Deterministic coresets for stochastic matrices with applications to scalable sparse pagerank

Harry Lang; Cenk Baykal; Najib Abu Samra; Tony Tannous; Dan Feldman; Daniela Rus

doi:10.1007/978-3-030-14812-6_25

Deterministic coresets for stochastic matrices with applications to scalable sparse pagerank

Harry Lang, Cenk Baykal, Najib Abu Samra, Tony Tannous, Dan Feldman, Daniela Rus

Department of Computer Science

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

The PageRank algorithm is used by search engines to rank websites in their search results. The algorithm outputs a probability distribution that a person randomly clicking on links will arrive at any particular page. Intuitively, a node in the center of the network should be visited with high probability even if it has few edges, and an isolated node that has many (local) neighbours will be visited with low probability. The idea of PageRank is to rank nodes according to a stable state and not according to the previous local measurement of inner/outer edges from a node that may be manipulated more easily than the corresponding entry in the stable state. In this paper we present a deterministic and completely parallelizable algorithm for computing an ε -approximation to the PageRank of a graph of n nodes. Typical inputs consist of millions of pages, but the average number of links per page is less than ten. Our algorithm takes advantage of this sparsity, assuming the out-degree of each node at most s, and terminates in O(ns/ε²) time. Beyond the input graph, which may be stored in read-only storage, our algorithm uses only O(n) memory. This is the first algorithm whose complexity takes advantage of sparsity. Real data exhibits an average out-degree of 7 while n is in the millions, so the advantage is immense. Moreover, our algorithm is simple and robust to floating point precision issues. Our sparse solution (core-set) is based on reducing the PageRank problem to an l₂ approximation of the Carathéodory problem, which independently has many applications such as in machine learning and game theory. We hope that our approach will be useful for many other applications for learning sparse data and graphs. Algorithm, analysis, and open code with experimental results are provided.

Original language	English
Title of host publication	Theory and Applications of Models of Computation - 15th Annual Conference, TAMC 2019, Proceedings
Editors	T. V. Gopal, Junzo Watada
Publisher	Springer Verlag
Pages	410-423
Number of pages	14
ISBN (Print)	9783030148119
DOIs	https://doi.org/10.1007/978-3-030-14812-6_25
State	Published - 2019
Event	15th Annual Conference on Theory and Applications of Models of Computation, TAMC 2019 - Kitakyushu, Japan Duration: 13 Apr 2019 → 16 Apr 2019

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	11436 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	15th Annual Conference on Theory and Applications of Models of Computation, TAMC 2019
Country/Territory	Japan
City	Kitakyushu
Period	13/04/19 → 16/04/19

Bibliographical note

Funding Information:
In this paper we present a deterministic and completely parallelizable algorithm for computing an ε-approximation to the PageRank of a graph of n nodes. Typical inputs consist of millions of pages, but the average number of links per page is less than ten. Our algorithm takes advantage of this sparsity, assuming the out-degree of each node at most s, and terminates in O(ns/ε2) time. Beyond the input graph, which may be stored in read-only storage, our algorithm uses only O(n) memory. This is the first algorithm whose complexity takes advantage of sparsity. Real data exhibits an average out-degree of 7 while n is in the millions, so the advantage is immense. Moreover, our algorithm is simple and robust to floating point precision issues. Our sparse solution (core-set) is based on reducing the PageRank problem to an ℓ2 approximation of the Carathéodory problem, which independently has many applications Lang, Baykal, and Rus thank NSF 1723943, NSF 1526815, and The Boeing Company. This research was supported by Grant No. 2014627 from the United States-Israel Binational Science Foundation (BSF) and by Grant No. 1526815 from the United States National Science Foundation (NSF). Dan Feldman is grateful for the support of the Simons Foundation for part of this work that was done while he was visiting the Simons Institute for the Theory of Computing. H. Lang and C. Baykal—contributed equally to this work.

Funding Information:
Lang, Baykal, and Rus thank NSF 1723943, NSF 1526815, and The Boeing Company. This research was supported by Grant No. 2014627 from the United States-Israel Binational Science Foundation (BSF) and by Grant No. 1526815 from the United States National Science Foundation (NSF). Dan Feldman is grateful for the support of the Simons Foundation for part of this work that was done while he was visiting the Simons Institute for the Theory of Computing

Publisher Copyright:
© Springer Nature Switzerland AG 2019.

