pith. sign in

arxiv: 1907.09979 · v1 · pith:G7D7XCSSnew · submitted 2019-07-23 · 📡 eess.SY · cs.SY

Efficient PageRank Computation via Distributed Algorithms with Web Clustering

Atsushi Suzuki , Hideaki Ishii This is my paper

Pith reviewed 2026-05-24 17:01 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords PageRankdistributed algorithmsweb clusteringexponential convergencegossip algorithmshyperlink graphslocal updates
0
0 comments X

The pith

A reinterpretation of PageRank yields distributed algorithms that converge exponentially even after web pages are grouped into clusters for local updates.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops distributed methods for computing PageRank values on the web graph, where each page updates its score through interactions with pages it links to. It begins with randomized gossip-style selections of pages for updates, moves to deterministic versions allowing simultaneous updates, and then adds a clustering step in which groups of pages repeatedly update among themselves before exchanging information across clusters. The central move is a reinterpretation of the PageRank equation that makes these local iterations converge at an exponential rate while sharply reducing the total messages passed between pages. Numerical tests on real web data are used to show faster convergence and lower communication volume than prior distributed approaches.

Core claim

By reinterpreting PageRank as the solution to a set of local fixed-point equations, the authors obtain a family of iterative algorithms whose convergence remains exponential after the introduction of a clustering scheme that confines most updates to intra-cluster exchanges.

What carries the argument

The reinterpretation of PageRank that permits iterative local updates whose convergence rate remains exponential even after clustering restricts most interactions to intra-cluster exchanges.

If this is right

  • Gossip-type randomized updates produce exponential convergence for PageRank.
  • The same exponential rate carries over to deterministic simultaneous-update versions.
  • The clustering scheme further accelerates convergence while limiting communication to mostly local exchanges.
  • Real-web-data experiments exhibit measurable gains in both speed and message count over existing distributed PageRank methods.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same local-update structure could be tested on other eigenvector-based centrality measures on directed graphs.
  • Different clustering heuristics might trade off intra-cluster speed against the quality of the final global scores.
  • The approach suggests checking whether the exponential rate survives when clusters are allowed to overlap or change over time.

Load-bearing premise

The reinterpretation of PageRank permits iterative local updates whose convergence rate remains exponential even after the clustering scheme restricts most interactions to intra-cluster exchanges.

What would settle it

A set of numerical runs on the same real web graphs in which the measured convergence rate after clustering becomes sub-exponential or the total number of inter-page messages fails to drop below that of the unclustered version.

Figures

Figures reproduced from arXiv: 1907.09979 by Atsushi Suzuki, Hideaki Ishii.

Figure 1
Figure 1. Figure 1: An example graph with seven nodes two stochastic matrices A and (1/n)1n1 T n , it is stochastic as well. It is now clear that x ∗ is the eigenvector corresponding the eigenvalue 1 of the link matrix M. For its computation, the PageRank vector x ∗ can be obtained by solving the linear equation (1) or (2). However, due to its large dimension, the computation must rely on simple algorithms. It is common to us… view at source ↗
Figure 2
Figure 2. Figure 2: Time responses of the synchronous algorithms for the [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: Time responses of the sums of errors for Algorithm 2 in [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: PageRank values of the pages in the large network: Mar [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Time responses of errors in asynchronous randomized [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: (a) Time responses of errors: Algorithm 2, aggregati [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
read the original abstract

PageRank is a well-known centrality measure for the web used in search engines, representing the importance of each web page. In this paper, we follow the line of recent research on the development of distributed algorithms for computation of PageRank, where each page computes its own PageRank value by interacting with pages connected over hyperlinks. Our approach is novel in that it is based on a reinterpretation of PageRank, which leads us to a set of algorithms with exponential convergence rates. We first employ gossip-type randomization for the page selections in the update iterations. Then, the algorithms are generalized to deterministic ones, allowing simultaneous updates by multiple pages. Finally, based on these algorithms, we propose a clustering-based scheme, in which groups of pages make updates by locally interacting among themselves many times to expedite the convergence. In comparison with other existing techniques, significant advantages can be exhibited in their convergence performance, as demonstrated via numerical examples using real web data, and also in the limited amount of communication required among pages.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper claims that a reinterpretation of PageRank yields distributed algorithms (gossip-type randomized, deterministic simultaneous, and clustering-based) with exponential convergence rates, superior performance on real web data, and reduced inter-page communication compared to prior methods.

