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Dynamic PageRank: Algorithms and Lower Bounds

Rajesh Jayaram, Jakub Łącki, Slobodan Mitrović, Krzysztof Onak, Piotr Sankowski

TL;DR

This work provides a comprehensive complexity characterization for maintaining explicit PageRank approximations in dynamic graphs. It establishes near-tight upper and lower bounds for additive maintenance and near-quadratic lower bounds for multiplicative maintenance, resolving a long-standing open question in the affirmative for adversarial update orders. It also presents a dynamic random-walk based approach that achieves near-optimal additive maintenance, and shows that in undirected graphs multiplicative PageRank can be maintained with polylogarithmic update time under an oblivious adversary. Together, these results clarify fundamental limits and guide the design of practical dynamic PageRank algorithms.

Abstract

We consider the PageRank problem in the dynamic setting, where the goal is to explicitly maintain an approximate PageRank vector $π\in \mathbb{R}^n$ for a graph under a sequence of edge insertions and deletions. Our main result is a complete characterization of the complexity of dynamic PageRank maintenance for both multiplicative and additive ($L_1$) approximations. First, we establish matching lower and upper bounds for maintaining additive approximate PageRank in both incremental and decremental settings. In particular, we demonstrate that in the worst-case $(1/α)^{Θ(\log \log n)}$ update time is necessary and sufficient for this problem, where $α$ is the desired additive approximation. On the other hand, we demonstrate that the commonly employed ForwardPush approach performs substantially worse than this optimal runtime. Specifically, we show that ForwardPush requires $Ω(n^{1-δ})$ time per update on average, for any $δ> 0$, even in the incremental setting. For multiplicative approximations, however, we demonstrate that the situation is significantly more challenging. Specifically, we prove that any algorithm that explicitly maintains a constant factor multiplicative approximation of the PageRank vector of a directed graph must have amortized update time $Ω(n^{1-δ})$, for any $δ> 0$, even in the incremental setting, thereby resolving a 13-year old open question of Bahmani et al.~(VLDB 2010). This sharply contrasts with the undirected setting, where we show that $\rm{poly}\ \log n$ update time is feasible, even in the fully dynamic setting under oblivious adversary.

Dynamic PageRank: Algorithms and Lower Bounds

TL;DR

This work provides a comprehensive complexity characterization for maintaining explicit PageRank approximations in dynamic graphs. It establishes near-tight upper and lower bounds for additive maintenance and near-quadratic lower bounds for multiplicative maintenance, resolving a long-standing open question in the affirmative for adversarial update orders. It also presents a dynamic random-walk based approach that achieves near-optimal additive maintenance, and shows that in undirected graphs multiplicative PageRank can be maintained with polylogarithmic update time under an oblivious adversary. Together, these results clarify fundamental limits and guide the design of practical dynamic PageRank algorithms.

Abstract

We consider the PageRank problem in the dynamic setting, where the goal is to explicitly maintain an approximate PageRank vector for a graph under a sequence of edge insertions and deletions. Our main result is a complete characterization of the complexity of dynamic PageRank maintenance for both multiplicative and additive () approximations. First, we establish matching lower and upper bounds for maintaining additive approximate PageRank in both incremental and decremental settings. In particular, we demonstrate that in the worst-case update time is necessary and sufficient for this problem, where is the desired additive approximation. On the other hand, we demonstrate that the commonly employed ForwardPush approach performs substantially worse than this optimal runtime. Specifically, we show that ForwardPush requires time per update on average, for any , even in the incremental setting. For multiplicative approximations, however, we demonstrate that the situation is significantly more challenging. Specifically, we prove that any algorithm that explicitly maintains a constant factor multiplicative approximation of the PageRank vector of a directed graph must have amortized update time , for any , even in the incremental setting, thereby resolving a 13-year old open question of Bahmani et al.~(VLDB 2010). This sharply contrasts with the undirected setting, where we show that update time is feasible, even in the fully dynamic setting under oblivious adversary.
Paper Structure (22 sections, 17 theorems, 20 equations, 1 figure, 3 algorithms)

This paper contains 22 sections, 17 theorems, 20 equations, 1 figure, 3 algorithms.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Additive Approximation
  4. Multiplicative Approximation
  5. Related Work
  6. Further related work.
  7. Impossibility of Non-Trivial Worst-Case Bounds
  8. Organization of the Paper
  9. Preliminaries
  10. Lower Bounds
  11. The Graph.
  12. Update Sequence
  13. Lower Bound for Maintaining Additive Approximation
  14. Lower Bound for Maintaining Multiplicative Approximation
  15. Lower bound for the ForwardPush algorithm
  16. ...and 7 more sections

Key Result

Theorem 1.1

Fix $\epsilon \in (0.01, 0.99)$. For any sufficiently large $n \geq 1$ and any $\alpha$ such that $1/\alpha = n^{o(1 / \log \log n)}$, any algorithm which explicitly maintains $\alpha$-additive approximation of PageRank must run in $n \cdot (1/\alpha)^{\Omega(\log \log n)}$ total time.

Figures (1)

  • Figure 1: An example illustrating that maintaining multiplicative approximation or even an $L_1\xspace$ approximation of PageRank in the worst case requires $\Theta(n / \log n)$ running time even after a single deletion/insertion of edge $uv$. For details, see \ref{['sec:impossibility']}.

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 1: bahmani2010fastlkacki2020walking
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.3
  • ...and 15 more