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A local updating algorithm for personalized PageRank via Chebyshev polynomials

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Abstract

The personalized PageRank algorithm is one of the most versatile tools for the analysis of networks. In spite of its ubiquity, maintaining personalized PageRank vectors when the underlying network constantly evolves is still a challenging task. To address this limitation, this work proposes a novel distributed algorithm to locally update personalized PageRank vectors when the graph topology changes. The proposed algorithm is based on the use of Chebyshev polynomials and a novel update equation that encompasses a large family of PageRank-based methods. In particular, the algorithm has the following advantages: (i) it has faster convergence speed than state-of-the-art alternatives for local personalized PageRank updating; and (ii) it can update the solution of recent extensions of personalized PageRank that rely on complex dynamical processes for which no updating algorithms have been developed. Experiments in a real-world temporal network of an autonomous system validate the effectiveness of the proposed algorithm.

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Data availability

The datasets generated during and/or analyzed during the current study are available in the Network Data Repository, [https://networkrepository.com/tech-as-topology.php]. The code to replicate the results is available at https://github.com/estbautista/PageRank_Updating_Chebyshev_Paper.

References

  • Page L, Brin S, Motwani R, Winograd T (1999) The pagerank citation ranking: Bringing order to the web. Technical report, Stanford InfoLab

    Google Scholar 

  • Ding C, He X, Husbands P, Zha H, Simon H (2003) Pagerank, hits and a unified framework for link analysis. In: Proceedings of the 2003 SIAM International Conference on Data Mining, pp. 249–253. SIAM

  • Haveliwala TH (2003) Topic-sensitive pagerank: A context-sensitive ranking algorithm for web search. IEEE Trans Knowl Data Eng 15(4):784–796

    Article  Google Scholar 

  • Chung F (2009) A local graph partitioning algorithm using heat kernel pagerank. Internet Math 6(3):315–330

    Article  MathSciNet  MATH  Google Scholar 

  • Andersen R, Chung F, Lang K (2006) Local graph partitioning using pagerank vectors. In: 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS’06), pp. 475–486. IEEE

  • Tabrizi SA, Shakery A, Asadpour M, Abbasi M, Tavallaie MA (2013) Personalized pagerank clustering: A graph clustering algorithm based on random walks. Physica A: Stat Mech Appl 392(22):5772–5785

    Article  MathSciNet  MATH  Google Scholar 

  • Avrachenkov K, Gonçalves P, Legout A, Sokol M (2012) Classification of content and users in bittorrent by semi-supervised learning methods. In: 2012 8th International Wireless Communications and Mobile Computing Conference (IWCMC), pp. 625–630. IEEE

  • Merkurjev E, Bertozzi AL, Chung F (2018) A semi-supervised heat kernel pagerank mbo algorithm for data classification. Commun Math Sci 16(5):1241–1265

    Article  MathSciNet  MATH  Google Scholar 

  • Dostal M, Nykl M, Ježek K (2014) Exploration of document classification with linked data and pagerank. In: Intelligent Distributed Computing VII, pp. 37–43. Springer

  • Avrachenkov K, Mishenin A, Gonçalves P, Sokol M (2012) Generalized optimization framework for graph-based semi-supervised learning. In: Proceedings of the 2012 SIAM International Conference on Data Mining, pp. 966–974. SIAM

  • Fontugne R, Bautista E, Petrie C, Nomura Y, Abry P, Gonçalves P, Fukuda K, Aben E (2019) Bgp zombies: An analysis of beacons stuck routes. In: International Conference on Passive and Active Network Measurement, pp. 197–209. Springer

  • Yoon M, Hooi B, Shin K, Faloutsos C (2019) Fast and accurate anomaly detection in dynamic graphs with a two-pronged approach. In: Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pp. 647–657

  • Yao Z, Mark P, Rabbat M (2012) Anomaly detection using proximity graph and pagerank algorithm. IEEE Trans Inf Forens Secur 7(4):1288–1300

    Article  Google Scholar 

  • Al_Janabi S, Kadiam, N (2019)Recommendation system of big data based on pagerank clustering algorithm. In: International Conference on Big Data and Networks Technologies, pp. 149–171. Springer

  • Zhang Y, Zhang N, Tang J (2009) A collaborative filtering tag recommendation system based on graph. ECML PKDD discovery challenge, 297–306

  • Nguyen P, Tomeo P, Di Noia T, Di Sciascio E (2015) An evaluation of simrank and personalized pagerank to build a recommender system for the web of data. In: Proceedings of the 24th International Conference on World Wide Web, pp. 1477–1482

  • Langville AN, Meyer CD (2004) Deeper inside pagerank. Internet Math 1(3):335–380

    Article  MathSciNet  MATH  Google Scholar 

  • Ipsen IC, Wills RS (2006) Mathematical properties and analysis of google’s pagerank. Bol Soc Esp Mat Apl 34:191–196

    MathSciNet  MATH  Google Scholar 

  • Brezinski C, Redivo-Zaglia M (2006) The pagerank vector: properties, computation, approximation, and acceleration. SIAM J Matrix Analy Appl 28(2):551–575

    Article  MathSciNet  MATH  Google Scholar 

  • Pretto L (2002) A theoretical analysis of google’s pagerank. In: International Symposium on String Processing and Information Retrieval, pp. 131–144. Springer

  • Bautista E, Abry P, Gonçalves P (2019) L\(^\gamma\)-pagerank for semi-supervised learning. Appl Netw Sci 4(1):1–20

    Article  Google Scholar 

  • De Nigris S, Bautista E, Abry P, Avrachenkov K, Gonçalves P (2017) Fractional graph-based semi-supervised learning. In: 2017 25th European Signal Processing Conference (EUSIPCO), pp. 356–360. IEEE