Significance. If the reinterpretation produces a provable geometric contraction that survives the clustering restriction and the numerical advantages are reproducible, the work would advance practical distributed computation of network centralities under communication constraints.

major comments (2)
  1. [Abstract] Abstract: the claim that the clustering-based scheme retains exponential convergence is unsupported by any contraction analysis or bound on the effective rate after restricting updates to intra-cluster exchanges; the provided text gives no indication that the geometric factor remains strictly less than 1 independently of the partition.
  2. [Abstract] Abstract: numerical examples are said to demonstrate significant advantages on real web data, yet no dataset description, cluster selection procedure, iteration counts, error norms, or baseline comparisons are supplied, rendering the performance claim unverifiable.
minor comments (1)
  1. The abstract refers to 'exponential convergence rates' without indicating dependence on graph properties or the base of the rate.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive comments. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that the clustering-based scheme retains exponential convergence is unsupported by any contraction analysis or bound on the effective rate after restricting updates to intra-cluster exchanges; the provided text gives no indication that the geometric factor remains strictly less than 1 independently of the partition.

    Authors: We agree that the manuscript text does not supply a contraction analysis or explicit bound for the clustering scheme after restricting to intra-cluster exchanges. The exponential rates are established for the gossip and deterministic algorithms; the clustering scheme is presented as an extension relying on repeated local updates. We will revise the paper to add a dedicated convergence analysis section proving that the effective contraction factor remains strictly less than 1 independently of the partition under the same assumptions used for the base algorithms. revision: yes

  2. Referee: [Abstract] Abstract: numerical examples are said to demonstrate significant advantages on real web data, yet no dataset description, cluster selection procedure, iteration counts, error norms, or baseline comparisons are supplied, rendering the performance claim unverifiable.

    Authors: We agree the abstract (and, per the comment, the supplied text) omits these details. The numerical examples appear in the full manuscript but lack explicit descriptions of the web graphs, clustering method, metrics, and baselines. We will revise by expanding the abstract with a brief mention of the data and adding a subsection in the experiments that specifies the datasets, cluster selection procedure, iteration counts, error norms, and baseline comparisons. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation from reinterpretation is self-contained

full rationale

The paper presents a reinterpretation of PageRank that directly yields gossip-type randomized updates and deterministic simultaneous updates, both claimed to exhibit exponential convergence, followed by a clustering extension that restricts most interactions to intra-cluster exchanges. No equations, parameters, or self-citations are shown that reduce the claimed convergence rates or clustering benefits to quantities defined by the inputs themselves. The approach is derived from the reinterpretation rather than fitted or renamed by construction, and the abstract and description treat the exponential rates as a consequence of the algorithmic structure, not a tautology. This is the normal case of an independent derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard assumptions about directed web graphs and convergence of linear iterations; no free parameters, new entities, or ad-hoc axioms are introduced in the provided abstract.

axioms (1)
  • domain assumption Web graph is a directed graph whose adjacency structure defines valid local update neighborhoods for PageRank iterations.
    Implicit throughout the description of hyperlink-based interactions and clustering.

pith-pipeline@v0.9.0 · 5698 in / 1198 out tokens · 22738 ms · 2026-05-24T17:01:49.478342+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages · 1 internal anchor

  1. [1]

    R. W. Aldhaheri and H. K. Khalil. Aggregation of the polic y iteration method for nearly completely decomposable Markov chains. IEEE Trans. Autom. Contr ., 36:178–187, 1991

    work page 1991

  2. [2]

    Avrachenkov, N

    K. Avrachenkov, N. Litvak, D. Nemirovsky, and N. Osipova . Monte Carlo methods in PageRank computation: When one iteration i s suffi- cient. SIAM J. Numer . Anal. , 45:890–904, 2007

    work page 2007

  3. [3]