ASJC Scopus subject areas

Theoretical Computer Science
General Computer Science

Access to Document

10.1007/978-3-030-14812-6_25

Cite this

Lang, H., Baykal, C., Samra, N. A., Tannous, T., Feldman, D., & Rus, D. (2019). Deterministic coresets for stochastic matrices with applications to scalable sparse pagerank. In T. V. Gopal, & J. Watada (Eds.), Theory and Applications of Models of Computation - 15th Annual Conference, TAMC 2019, Proceedings (pp. 410-423). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 11436 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-030-14812-6_25

Lang, Harry ; Baykal, Cenk ; Samra, Najib Abu et al. / Deterministic coresets for stochastic matrices with applications to scalable sparse pagerank. Theory and Applications of Models of Computation - 15th Annual Conference, TAMC 2019, Proceedings. editor / T. V. Gopal ; Junzo Watada. Springer Verlag, 2019. pp. 410-423 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

@inproceedings{673c5868a8a64400847ebc4f2ebf7c1c,

title = "Deterministic coresets for stochastic matrices with applications to scalable sparse pagerank",

abstract = "The PageRank algorithm is used by search engines to rank websites in their search results. The algorithm outputs a probability distribution that a person randomly clicking on links will arrive at any particular page. Intuitively, a node in the center of the network should be visited with high probability even if it has few edges, and an isolated node that has many (local) neighbours will be visited with low probability. The idea of PageRank is to rank nodes according to a stable state and not according to the previous local measurement of inner/outer edges from a node that may be manipulated more easily than the corresponding entry in the stable state. In this paper we present a deterministic and completely parallelizable algorithm for computing an ε -approximation to the PageRank of a graph of n nodes. Typical inputs consist of millions of pages, but the average number of links per page is less than ten. Our algorithm takes advantage of this sparsity, assuming the out-degree of each node at most s, and terminates in O(ns/ε2) time. Beyond the input graph, which may be stored in read-only storage, our algorithm uses only O(n) memory. This is the first algorithm whose complexity takes advantage of sparsity. Real data exhibits an average out-degree of 7 while n is in the millions, so the advantage is immense. Moreover, our algorithm is simple and robust to floating point precision issues. Our sparse solution (core-set) is based on reducing the PageRank problem to an l2 approximation of the Carath{\'e}odory problem, which independently has many applications such as in machine learning and game theory. We hope that our approach will be useful for many other applications for learning sparse data and graphs. Algorithm, analysis, and open code with experimental results are provided.",

author = "Harry Lang and Cenk Baykal and Samra, {Najib Abu} and Tony Tannous and Dan Feldman and Daniela Rus",

note = "Funding Information: In this paper we present a deterministic and completely parallelizable algorithm for computing an ε-approximation to the PageRank of a graph of n nodes. Typical inputs consist of millions of pages, but the average number of links per page is less than ten. Our algorithm takes advantage of this sparsity, assuming the out-degree of each node at most s, and terminates in O(ns/ε2) time. Beyond the input graph, which may be stored in read-only storage, our algorithm uses only O(n) memory. This is the first algorithm whose complexity takes advantage of sparsity. Real data exhibits an average out-degree of 7 while n is in the millions, so the advantage is immense. Moreover, our algorithm is simple and robust to floating point precision issues. Our sparse solution (core-set) is based on reducing the PageRank problem to an ℓ2 approximation of the Carath{\'e}odory problem, which independently has many applications Lang, Baykal, and Rus thank NSF 1723943, NSF 1526815, and The Boeing Company. This research was supported by Grant No. 2014627 from the United States-Israel Binational Science Foundation (BSF) and by Grant No. 1526815 from the United States National Science Foundation (NSF). Dan Feldman is grateful for the support of the Simons Foundation for part of this work that was done while he was visiting the Simons Institute for the Theory of Computing. H. Lang and C. Baykal—contributed equally to this work. Funding Information: Lang, Baykal, and Rus thank NSF 1723943, NSF 1526815, and The Boeing Company. This research was supported by Grant No. 2014627 from the United States-Israel Binational Science Foundation (BSF) and by Grant No. 1526815 from the United States National Science Foundation (NSF). Dan Feldman is grateful for the support of the Simons Foundation for part of this work that was done while he was visiting the Simons Institute for the Theory of Computing Publisher Copyright: {\textcopyright} Springer Nature Switzerland AG 2019.; 15th Annual Conference on Theory and Applications of Models of Computation, TAMC 2019 ; Conference date: 13-04-2019 Through 16-04-2019",

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doi = "10.1007/978-3-030-14812-6_25",

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Lang, H, Baykal, C, Samra, NA, Tannous, T, Feldman, D & Rus, D 2019, Deterministic coresets for stochastic matrices with applications to scalable sparse pagerank. in TV Gopal & J Watada (eds), Theory and Applications of Models of Computation - 15th Annual Conference, TAMC 2019, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 11436 LNCS, Springer Verlag, pp. 410-423, 15th Annual Conference on Theory and Applications of Models of Computation, TAMC 2019, Kitakyushu, Japan, 13/04/19. https://doi.org/10.1007/978-3-030-14812-6_25

