Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences


The class of Sarymsakov square stochastic matrices is the largest subset of the set of stochastic, indecomposable, aperiodic (SIA) matrices that is closed under matrix multiplication and any infinitely long left-product of the elements from any of its compact subsets converges to a rank-one (stable) matrix. In this paper, we introduce a new class of the so-called Sarymsakov cubic stochastic matrices and study the consensus problem in the multi-agent system in which an opinion sharing dynamics is presented by quadratic stochastic operators associated with Sarymsakov cubic stochastic matrices.

First Page


Last Page



1. Berger R.L. A necessary and sufficient condition for reaching a consensus using DeGroot's method. J. Amer. Stat. Assoc., Vol. 76, Issue 374, 415–418 (1981).

2. Bernstein S. Solution of a mathematical problem connected with the theory of heredity. Annals of Mathematical Statistics, Vol. 13, No. 1, 53–61 (1942).

3. Chatterjee S., Seneta E. Towards consensus: some convergence theorems on repeated averaging. J. Appl. Prob., Vol. 14, Issue 1, 89–97 (1977).

4. De Groot M.H. Reaching a consensus. J. Amer. Stat. Assoc., Vol 69, Issue 345, 118–121 (1974).

5. Ganihodzhaev N.N. On stochastic processes generated by quadratic operators. Journal of Theoretical Probability, Vol. 4, Issue 4, 639–653 (1991).

6. Ganikhodzhaev R., Mukhamedov F., Rozikov U. Quadratic stochastic operators and processes: Results and Open Problems. Inf. Dim. Anal. Quan. Prob. Rel. Top., Vol. 14, Issue 2, 279–335 (2011).

7. Hartfiel D.J., Seneta E. A note on semigroups of regular stochastic matrices. Linear Algebra and its Applications, Vol. 141, 47–51 (1990).

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

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

10. Kesten H. Quadratic transformations: A model for population growth. I. Advances in Appl. Probability, Vol. 2, 1–82 (1970).

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

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

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

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

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

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

17. Lyubich Y. Mathematical Structures in Population Genetics. Springer, (1992).

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

19. Saburov M. Quadratic stochastic Sarymsakov operators. Journal of Physics: Conference Series, Vol. 697, 012015 (2016).

20. Saburov M. Ergodicity of p-majorizing quadratic stochastic operators. Markov Processes Relat. Fields, Vol. 24, Issue 1, 131–150 (2018).

21. Saburov M. Ergodicity of p-majorizing nonlinear Markov operators on the finite dimensional space. Linear Algebra and its Applications, Vol. 578, 53–74 (2019).

22. Saburov M., Saburov Kh. Reaching a consensus in multi-agent systems: A time invariant nonlinear rule. Journal of Education and Vocational Research, Vol. 4, Issue 5, 130–133 (2013).

23. Saburov M., Saburov Kh. Mathematical models of nonlinear uniform consensus. ScienceAsia, Vol. 40, Issue 4, 306–312 (2014).

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

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

26. Saburov M., Saburov Kh. Reaching consensus via polynomial stochastic operators: A general study. Springer Proceedings in Mathematics and Statistics, Vol. 212, 219–230 (2017).

27. Saburov M., Saburov Kh. Mathematical models of nonlinear uniformly consensus II. Journal of Applied Nonlinear Dynamics, Vol. 7, Issue 1, 95–104 (2018).

28. Sarymsakov T. Inhomogeneous Markov chains. Theory of Prob and Appl, Vol. 6, Issue 2, 178–185 (1961).

29. Sarymsakov T., Ganikhodjaev N. Analytic methods in the theory of quadratic stochastic processes. J Theoretical Prob, Vol. 3, 51–70 (1990).

30. Seneta E. Coefficients of ergodicity: Structure and applications. Adv. in Prob, Vol. 11, 576–590 (1979).

31. Seneta E. Nonnegative Matrices and Markov Chains. Springer-Verlag, New York, (1981).

32. Ulam S. A collection of mathematical problems. Interscience Publishers, New-York, London, (1960).

33. Wolfowitz J. Products of indecomposable, aperiodic, stochastic matrices. Proc. of the Amer. Math. Soc., Vol. 14, Issue 5, 733–737 (1963).

34. Xia W. Distributed Algorithms for Interacting Autonomous Agents. PhD Thesis, (2013).

35. Xia W., Cao M. Sarymsakov matrices and their application in coordinating multi-agent systems. Proceedings of the 31st Chinese Control Conference, Hefei, 6321–6326 (2012).

36. Xia W., Cao M. Sarymsakov matrices and asynchronous implementation of distributed coordination algorithms. IEEE Transcations on Automatic Control, Vol. 59, Issue 8, 2228–2233 (2014).

37. Xia W., Liu J., Cao M., Johansson K.H., Basar T. Products of generalized stochastic Sarymsakov matrices, in Proc. of the 54th IEEE Conference on Decision and Control, 3621–3626 (2015).

38. Xia W., Liu J., Cao M., Johansson K.H., Basar T. Generalized Sarymsakov Matrices. IEEE Transactions on Automatic Control, Vol. 64 , Issue 8, 3085–3100 (2019).



To view the content in your browser, please download Adobe Reader or, alternately,
you may Download the file to your hard drive.

NOTE: The latest versions of Adobe Reader do not support viewing PDF files within Firefox on Mac OS and if you are using a modern (Intel) Mac, there is no official plugin for viewing PDF files within the browser window.