Encyclopedia of Systems and Control

Living Edition
| Editors: John Baillieul, Tariq Samad

Markov Chains and Ranking Problems in Web Search

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DOI: https://doi.org/10.1007/978-1-4471-5102-9_135-2


Markov chains refer to stochastic processes whose states change according to transition probabilities determined only by the states of the previous time step. They have been crucial for modeling large-scale systems with random behavior in various fields such as control, communications, biology, optimization, and economics. In this entry, we focus on their recent application to the area of search engines, namely, the PageRank algorithm employed at Google, which provides a measure of importance for each page in the web. We present several researches carried out with control theoretic tools such as aggregation, distributed randomized algorithms, and PageRank optimization. Due to the large size of the web, computational issues are the underlying motivation of these studies.


Aggregation Distributed randomized algorithms Markov chains Optimization PageRank Search engines World Wide Web 
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  1. Aldhaheri R, Khalil H (1991) Aggregation of the policy iteration method for nearly completely decomposable Markov chains. IEEE Trans Autom Control 36: 178–187MathSciNetCrossRefGoogle Scholar
  2. Bertsekas D, Tsitsiklis J (1989) Parallel and distributed computation: numerical methods. Prentice-Hall, Englewood CliffszbMATHGoogle Scholar
  3. Brin S, Page L (1998) The anatomy of a large-scale hypertextual web search engine. Comput Netw ISDN Syst 30:107–117CrossRefGoogle Scholar
  4. Csáji B, Jungers R, Blondel V (2014) PageRank optimization by edge selection. Discrete Appl Math 169:73–87MathSciNetCrossRefGoogle Scholar
  5. de Kerchove C, Ninove L, Van Dooren P (2008) Influence of the outlinks of a page on its PageRank. Linear Algebra Appl 429:1254–1276MathSciNetCrossRefGoogle Scholar
  6. Fercoq O, Akian M, Bouhtou M, Gaubert S (2013) Ergodic control and polyhedral approaches to PageRank optimization. IEEE Trans Autom Control 58: 134–148MathSciNetCrossRefGoogle Scholar
  7. Horn R, Johnson C (1985) Matrix analysis. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  8. Ishii H, Suzuki A (2018) Distributed randomized algorithms for PageRank computation: recent advances. In: Basar T (ed) Uncertainty in complex networked systems. Birkhäuser, Basel, pp 419–447CrossRefGoogle Scholar
  9. Ishii H, Tempo R (2010) Distributed randomized algorithms for the PageRank computation. IEEE Trans Autom Control 55:1987–2002MathSciNetCrossRefGoogle Scholar
  10. Ishii H, Tempo R, Bai EW (2012) A web aggregation approach for distributed randomized PageRank algorithms. IEEE Trans Autom Control 57:2703–2717MathSciNetCrossRefGoogle Scholar
  11. Ishii H, Tempo R (2014) The PageRank problem, multi-agent consensus and web aggregation: a systems and control viewpoint. IEEE Control Syst Mag 34(3):34–53CrossRefGoogle Scholar
  12. Kumar P, Varaiya P (1986) Stochastic systems: estimation, identification, and adaptive control. Prentice Hall, Englewood CliffszbMATHGoogle Scholar
  13. Lagoa C, Zaccarian L, Dabbene F (2017) A distributed algorithm with consistency for PageRank-like linear algebraic systems. In: Proceedings of 20th IFAC World Congress, pp 5339–5344Google Scholar
  14. Langville A, Meyer C (2006) Google’s PageRank and beyond: the science of search engine rankings. Princeton University Press, PrincetonCrossRefGoogle Scholar
  15. Meyer C (1989) Stochastic complementation, uncoupling Markov chains, and the theory of nearly reducible systems. SIAM Rev 31:240–272MathSciNetCrossRefGoogle Scholar
  16. Norris J (1997) Markov chains. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  17. Phillips R, Kokotovic P (1981) A singular perturbation approach to modeling and control of Markov chains. IEEE Trans Autom Control 26:1087–1094CrossRefGoogle Scholar
  18. Puterman M (1994) Markov decision processes: discrete stochastic dynamic programming. Wiley, New YorkCrossRefGoogle Scholar
  19. You K, Tempo R, Qiu L (2017) Distributed algorithms for computation of centrality measures in complex networks. IEEE Trans Autom Control 62:2080–2094MathSciNetCrossRefGoogle Scholar
  20. Zhao W, Chen H, Fang H (2013) Convergence of distributed randomized PageRank algorithms. IEEE Trans Autom Control 58:3255–3259CrossRefGoogle Scholar

Authors and Affiliations

  1. 1.Department of Computer ScienceTokyo Institute of TechnologyYokohamaJapan
  2. 2.CNR-IEIIT, Politecnico di TorinoTorinoItaly

Section editors and affiliations

  • Fabrizio Dabbene
    • 1
  1. 1.Systems Modeling & Control GroupCNR-IEIITTorinoItaly