  • Mai X, Couillet R (2017) The counterintuitive mechanism of graph-based semi-supervised learning in the big data regime. In: 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 2821–2825. IEEE

  • Zhou X, Belkin M (2011) Semi-supervised learning by higher order regularization. In: Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics, pp. 892–900. JMLR Workshop and Conference Proceedings

  • Girault B, Gonçalves P, Fleury E, Mor AS (2014) Semi-supervised learning for graph to signal mapping: A graph signal wiener filter interpretation. In: 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 1115–1119. IEEE

  • Haveliwala T (1999) Ecient computation of pagerank. Technical report, Stanford

    Google Scholar 

  • Kamvar S, Haveliwala T, Golub G (2004) Adaptive methods for the computation of pagerank. Linear Algebra Appl 386:51–65

    Article  MathSciNet  MATH  Google Scholar 

  • Fujiwara Y, Nakatsuji M, Yamamuro T, Shiokawa H, Onizuka M (2012) Efficient personalized pagerank with accuracy assurance. In: Proceedings of the 18th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 15–23

  • Bahmani B, Chakrabarti K, Xin D (2011) Fast personalized pagerank on mapreduce. In: Proceedings of the 2011 ACM SIGMOD International Conference on Management of Data, pp. 973–984

  • Maehara T, Akiba T, Iwata Y, Kawarabayashi K-i (2014) Computing personalized pagerank quickly by exploiting graph structures. Proc VLDB Endowment 7(12):1023–1034

    Article  Google Scholar 

  • Avrachenkov K, Litvak N, Nemirovsky D, Osipova N (2007) Monte carlo methods in pagerank computation: When one iteration is sufficient. SIAM J Numer Analy 45(2):890–904

    Article  MathSciNet  MATH  Google Scholar 

  • Berkhin P (2006) Bookmark-coloring algorithm for personalized pagerank computing. Internet Math 3(1):41–62

    Article  MathSciNet  MATH  Google Scholar 

  • Ohsaka N, Maehara T, Kawarabayashi K-i (2015) Efficient pagerank tracking in evolving networks. In: Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 875–884

  • Bahmani B, Chowdhury A, Goel A (2010) Fast incremental and personalized pagerank. Proc. VLDB Endow. 4(3):173–184. https://doi.org/10.14778/1929861.1929864

    Article  Google Scholar 

  • Yoon M, Jin W, Kang U (2018) Fast and accurate random walk with restart on dynamic graphs with guarantees. In: Proceedings of the 2018 World Wide Web Conference, pp. 409–418

  • Zhang H, Lofgren P, Goel A (2016) Approximate personalized pagerank on dynamic graphs. In: Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 1315–1324

  • Yoon M, Gervet T, Hooi B, Faloutsos C (2020) Autonomous graph mining algorithm search with best speed/accuracy trade-off. In: 2020 IEEE International Conference on Data Mining (ICDM), 751–760

  • Bautista Ruiz E (2019) Laplacian Powers for Graph-Based Semi-Supervised Learning. Theses, Université de Lyon. https://tel.archives-ouvertes.fr/tel-02476246

  • Shuman DI, Vandergheynst P, Frossard P (2011) Chebyshev polynomial approximation for distributed signal processing. In: 2011 International Conference on Distributed Computing in Sensor Systems and Workshops (DCOSS), pp. 1–8. IEEE

  • Cheng C, Jiang J, Emirov N, Sun Q (2019) Iterative chebyshev polynomial algorithm for signal denoising on graphs. In: 2019 13th International Conference on Sampling Theory and Applications (SampTA), pp. 1–5. IEEE

  • Tseng C-C, Lee S-L (2021) Minimax design of graph filter using chebyshev polynomial approximation. IEEE Trans Circ Syst II: Express Briefs 68(5):1630–1634

    Google Scholar 

  • Tian D, Mansour H, Knyazev A, Vetro A (2014) Chebyshev and conjugate gradient filters for graph image denoising. In: 2014 IEEE International Conference on Multimedia and Expo Workshops (ICMEW), pp. 1–6. IEEE

  • Defferrard M, Bresson X, Vandergheynst P (2016) Convolutional neural networks on graphs with fast localized spectral filtering. Adv Neural Inf Process Syst 29:3844–3852

    Google Scholar 

  • Yan B, Wang G, Yu J, Jin X, Zhang H (2021) Spatial-temporal chebyshev graph neural network for traffic flow prediction in iot-based its. IEEE Internet Things J

  • Tremblay N, Gonçalves P, Borgnat P (2018) Design of graph filters and filterbanks. In: Cooperative and Graph Signal Processing, pp. 299–324. Elsevier

  • Giovanidis A, Baynat B, Magnien C, Vendeville A (2021) Ranking online social users by their influence. IEEE/ACM Transactions on Networking

  • Rossi RA, Ahmed NK (2015) The network data repository with interactive graph analytics and visualization. In: AAAI. http://networkrepository.com

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Acknowledgements

The authors would like to thank P. Gonçalves and P. Abry for helpful discussions.

Funding

This work is funded in part by the ANR (French National Agency of Research) under the Limass (ANR-19-CE23-0010) and FiT LabCom grants.

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Authors and Affiliations

  1. Sorbonne Université, CNRS, LIP6, F-75005, Paris, France

    Esteban Bautista & Matthieu Latapy

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  1. Esteban Bautista
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  2. Matthieu Latapy
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Correspondence to Esteban Bautista.

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Bautista, E., Latapy, M. A local updating algorithm for personalized PageRank via Chebyshev polynomials. Soc. Netw. Anal. Min. 12, 31 (2022). https://doi.org/10.1007/s13278-022-00860-5

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