factor. When $1/ε^2\ll m$, this implies the existence of an algorithm that computes an $\ epsilon $-approximation of the PPR vector with an overall time complexity of $\tilde{\mathcal{O}}\left(R^2 / (\sqrtαε^2)\right)$, independent of the underlying graph size. Our result resolves an open conjecture from existing literature. Experimental results on real-world graphs validate the efficiency of our methods, demonstrating significant convergence in the early stages.

Accelerated Evolving Set Processes for Local PageRank Computation

TL;DR

This work proposes a novel framework based on nested evolving set processes to accelerate Personalized PageRank (PPR) computation, and shows that the time complexity of such localized methods is upper bounded by to obtain an -approximation of the PPR vector.

Abstract

This work proposes a novel framework based on nested evolving set processes to accelerate Personalized PageRank (PPR) computation. At each stage of the process, we employ a localized inexact proximal point iteration to solve a simplified linear system. We show that the time complexity of such localized methods is upper bounded by to obtain an -approximation of the PPR vector, where denotes the number of edges in the graph and is a constant defined via nested evolving set processes. Furthermore, the algorithms induced by our framework require solving only such linear systems, where is the damping factor. When , this implies the existence of an algorithm that computes an -approximation of the PPR vector with an overall time complexity of , independent of the underlying graph size. Our result resolves an open conjecture from existing literature. Experimental results on real-world graphs validate the efficiency of our methods, demonstrating significant convergence in the early stages.

Paper Structure

This paper contains 24 sections, 13 theorems, 115 equations, 12 figures, 4 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem reformulations and properties
  4. Inexact accelerated proximal point framework
  5. Accelerated Evolving Set Processes
  6. Nested evolving set process
  7. Localized inexact proximal operators
  8. Time complexity analysis and AESP-PPR
  9. Discussions and related problems
  10. Experiments
  11. Conclusions and Discussions
  12. Missing Proofs
  13. Proofs of Lemmas \ref{['lemma:properties-ppv']} and \ref{['lemma:properties-f']}
  14. The proof of Lemma \ref{['lemma:inner-loop-eps-t']}
  15. Proofs of Theorems \ref{['thm:convergence-loc-gd']} and \ref{['thm:convergence-local-appr']}
  16. ...and 9 more sections

Key Result

Lemma 2.1

Define the PPR matrix $\bm \Pi_{\alpha} = \alpha \left(\frac{1+\alpha}{2} {\bm{I}} - \frac{1-\alpha}{2} {\bm{A}}{\bm{D}}^{-1} \right)^{-1}$. Let the estimate-residual pair $({\bm{p}},{\bm{r}})$ for Eq. equ:PPR satisfy ${\bm{r}} = {\bm{e}}_s - \bm \Pi_\alpha^{-1}{\bm{p}}$. Then,

Figures (12)

  • Figure 1: The comparison of volumes of ESP for APPR and Ours.
  • Figure 2: Convergence of $\log\|{\bm{D}}^{-1}(\hat{\bm \pi} -\bm \pi)\|_\infty$ for AESP-LocAPPR with three different initializations for ${\bm{z}}_t^{(0)}$ as a function of total operations and running times on the com-dblp graph.
  • Figure 3: $C_{h_t}^0/\epsilon_t$, $\overline{\operatorname{vol}}({\mathcal{S}}_t)/\overline{\gamma_t}$ and $R$ of AESP-LocAPPR on 19 graphs (in ascending order of $n$) when ${\bm{z}}_t^{(0)} = {\bm{y}}^{(t-1)}$.
  • Figure 4: Performance of estimation error reduction, $\log\|{\bm{D}}^{-1}(\hat{\bm \pi} -\bm \pi)\|_\infty$, as a function of operations ${\mathcal{T}}$, on the graph ogb-mag240m, ogbn-papers100M, com-friendster and wiki-en21 with $\alpha = 0.01$ and $\epsilon = 10^{-6}$ where the graph can scale up to $n=244M$ and $m=1.728B$.
  • Figure 5: Speedup of AESP-based methods over standard local solvers (LocAPPR, LocGD) as a function of $\alpha$, on the com-dblp graph with $\epsilon = 0.1/n$ and $\alpha \in (10^{-3},10^{-1})$.
  • ...and 7 more figures

Theorems & Definitions (26)

  • Lemma 2.1: Properties of $\bm \pi$
  • Lemma 2.2: Properties of ${\bm{x}}_f^*$ and ${\bm{x}}_\psi^*$
  • Definition 3.1: Nested evolving set process (ESP)
  • Lemma 3.2
  • Theorem 3.3: Convergence of LocGD
  • Lemma 3.4: Outer-loop iteration complexity of AESP
  • Theorem 3.5: Time complexity of AESP
  • Theorem 3.6: Time complexity of AESP-PPR
  • proof
  • proof
  • ...and 16 more