Contemporary Mathematics. Fundamental DirectionsContemporary Mathematics. Fundamental DirectionsСовременная математика. Фундаментальные направления2413-36392949-0618Peoples’ Friendship University of Russia named after Patrice Lumumba (RUDN University)3085510.22363/2413-3639-2022-68-1-110-126ArticlesСтатьиResearch ArticleApplications of Quadratic Stochastic Operators to Nonlinear Consensus ProblemsПриложения квадратичных стохастических операторов к нелинейным проблемам консенсусаSaburovM.СабуровМ.msaburov@gmail.comSaburovKh.СабуровХ.khikmatdr@gmail.comAmerican University of the Middle EastАмериканский университет Ближнего востокаNational University of UzbekistanНациональный университет Узбекистана им. М. Улугбека20042022681Science — Technology — Education — Mathematics — MedicineНаука — технология — образование — математика — медицина11012620042022Copyright ©; 2022, Contemporary Mathematics. Fundamental DirectionsCopyright ©; 2022, Современная математика. Фундаментальные направления2022Contemporary Mathematics. Fundamental DirectionsСовременная математика. Фундаментальные направленияhttps://creativecommons.org/licenses/by-nc-nd/4.0/deed.enhttps://journals.rudn.ru/CMFD/article/view/30855

Historically, the idea of reaching consensus through repeated averaging was introduced by De Groot for a structured time-invariant and synchronous environment. Since that time, the consensus, which is the most ubiquitous phenomenon of multi-agent systems, becomes popular in the various scientific fields such as biology, physics, control engineering and social science. In this paper, we give an overview of the recent development of applications of quadratic stochastic operators to nonlinear consensus problems. We also present some refinement and improvement of the previous results.

Исторически идея достижения консенсуса путем повторных усреднений была предложена Де Грутом для структурированной синхронной среды, инвариантной по времени. С того времени консенсус, будучи наиболее общим феноменом многоагентных систем, становится популярным в разнообразных научных областях, таких как биология, физика, инженерия управления и социальные науки. В данной работе мы даем обзор недавнего развития приложения квадратичных стохастических операторов к нелинейным задачам консенсуса. Мы также даем некоторые уточнения и улучшения предыдущих результатов.

