Skip to main content

Reaching Consensus via Polynomial Stochastic Operators: A General Study

Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 212)

Abstract

In this paper, we consider a nonlinear protocol for a structured time-varying synchronous multi-agent system in which an opinion sharing dynamics is presented by non-autonomous polynomial stochastic operators associated with high-order stochastic hyper-matrices. We show that the proposed nonlinear protocol generates the Krause mean process. We provide a criterion to establish a consensus in the multi-agent system under the proposed nonlinear protocol.

Keywords

  • Krause mean process
  • Markov chain with memory
  • Stochastic hyper-matrices
  • Polynomial stochastic operators
  • Consensus

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-981-10-6409-8_14
  • Chapter length: 12 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   129.00
Price excludes VAT (USA)
  • ISBN: 978-981-10-6409-8
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   169.99
Price excludes VAT (USA)
Hardcover Book
USD   169.99
Price excludes VAT (USA)

References

  1. Berchtold, A., Raftery, A.: The mixture transition distribution model for high-order Markov chains and non-Gaussian time series. Stat. Sci. 7, 328–356 (2002)

    MATH  MathSciNet  Google Scholar 

  2. Berger, R.L.: A necessary and sufficient condition for reaching a consensus using DeGroot’s method. J. Amer. Stat. Assoc. 76, 415–418 (1981)

    CrossRef  MATH  MathSciNet  Google Scholar 

  3. Chatterjee, S., Seneta, E.: Towards consensus: some convergence theorems on repeated averaging. J. Appl. Prob. 14, 89–97 (1977)

    CrossRef  MATH  MathSciNet  Google Scholar 

  4. De Groot, M.H.: Reaching a consensus. J. Amer. Stat. Assoc. 69, 118–121 (1974)

    CrossRef  MATH  Google Scholar 

  5. Hegselmann, R., Krause, U.: Opinion dynamics and bounded confidence: models, analysis and simulation. J. Art. Soc. Soc. Sim. 5(3), 1–33 (2002)

    Google Scholar 

  6. Hegselmann, R., Krause, U.: Opinion dynamics driven by various ways of averaging. Comp. Econ. 25, 381–405 (2005)

    CrossRef  MATH  Google Scholar 

  7. Krause, U.: A discrete nonlinear and non-autonomous model of consensus formation. In: Elaydi, S., et al. (eds.) Communications in Difference Equations, pp. 227–236. Gordon and Breach, Amsterdam (2000)

    CrossRef  Google Scholar 

  8. Krause, U.: Compromise, consensus, and the iteration of means. Elem. Math. 64, 1–8 (2009)

    CrossRef  MATH  MathSciNet  Google Scholar 

  9. Krause, U.: Markov chains, Gauss soups, and compromise dynamics. J. Cont. Math. Anal. 44(2), 111–116 (2009)

    CrossRef  MATH  MathSciNet  Google Scholar 

  10. Krause, U.: Opinion dynamics – local and global. In: Liz, E., Manosa, V. (eds.) Proceedings of the Workshop Future Directions in Difference Equations, pp. 113–119. Universidade de Vigo, Vigo (2011)

    Google Scholar 

  11. Krause, U.: Positive Dynamical Systems in Discrete Time: Theory, Models, and Applications. Walter de Gruyter (2015)

    Google Scholar 

  12. Kolokoltsov, V.: Nonlinear Markov Processes and Kinetic Equations. Cambridge University Press (2010)

    Google Scholar 

  13. Lyubich, Y.I.: Mathematical Structures in Population Genetics. Springer (1992)

    Google Scholar 

  14. Lu, J., Yu, X., Chen, G., Yu, W.: Complex Systems and Networks: Dynamics. Springer, Controls and Applications (2016)

    CrossRef  MATH  Google Scholar 

  15. Raftery, A.: A model of high-order Markov chains. J. Roy. Stat. Soc. 47, 528–539 (1985)

    MATH  MathSciNet  Google Scholar 

  16. Saburov, M.: Ergodicity of nonlinear Markov operators on the finite dimensional space. Non. Anal. Theo. Met. Appl. 143, 105–119 (2016)

    CrossRef  MATH  MathSciNet  Google Scholar 

  17. Saburov, M., Saburov, Kh: Reaching a consensus in multi-agent systems: A time invariant nonlinear rule. J. Educ. Vocat. Res. 4(5), 130–133 (2013)

    MATH  Google Scholar 

  18. Saburov, M., Saburov, Kh: Mathematical models of nonlinear uniform consensus. Sci. Asia 40(4), 306–312 (2014)

    CrossRef  MATH  Google Scholar 

  19. Saburov, M., Saburov, Kh: Reaching a nonlinear consensus: polynomial stochastic operators. Inter. J. Cont. Auto. Sys. 12(6), 1276–1282 (2014)

    CrossRef  MATH  MathSciNet  Google Scholar 

  20. Saburov, M., Saburov, Kh: Reaching a nonlinear consensus: a discrete nonlinear time-varying case. Inter. J. Sys. Sci. 47(10), 2449–2457 (2016)

    CrossRef  MATH  MathSciNet  Google Scholar 

  21. Saburov, M., Yusof, N.A.: Counterexamples to the conjecture on stationary probability vectors of the second-order Markov chains. Linear Algebra Appl. 507, 153–157 (2016)

    CrossRef  MATH  MathSciNet  Google Scholar 

  22. Seneta, E.: Nonnegative Matrices and Markov Chains. Springer-Verlag (1981)

    Google Scholar 

Download references

Acknowledgements

This work has been done under the MOHE grant FRGS14-141-0382. The first Author (M.S.) is grateful to the Junior Associate scheme of the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. He is also indebted to Professor Hideaki Matsunaga for his kind hospitality during the conference ICDEA2016, Osaka, Japan.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Mansoor Saburov .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2017 Springer Nature Singapore Pte Ltd.

About this paper

Verify currency and authenticity via CrossMark

Cite this paper

Saburov, M., Saburov, K. (2017). Reaching Consensus via Polynomial Stochastic Operators: A General Study. In: Elaydi, S., Hamaya, Y., Matsunaga, H., Pötzsche, C. (eds) Advances in Difference Equations and Discrete Dynamical Systems. ICDEA 2016. Springer Proceedings in Mathematics & Statistics, vol 212. Springer, Singapore. https://doi.org/10.1007/978-981-10-6409-8_14

Download citation