Dynamic PageRank: Algorithms and Lower Bounds

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

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Author Details

Rajesh Jayaram
  • Google Research, New York, NY, USA
Jakub Łącki
  • Google Research, New York, NY, USA
Slobodan Mitrović
  • University of California Davis, CA, USA
Krzysztof Onak
  • Boston University, USA
Piotr Sankowski
  • IDEAS NCBR, University of Warsaw, Poland
  • MIM Solutions, Warsaw, Poland

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Rajesh Jayaram, Jakub Łącki, Slobodan Mitrović, Krzysztof Onak, and Piotr Sankowski. Dynamic PageRank: Algorithms and Lower Bounds. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 90:1-90:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


We consider the PageRank problem in the dynamic setting, where the goal is to explicitly maintain an approximate PageRank vector π ∈ ℝⁿ 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 (L₁) 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 poly log n update time is feasible, even in the fully dynamic setting under oblivious adversary.

Subject Classification

ACM Subject Classification
  • Theory of computation → Dynamic graph algorithms
  • Information systems → Page and site ranking
  • Theory of computation → Random walks and Markov chains
  • Mathematics of computing → Graph algorithms
  • PageRank
  • dynamic algorithms
  • graph algorithms


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