Revisiting Local Computation of PageRank: Simple and Optimal
Pages 911 - 922
Abstract
We revisit ApproxContributions, the classic local graph exploration algorithm proposed by Andersen, Borgs, Chayes, Hopcroft, Mirrokni, and Teng (WAW ’07, Internet Math. ’08) for computing an є-approximation of the PageRank contribution vector for a target node t on a graph with n nodes and m edges. We give a worst-case complexity bound of it as O(nπ(t)/є·min(Δin,Δout,√m)), where π(t) is the PageRank score of t, and Δin and Δout are the maximum in-degree and out-degree of the graph, resp. We also give a lower bound of Ω(min(Δin/δ,Δout/δ,√m/δ,m)) for detecting t’s δ-contributing set, showing that the simple ApproxContributions algorithm is already optimal.
As ApproxContributions has become a cornerstone for computing random-walk probabilities, our results and techniques can be applied to derive better bounds for various relevant problems. In particular, we investigate the computational complexity of locally estimating a node’s PageRank centrality. We improve the best-known upper bound of O(n2/3·min(Δout1/3,m1/6)) given by Bressan, Peserico, and Pretto (SICOMP ’23) to O(n1/2·min(Δin1/2,Δout1/2,m1/4)) by combining ApproxContributions with Monte Carlo sampling. We also improve their lower bound of Ω(min(n1/2Δout1/2,n1/3m1/3)) to Ω(n1/2·min(Δin1/2,Δout1/2,m1/4)) if min(Δin,Δout)=Ω(n1/3), and to Ω(n1/2−γ(min(Δin,Δout))1/2+γ) otherwise, where γ>0 is an arbitrarily small constant. Our matching upper and lower bounds resolve the open problem of whether one can tighten the bounds given by Bressan, Peserico, and Pretto (FOCS ’18, SICOMP ’23). Remarkably, the techniques and analyses for proving all our results are surprisingly simple.
References
[1]
Reid Andersen, Christian Borgs, Jennifer T. Chayes, John E. Hopcroft, Kamal Jain, Vahab S. Mirrokni, and Shang-Hua Teng. 2008. Robust pagerank and locally computable spam detection features. In Proc. 4th Int. Workshop Adversarial Inf. Retrieval Web. 69–76.
[2]
Reid Andersen, Christian Borgs, Jennifer T. Chayes, John E. Hopcroft, Vahab S. Mirrokni, and Shang-Hua Teng. 2007. Local computation of pagerank contributions. In Proc. 5th Int. Workshop Algorithms Models Web Graph, Vol. 4863. 150–165.
[3]
Reid Andersen, Christian Borgs, Jennifer T. Chayes, John E. Hopcroft, Vahab S. Mirrokni, and Shang-Hua Teng. 2008. Local computation of pagerank contributions. Internet Math. 5, 1 (2008), 23–45.
[4]
Reid Andersen, Fan R. K. Chung, and Kevin J. Lang. 2006. Local graph partitioning using pagerank vectors. In Proc. 47th Annu. IEEE Symp. Found. Comput. Sci. 475–486.
[5]
Reid Andersen, Fan R. K. Chung, and Kevin J. Lang. 2007. Using pagerank to locally partition a graph. Internet Math. 4, 1 (2007), 35–64.
[6]
Konstantin Avrachenkov, Nelly Litvak, Danil Nemirovsky, and Natalia Osipova. 2007. Monte carlo methods in pagerank computation: when one iteration is sufficient. SIAM J. Numer. Anal. 45, 2 (2007), 890–904.
[7]
Siddhartha Banerjee and Peter Lofgren. 2015. Fast bidirectional probability estimation in markov models. In Advances Neural Inf. Process. Syst. 28. 1423–1431.
[8]
Ziv Bar-Yossef and Li-Tal Mashiach. 2008. Local approximation of pagerank and reverse pagerank. In Proc. 17th ACM Int. Conf. Inf. Knowl. Manage. 279–288.
[9]
Aleksandar Bojchevski, Johannes Klicpera, Bryan Perozzi, Amol Kapoor, Martin Blais, Benedek Rózemberczki, Michal Lukasik, and Stephan Günnemann. 2020. Scaling graph neural networks with approximate pagerank. In Proc. 26th ACM SIGKDD Int. Conf. Knowl. Discovery Data Mining. 2464–2473.
[10]
Christian Borgs, Michael Brautbar, Jennifer T. Chayes, and Shang-Hua Teng. 2012. A sublinear time algorithm for pagerank computations. In Proc. 9th Int. Workshop Algorithms Models Web Graph, Vol. 7323. 41–53.
[11]
Christian Borgs, Michael Brautbar, Jennifer T. Chayes, and Shang-Hua Teng. 2014. Multiscale matrix sampling and sublinear-time pagerank computation. Internet Math. 10, 1-2 (2014), 20–48.
[12]
Marco Bressan, Enoch Peserico, and Luca Pretto. 2013. The power of local information in pagerank. In Proc. 22nd Int. World Wide Web Conf. 179–180.