    Bıyık and M

    E. Bıyık and M. Arcak. Area aggregation and time-scale mo deling for sparse nonlinear networks. Syst. & Cont. Lett. , 57:142–149, 2008

    work page 2008

  4. [4]

    Brin and L

    S. Brin and L. Page. The anatomy of a large-scale hypertex tual Web search engine. Computer Networks & ISDN Systems , 30:107–117, 1998

    work page 1998

  5. [5]

    A. Z. Broder, R. Lempel, F. Maghoul, and J. Pedersen. Effic ient PageRank approximation via graph aggregation. Inform. Retrieval , 9:123–138, 2006

    work page 2006

  6. [6]

    F. Bullo. Lectures on Network Systems . CreateSpace, 2018

    work page 2018

  7. [7]

    Charalambous, C

    T. Charalambous, C. Hadjicostis, M. Rabbat, and M. Johan sson. Totally asynchronous distributed estimation of eigenvector centr ality in digraphs with application to the PageRank problem. In Proc. 55th IEEE Conf. on Decision and Control , pages 25–30, 2016

    work page 2016

  8. [8]

    B. C. Cs´ aji, R. M. Jungers, and V . D. Blondel. PageRank op timization by edge selection. Discrete Applied Mathematics , 169:73–87, 2014

    work page 2014

  9. [9]

    Fully distributed PageRank computation with exponential convergence

    L. Dai and N. Freris. Fully distributed PageRank computa tion with exponential convergence. arXiv:1705.09927, 2017

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  10. [10]

    Fercoq, M

    O. Fercoq, M. Akian, M. Bouhtou, and S. Gaubert. Ergodic control and polyhedral approaches to PageRank optimization. IEEE Trans. Autom. Contr ., 2013

    work page 2013

  11. [11]

    Frasca, H

    P . Frasca, H. Ishii, C. Ravazzi, and R. Tempo. Distribut ed randomized algorithms for opinion formation, centrality computation and power systems estimation: A tutorial overview. European J. Control , 24:2– 13, 2009

    work page 2009

  12. [12]

    D. F. Gleich. PageRank beyond the Web. SIAM Review, 57(3):321–363, 2015

    work page 2015

  13. [13]

    Academic We b Link- database Project, Univ

    Statistical Cybermetrics Research Group. Academic We b Link- database Project, Univ. Wolverhampton, U.K. [Online] Avai lable: http://cybermetrics.wlv.ac.uk/database/, 2006

    work page 2006

  14. [14]

    R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge Univ. Press, 1985

    work page 1985

  15. [15]

    Ishii and A

    H. Ishii and A. Suzuki. Distributed randomized algorit hms for pagerank computation: Recent advances. In T. Bas ¸ar, editor, Uncertainty in Complex Networked Systems: In Honor of Roberto Tempo , pages 419–

    work page

  16. [16]

    Ishii and R

    H. Ishii and R. Tempo. Distributed randomized algorith ms for the PageRank computation. IEEE Trans. Autom. Contr . , 55:1987–2002, 2010

    work page 1987

  17. [17]

    Ishii and R

    H. Ishii and R. Tempo. The PageRank problem, multi-agen t consensus and web aggregation: A systems and control viewpoint. IEEE Control Systems Magazine , 34:34–53, 2014

    work page 2014

  18. [18]

    Ishii, R

    H. Ishii, R. Tempo, and E.-W. Bai. PageRank computation via a distributed randomized approach with lossy communication . Syst. & Cont. Lett. , 61:1221–1228, 2012

    work page 2012

  19. [19]

    Ishii, R

    H. Ishii, R. Tempo, and E.-W. Bai. A web aggregation appr oach for distributed randomized PageRank algorithms. IEEE Trans. Autom. Contr ., 57:2703–2717, 2012

    work page 2012

  20. [20]