Жамилов У. У., Розиков У. А. О динамике строго невольтерровских квадратичных стохастических операторов на двумерном симплексе// Мат. сб. - 2009. -200, № 9. - С. 81-94.Розиков У. А., Зада А. Об l-вольтерровских квадратичных стохастических операторах// Докл. РАН. - 2009. -424, № 2. - С. 168-170.Berger R. L. A necessary and sufficient condition for reaching a consensus using DeGroot’s method// J. Am. Statist. Assoc. - 1981. -76. - С. 415-418.Bernstein S. Solution of a mathematical problem connected with the theory of heredity// Ann. Math. Stat. - 1942. -13. - С. 53-61.Chatterjee S., Seneta E. Towards consensus: some convergence theorems on repeated averaging// J. Appl. Probab. - 1977. -14. - С. 89-97.De Groot M. H. Reaching a consensus// J. Am. Statist. Assoc. - 1974. -69. - С. 118-121.Ganihodzhaev N. On stochastic processes generated by quadratic operators// J. Theoret. Probab. - 1991. - 4. - С. 639-653.Ganikhodjaev N., Akin H., Mukhamedov F. On the ergodic principle for Markov and quadratic stochastic processes and its relations// Linear Algebra App. - 2006. -416. - С. 730-741.Ganikhodzhaev R., Mukhamedov F., Rozikov U. Quadratic stochastic operators and processes: results and open problems// Inf. Dim. Anal. Quan. Prob. Rel. Top. - 2011. -14, № 2. - С. 279-335.Hegselmann R., Krause U. Opinion dynamics and bounded confidence: models, analysis and simulation// J. Art. Soc. Social Sim. - 2002. -5, № 3. - С. 1-33.Hegselmann R., Krause U. Opinion dynamics driven by various ways of averaging// Comput. Econ. - 2005. -25. - С. 381-405.Jadbabaie A., Lin J., Morse A. S. Coordination of groups of mobile autonomous agents using nearest neighbor rules// IEEE Trans. Automat. Control. - 2003. -48, № 6. - С. 985-1001.Jamilov U., Ladra M. Non-ergodicity of uniform quadratic stochastic operators// Qual. Theory Dyn. Sys. - 2016. -15, № 1. - С. 257-271.Jamilov U., Ladra M., Mukhitdinov R. On the equiprobable strictly non-Volterra quadratic stochastic operators// Qual. Theory Dyn. Sys. - 2017. - 16, № 3. - С. 645-655.Kesten H. Quadratic transformations: a model for population growth I// Adv. Appl. Probab. - 1970. -2. - С. 1-82.Kolokoltsov V. Nonlinear Markov processes and kinetic equations. - Cambridge: Cambridge University Press, 2010.Krause U. A discrete nonlinear and non-autonomous model of consensus formation// В сб.: «Communications in difference equations». - Amsterdam: Gordon and Breach, 2000. - С. 227-236.Krause U. Compromise, consensus, and the iteration of means// Elem. Math. - 2009. -64. - С. 1-8.Krause U. Markov chains, Gauss soups, and compromise dynamics// J. Cont. Math. Anal. - 2009. -44, № 2. - С. 111-116.Krause U. Opinion dynamics - local and global// В сб.: «Proceedings of the Workshop “Future Directions in Difference Equations”». - Vigo: Universidade de Vigo, 2011. - С. 113-119.Krause U. Positive dynamical systems in discrete time: theory, models, and applications. - Berlin: De Gruyter, 2015.Lyubich Y. I. Mathematical structures in population genetics. - Berlin etc.: Springer, 1992.Moreau L. Stability of multiagent systems with time-dependent communication links// IEEE Trans. Automat. Control. - 2005. -50, № 2. - С. 169-182.Mukhamedov F., Ganikhodjaev N. Quantum quadratic operators and processes. - Cham: Springer, 2015.Pulka M. On the mixing property and the ergodic principle for non-homogeneous Markov chains// Linear Algebra App. - 2011. -434. - С. 1475-1488.Rozikov U. Population dynamics: algebraic and probabilistic approach. - World Scientific, 2020.Rozikov U., Jamilov U. F-Quadratic stochastic operators// Math. Notes. - 2008. -83, № 4. - С. 606-612.Saburov M. Ergodicity of nonlinear Markov operators on the finite dimensional space// Nonlinear Anal. - 2016. -143. - С. 105-119.Saburov M. Quadratic stochastic Sarymsakov operators// J. Phys. Conf. Ser. - 2016. -697. - 012015.Saburov M. On regularity of diagonally positive quadratic doubly stochastic operators// Results Math. - 2017. -72. - С. 1907-1918.Saburov M. On regularity of positive quadratic doubly stochastic operators// Math Notes. - 2018. -103, № 2. - С. 328-333.Saburov M. Ergodicity of p-majorizing quadratic stochastic operators// Markov Process. Related Fields. - 2018. -24, № 1. - С. 131-150.Saburov M. Ergodicity of p-majorizing nonlinear Markov operators on the finite dimensional space// Linear Algebra Appl. - 2019. -578. - С. 53-74.Saburov M., Saburov Kh. Reaching a consensus in multi-agent systems: a time invariant nonlinear rule// J. Educ. Vocational Research. - 2013. - 4, № 5. - С. 130-133.Saburov M., Saburov Kh. Mathematical models of nonlinear uniform consensus// Sci. Asia. - 2014. -40, № 4. - С. 306-312.Saburov M., Saburov Kh. Reaching a nonlinear consensus: polynomial stochastic operators// Inter. J. Cont. Auto. Sys. - 2014. -12, № 6. - С. 1276-1282.Saburov M., Saburov Kh. Reaching a nonlinear consensus: a discrete nonlinear time-varying case// Inter. J. Sys. Sci. - 2016. -47, № 10. - С. 2449-2457.Saburov M., Saburov Kh. Reaching consensus via polynomial stochastic operators: a general study// Springer Proc. Math. Statist. - 2017. -212. - С. 219-230.Saburov M., Saburov Kh. Mathematical models of nonlinear uniformly consensus II// J. Appl. Nonlinear Dynamics. - 2018. -7, № 1. - С. 95-104.Saburov M., Yusof N. A. Counterexamples to the conjecture on stationary probability vectors of the secondorder Markov chains// Linear Algebra Appl. - 2016. -507. - С. 153-157.Saburov M., Yusof N. The structure of the fixed point set of quadratic operators on the simplex// Fixed Point Theory. - 2018. -19, № 1. - С. 383-396.Saburov M., Yusof N. On uniqueness of fixed points of quadratic stochastic operators on a 2D simplex// Methods Funct. Anal. Topol. - 2018. - 24, № 3. - С. 255-264.Sarymsakov T., Ganikhodjaev N. Analytic methods in the theory of quadratic stochastic processes// J. Theor. Probab. - 1990. -3. - С. 51-70.Seneta E. Nonnegative matrices and Markov chains. - New York-Heidelberg-Berlin: Springer, 1981.Touri B., Nedic´ A. Product of random stochastic matrices// IEEE Trans. Automat. Control. - 2014. - 59, № 2. - С. 437-448.Tsitsiklis J., Bertsekas D., Athans M. Distributed asynchronous deterministic and stochastic gradient optimization algorithms// IEEE Trans. Automat. Control. - 1986. -31, № 9. - С. 803-812.Ulam S. A collection of mathematical problems. - New York-London: Interscience Publishers, 1960.