Deterministic coresets for stochastic matrices with applications to scalable sparse pagerank. / Lang, Harry; Baykal, Cenk; Samra, Najib Abu et al.
Theory and Applications of Models of Computation - 15th Annual Conference, TAMC 2019, Proceedings. ed. / T. V. Gopal; Junzo Watada. Springer Verlag, 2019. p. 410-423 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 11436 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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N1 - Funding Information: In this paper we present a deterministic and completely parallelizable algorithm for computing an ε-approximation to the PageRank of a graph of n nodes. Typical inputs consist of millions of pages, but the average number of links per page is less than ten. Our algorithm takes advantage of this sparsity, assuming the out-degree of each node at most s, and terminates in O(ns/ε2) time. Beyond the input graph, which may be stored in read-only storage, our algorithm uses only O(n) memory. This is the first algorithm whose complexity takes advantage of sparsity. Real data exhibits an average out-degree of 7 while n is in the millions, so the advantage is immense. Moreover, our algorithm is simple and robust to floating point precision issues. Our sparse solution (core-set) is based on reducing the PageRank problem to an ℓ2 approximation of the Carathéodory problem, which independently has many applications Lang, Baykal, and Rus thank NSF 1723943, NSF 1526815, and The Boeing Company. This research was supported by Grant No. 2014627 from the United States-Israel Binational Science Foundation (BSF) and by Grant No. 1526815 from the United States National Science Foundation (NSF). Dan Feldman is grateful for the support of the Simons Foundation for part of this work that was done while he was visiting the Simons Institute for the Theory of Computing. H. Lang and C. Baykal—contributed equally to this work. Funding Information: Lang, Baykal, and Rus thank NSF 1723943, NSF 1526815, and The Boeing Company. This research was supported by Grant No. 2014627 from the United States-Israel Binational Science Foundation (BSF) and by Grant No. 1526815 from the United States National Science Foundation (NSF). Dan Feldman is grateful for the support of the Simons Foundation for part of this work that was done while he was visiting the Simons Institute for the Theory of Computing Publisher Copyright: © Springer Nature Switzerland AG 2019.

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N2 - The PageRank algorithm is used by search engines to rank websites in their search results. The algorithm outputs a probability distribution that a person randomly clicking on links will arrive at any particular page. Intuitively, a node in the center of the network should be visited with high probability even if it has few edges, and an isolated node that has many (local) neighbours will be visited with low probability. The idea of PageRank is to rank nodes according to a stable state and not according to the previous local measurement of inner/outer edges from a node that may be manipulated more easily than the corresponding entry in the stable state. In this paper we present a deterministic and completely parallelizable algorithm for computing an ε -approximation to the PageRank of a graph of n nodes. Typical inputs consist of millions of pages, but the average number of links per page is less than ten. Our algorithm takes advantage of this sparsity, assuming the out-degree of each node at most s, and terminates in O(ns/ε2) time. Beyond the input graph, which may be stored in read-only storage, our algorithm uses only O(n) memory. This is the first algorithm whose complexity takes advantage of sparsity. Real data exhibits an average out-degree of 7 while n is in the millions, so the advantage is immense. Moreover, our algorithm is simple and robust to floating point precision issues. Our sparse solution (core-set) is based on reducing the PageRank problem to an l2 approximation of the Carathéodory problem, which independently has many applications such as in machine learning and game theory. We hope that our approach will be useful for many other applications for learning sparse data and graphs. Algorithm, analysis, and open code with experimental results are provided.

AB - The PageRank algorithm is used by search engines to rank websites in their search results. The algorithm outputs a probability distribution that a person randomly clicking on links will arrive at any particular page. Intuitively, a node in the center of the network should be visited with high probability even if it has few edges, and an isolated node that has many (local) neighbours will be visited with low probability. The idea of PageRank is to rank nodes according to a stable state and not according to the previous local measurement of inner/outer edges from a node that may be manipulated more easily than the corresponding entry in the stable state. In this paper we present a deterministic and completely parallelizable algorithm for computing an ε -approximation to the PageRank of a graph of n nodes. Typical inputs consist of millions of pages, but the average number of links per page is less than ten. Our algorithm takes advantage of this sparsity, assuming the out-degree of each node at most s, and terminates in O(ns/ε2) time. Beyond the input graph, which may be stored in read-only storage, our algorithm uses only O(n) memory. This is the first algorithm whose complexity takes advantage of sparsity. Real data exhibits an average out-degree of 7 while n is in the millions, so the advantage is immense. Moreover, our algorithm is simple and robust to floating point precision issues. Our sparse solution (core-set) is based on reducing the PageRank problem to an l2 approximation of the Carathéodory problem, which independently has many applications such as in machine learning and game theory. We hope that our approach will be useful for many other applications for learning sparse data and graphs. Algorithm, analysis, and open code with experimental results are provided.

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Lang H, Baykal C, Samra NA, Tannous T, Feldman D, Rus D. Deterministic coresets for stochastic matrices with applications to scalable sparse pagerank. In Gopal TV, Watada J, editors, Theory and Applications of Models of Computation - 15th Annual Conference, TAMC 2019, Proceedings. Springer Verlag. 2019. p. 410-423. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-030-14812-6_25

Deterministic coresets for stochastic matrices with applications to scalable sparse pagerank

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