[13]
Marco Bressan, Enoch Peserico, and Luca Pretto. 2015. Simple set cardinality estimation through random sampling. CoRR abs/1512.07901 (2015).
[14]
Marco Bressan, Enoch Peserico, and Luca Pretto. 2018. Sublinear algorithms for local graph centrality estimation. In Proc. 59th Annu. IEEE Symp. Found. Comput. Sci. 709–718.
[15]
Marco Bressan, Enoch Peserico, and Luca Pretto. 2023. Sublinear algorithms for local graph-centrality estimation. SIAM J. Comput. 52, 4 (2023), 968–1008.
[16]
Marco Bressan and Luca Pretto. 2011. Local computation of pagerank: the ranking side. In Proc. 20th ACM Int. Conf. Inf. Knowl. Manage. 631–640.
[17]
Sergey Brin and Lawrence Page. 1998. The anatomy of a large-scale hypertextual web search engine. Comput. Netw. 30, 1-7 (1998), 107–117.
[18]
Ming Chen, Zhewei Wei, Bolin Ding, Yaliang Li, Ye Yuan, Xiaoyong Du, and Ji-Rong Wen. 2020. Scalable graph neural networks via bidirectional propagation. In Advances Neural Inf. Process. Syst. 33. 14556–14566.
[19]
Yen-Yu Chen, Qingqing Gan, and Torsten Suel. 2004. Local methods for estimating pagerank values. In Proc. 13th ACM Int. Conf. Inf. Knowl. Manage. 381–389.
[20]
Ashish Chiplunkar, Michael Kapralov, Sanjeev Khanna, Aida Mousavifar, and Yuval Peres. 2018. Testing graph clusterability: algorithms and lower bounds. In Proc. 59th Annu. IEEE Symp. Found. Comput. Sci. 497–508.
[21]
Fan Chung. 2007. The heat kernel as the pagerank of a graph. Proc. Nat. Acad. Sci. 104, 50 (2007), 19735–19740.
[22]
Artur Czumaj, Pan Peng, and Christian Sohler. 2015. Testing cluster structure of graphs. In Proc. 47th Annu. ACM Symp. Theory Comput. 723–732.
[23]
Paul Dagum, Richard M. Karp, Michael Luby, and Sheldon M. Ross. 2000. An optimal algorithm for monte carlo estimation. SIAM J. Comput. 29, 5 (2000), 1484–1496.
[24]
Dániel Fogaras, Balázs Rácz, Károly Csalogány, and Tamás Sarlós. 2005. Towards scaling fully personalized pagerank: algorithms, lower bounds, and experiments. Internet Math. 2, 3 (2005), 333–358.
[25]
Kimon Fountoulakis, Farbod Roosta-Khorasani, Julian Shun, Xiang Cheng, and Michael W. Mahoney. 2019. Variational perspective on local graph clustering. Math. Program. 174, 1-2 (2019), 553–573.
[26]
David F. Gleich and Marzia Polito. 2007. Approximating personalized pagerank with minimal use of web graph data. Internet Math. 3, 3 (2007), 257–294.
[27]
Oded Goldreich, Shafi Goldwasser, and Dana Ron. 1998. Property testing and its connection to learning and approximation. J. ACM 45, 4 (1998), 653–750.
[28]
Oded Goldreich and Dana Ron. 2002. Property testing in bounded degree graphs. Algorithmica 32, 2 (2002), 302–343.
[29]
Wentian Guo, Yuchen Li, Mo Sha, and Kian-Lee Tan. 2017. Parallel personalized pagerank on dynamic graphs. Proc. VLDB Endowment 11, 1 (2017), 93–106.
[30]
Taher H. Haveliwala. 2003. Topic-sensitive pagerank: A context-sensitive ranking algorithm for web search. IEEE Trans. Knowl. Data Eng. 15, 4 (2003), 784–796.
[31]
Mark Jerrum, Leslie G. Valiant, and Vijay V. Vazirani. 1986. Random generation of combinatorial structures from a uniform distribution. Theor. Comput. Sci. 43 (1986), 169–188.
[32]
Satyen Kale and C. Seshadhri. 2011. An expansion tester for bounded degree graphs. SIAM J. Comput. 40, 3 (2011), 709–720.
[33]
Johannes Klicpera, Aleksandar Bojchevski, and Stephan Günnemann. 2019. Predict then Propagate: graph neural networks meet personalized pagerank. In Proc. 7th Int. Conf. Learn. Representations. https://openreview.net/forum?id=H1gL-2A9Ym
[34]
Meihao Liao, Rong-Hua Li, Qiangqiang Dai, and Guoren Wang. 2022. Efficient personalized pagerank computation: A spanning forests sampling based approach. In Proc. 2022 ACM SIGMOD Int. Conf. Manage. Data. 2048–2061.
[35]
Peter Lofgren, Siddhartha Banerjee, and Ashish Goel. 2015. Bidirectional pagerank estimation: from average-case to worst-case. In Proc. 12th Int. Workshop Algorithms Models Web Graph, Vol. 9479. 164–176.