    C. M. Lagoa, L. Zaccarian, and F. Dabbene. A distributed algorithm with consistency for PageRank-like linear algebraic syste ms. In Proc. 20th IF AC W orld Congress, pages 5339–5344, 2017

    work page 2017

  21. [21]

    A. N. Langville and C. D. Meyer. Google’s PageRank and Beyond: The Science of Search Engine Rankings . Princeton University Press, 2006

    work page 2006

  22. [22]

    Lei and H.-F

    J. Lei and H.-F. Chen. Distributed randomized PageRank algorithm based on stochastic approximation. IEEE Trans. Autom. Contr ., 60:1641– 1646, 2015

    work page 2015

  23. [23]

    J. M. Maestre, H. Ishii, and E. Algaba. Node aggregation for enhancing PageRank. IEEE Access , 5:19799–19811, 2017

    work page 2017

  24. [24]

    Mesbahi and M

    M. Mesbahi and M. Egerstedt. Graph Theoretic Methods in Multiagent Networks. Princeton University Press, 2010

    work page 2010

  25. [25]

    Montijano, G

    E. Montijano, G. Oliva, and A. Gasparri. Distributed es timation of node centrality with application to agreement problems in socia l networks. In Proc. 57th IEEE Conf. on Decision and Control , pages 5245–5250, 2018

    work page 2018

  26. [26]

    A. V . Nazin and B. T. Polyak. Randomized algorithm to det ermine the eigenvector of a stochastic matrix with application to the P ageRank problem. Automation and Remote Control , 72:342–352, 2011

    work page 2011

  27. [27]

    B. T. Polyak and A. Tremba. Regularization-based solut ion of the PageRank problem for large matrices. Automation and Remote Control , 73:1877–1894, 2012

    work page 2012

  28. [28]

    Ravazzi, P

    C. Ravazzi, P . Frasca, R. Tempo, and H. Ishii. Ergodic ra ndomized algorithms and dynamics over networks. IEEE Trans. Control of Network Syst. , 2:78–87, 2015

    work page 2015

  29. [29]

    Sarma, A

    A. Sarma, A. Molla, G. Pandurangan, and E. Upfal. Fast di stributed PageRank computation. Theoretical Computer Science , 561:113–121, 2015

    work page 2015

  30. [30]

    S. Shi, J. Y u, G. Y ang, and D. Wang. Distributed page rank ing in structured P2P networks. In Proc. Int. Conf. Parallel Processing , pages 179–186, 2003

    work page 2003

  31. [31]

    Suzuki and H

    A. Suzuki and H. Ishii. Distributed randomized algorit hms for PageRank based on a novel interpretation. In Proc. American Control Conf. , pages 472–477, 2018

    work page 2018

  32. [32]

    Suzuki and H

    A. Suzuki and H. Ishii. PageRank computation via web agg regation in distributed randomized algorithms. Submitted for confere nce publica- tion, 2019

    work page 2019

  33. [33]

    Tempo, G

    R. Tempo, G. Calafiore, and F. Dabbene. Randomized Algorithms for Analysis and Control of Uncertain Systems, with Applicatio ns, Second Edition. Springer, London, 2013

    work page 2013

  34. [34]

    Wang and C

    W. Wang and C. Y . Tang. Distributed estimation of closen ess centrality. In Proc. 54th IEEE Conf. on Decision and Control , pages 4860–4865, 2015

    work page 2015

  35. [35]

    K. Y ou, R. Tempo, and L. Qiu. Distributed algorithms for computation of centrality measures in complex networks. IEEE Trans. Autom. Contr ., 62:2080–2094, 2017

    work page 2080

  36. [36]

    Zhao, H.-F

    W. Zhao, H.-F. Chen, and H. Fang. Convergence of distrib uted randomized PageRank algorithms. IEEE Trans. Autom. Contr ., 50:1177– 1181, 2013

    work page 2013

  37. [37]

    Y . Zhu, S. Y e, and X. Li. Distributed PageRank computati on based on iterative aggregation-disaggregation methods. In Proc. 14th ACM Conf. on Information and Knowledge Management , pages 578–585, 2005

    work page 2005