[36]
Peter Lofgren, Siddhartha Banerjee, and Ashish Goel. 2016. Personalized pagerank estimation and search: A bidirectional approach. In Proc. 9th ACM Int. Conf. Web Search Data Mining. 163–172.
[37]
Peter Lofgren, Siddhartha Banerjee, Ashish Goel, and Seshadhri Comandur. 2014. FAST-PPR: scaling personalized pagerank estimation for large graphs. In Proc. 20th ACM SIGKDD Int. Conf. Knowl. Discovery Data Mining. 1436–1445.
[38]
Peter Lofgren and Ashish Goel. 2013. Personalized pagerank to a target node. CoRR abs/1304.4658 (2013).
[39]
Dingheng Mo and Siqiang Luo. 2023. Single-source personalized pageranks with workload robustness. IEEE Trans. Knowl. Data Eng. 35, 6 (2023), 6320–6334.
[40]
Daniel A. Spielman and Nikhil Srivastava. 2011. Graph sparsification by effective resistances. SIAM J. Comput. 40, 6 (2011), 1913–1926.
[41]
Daniel A. Spielman and Shang-Hua Teng. 2004. Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems. In Proc. 36th Annu. ACM Symp. Theory Comput. 81–90.
[42]
Hanzhi Wang, Mingguo He, Zhewei Wei, Sibo Wang, Ye Yuan, Xiaoyong Du, and Ji-Rong Wen. 2021. Approximate graph propagation. In Proc. 27th ACM SIGKDD Int. Conf. Knowl. Discovery Data Mining. 1686–1696.
[43]
Hanzhi Wang and Zhewei Wei. 2023. Estimating single-node pagerank in ??? time. Proc. VLDB Endowment 16, 11 (2023), 2949–2961.
[44]
Hanzhi Wang, Zhewei Wei, Junhao Gan, Sibo Wang, and Zengfeng Huang. 2020. Personalized pagerank to a target node, revisited. In Proc. 26th ACM SIGKDD Int. Conf. Knowl. Discovery Data Mining. 657–667.
[45]
Hanzhi Wang, Zhewei Wei, Ji-Rong Wen, and Mingji Yang. 2024. Revisiting local computation of PageRank: simple and optimal. CoRR abs/2403.12648 (2024).
[46]
Sibo Wang, Youze Tang, Xiaokui Xiao, Yin Yang, and Zengxiang Li. 2016. HubPPR: Effective indexing for approximate personalized pagerank. Proc. VLDB Endowment 10, 3 (2016), 205–216.
[47]
Sibo Wang and Yufei Tao. 2018. Efficient algorithms for finding approximate heavy hitters in personalized pageranks. In Proc. 2018 ACM SIGMOD Int. Conf. Manage. Data. 1113–1127.
[48]
Zhewei Wei, Xiaodong He, Xiaokui Xiao, Sibo Wang, Yu Liu, Xiaoyong Du, and Ji-Rong Wen. 2019. PRSim: Sublinear time simrank computation on large power-law graphs. In Proc. 2019 ACM SIGMOD Int. Conf. Manage. Data. 1042–1059.
[49]
Zhewei Wei, Xiaodong He, Xiaokui Xiao, Sibo Wang, Shuo Shang, and Ji-Rong Wen. 2018. TopPPR: Top-k personalized pagerank queries with precision guarantees on large graphs. In Proc. 2018 ACM SIGMOD Int. Conf. Manage. Data. 441–456.
[50]
Yuan Yin and Zhewei Wei. 2019. Scalable graph embeddings via sparse transpose proximities. In Proc. 25th ACM SIGKDD Int. Conf. Knowl. Discovery Data Mining. 1429–1437.
[51]
Hongyang Zhang, Peter Lofgren, and Ashish Goel. 2016. Approximate personalized pagerank on dynamic graphs. In Proc. 22nd ACM SIGKDD Int. Conf. Knowl. Discovery Data Mining. 1315–1324.
[52]
Yanping Zheng, Hanzhi Wang, Zhewei Wei, Jiajun Liu, and Sibo Wang. 2022. Instant graph neural networks for dynamic graphs. In Proc. 28th ACM SIGKDD Int. Conf. Knowl. Discovery Data Mining. 2605–2615.
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- Revisiting Local Computation of PageRank: Simple and Optimal
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June 2024
2049 pages
ISBN:9798400703836
DOI:10.1145/3618260
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Published: 11 June 2024
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- National Natural Science Foundation of China
- Beijing Natural Science Foundation
- Beijing Outstanding Young Scientist Program
- Alibaba Group through Alibaba Innovative Research Program
- Huawei-Renmin University joint program on Information Retrieval
- the fund for building world-class universities (disciplines) of Renmin University of China
- Engineering Research Center of Next-Generation Intelligent Search and Recommendation, Ministry of Education
- Intelligent Social Governance Interdisciplinary Platform, Major Innovation & Planning Interdisciplinary Platform for the ?Double-First Class? Initiative, Public Policy and Decision-making Research Lab
- Public Computing Cloud, Renmin University of China
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