This paper proposes nonlinear operator of extreme doubly stochastic quadratic operator (EDSQO) for convergence algorithm aimed at solving consensus problem (CP) of discrete-time for multi-agent systems (MAS) on n-dimensional simplex. The first part undertakes systematic review of consensus problems. Convergence was generated via extreme doubly stochastic quadratic operators (EDSQOs) in the other part. However, this work was able to formulate convergence algorithms from doubly stochastic matrices, majorization theory, graph theory and stochastic analysis. We develop two algorithms: 1) the nonlinear algorithm of extreme doubly stochastic quadratic operator (NLAEDSQO) to generate all the convergent EDSQOs and 2) the nonlinear convergence algorithm (NLCA) of EDSQOs to investigate the optimal consensus for MAS. Experimental evaluation on convergent of EDSQOs yielded an optimal consensus for MAS. Comparative analysis with the convergence of EDSQOs and DeGroot model were carried out. The comparison was based on the complexity of operators, number of iterations to converge and the time required for convergences. This research proposed algorithm on convergence which is faster than the DeGroot linear model.
 W. Ren, R. W. Beard, and E. M. Atkins, A survey of consensus problems in multi-agent coordination, in American Control Conference, 2005. Proceedings of the 2005, pp. 1859–1864, IEEE, 2005.
 Z. Lin, B. Francis, and M. Maggiore, State agreement for continuous-time coupled nonlinear systems, SIAM Journal on Control and Optimization, vol. 46, no. 1, pp. 288–307, 2007.
 E. Lovisari and S. Zampieri, Performance metrics in the average consensus problem: a tutorial, Annual Reviews in Control, vol. 36, no. 1, pp. 26–41, 2012.
 E. Eisenberg and D. Gale, Consensus of subjective probabilities: The pari-mutuel method, The Annals of Mathematical Statistics, pp. 165–168, 1959.
 M. H. DeGroot, Reaching a consensus, Journal of the American Statistical Association, vol. 69, no. 345, pp. 118–121, 1974.
 R. L. Berger, A necessary and sufficient condition for reaching a consensus using degroot’s method, Journal of the American Statistical Association, vol. 76, no. 374, pp. 415–418, 1981.
 I. Matei, J. S. Baras, and C. Somarakis, Convergence results for the linear consensus problem under markovian random graphs, SIAM Journal on Control and Optimization, vol. 51, no. 2, pp. 1574–1591, 2013.
 H. J. LeBlanc, H. Zhang, S. Sundaram, and X. Koutsoukos, Consensus of multi-agent networks in the presence of adversaries using only local information, in Proceedings of the 1st international conference on High Confidence Networked Systems, pp. 1–10, ACM, 2012.
 P. Lin and W. Ren, Constrained consensus in unbalanced networks with communication delays, Automatic Control, IEEE Transactions on, vol. 59, no. 3, pp. 775–781, 2014.
 S. Bernstein, Solution of a mathematical problem connected with the theory of heredity, The Annals of Mathematical Statistics, vol. 13, no. 1, pp. 53–61, 1942.
 S. Vallander, On the limit behavior of iteration sequence of certain quadratic transformations, in Soviet Math. Doklady, vol. 13, pp. 123–126, 1972.
 I. Olkin and A. W. Marshall, Inequalities: Theory of majorization and its applications, Academic, New York, 1979.
 R. N. Ganikhodzhaev, On the definition of bistochastic quadratic operators, Russian Mathematical Surveys, vol. 48, no. 4, pp. 244–246, 1993.
 T. Ando, Majorization, doubly stochastic matrices, and comparison of eigenvalues, Linear Algebra and Its Applications, vol. 118, pp. 163–248, 1989.
 Y. I. Lyubich, D. Vulis, A. Karpov, and E. Akin, Mathematical structures in population genetics, Biomathematics(Berlin), 1992.
 R. Ganikhodzhaev, Quadratic stochastic operators, lyapunov functions, and tournaments, Russian Academy of Sciences. Sbornik Mathematics, vol. 76, no. 2, p. 489, 1993.
 R. Ganikhodzhaev and U. Rozikov, Quadratic stochastic operators: Results and open problems, arXiv preprint arXiv:0902.4207, 2009.
 R. Ganikhodzhaev and F. Shahidi, Doubly stochastic quadratic operators and birkhoffs problem, Linear Algebra and its Applications, vol. 432, no. 1, pp. 24–35, 2010.
 R. Abdulghafor, S. Turaev, A. Abubakar, and A. Zeki, The extreme doubly stochastic quadratic operators on two dimensional simplex, in Advanced Computer Science Applications and Technologies (ACSAT), 2015 4th International Conference on, pp. 192–197, IEEE, 2015.
 F. Shahidi, On dissipative quadratic stochastic operators, Applied Mathematics and Information Sciences, vol. 2, pp. 211–223, 2008.
 F. Shahidi, On the extreme points of the set of bistochastic operators, Mathematical Notes, vol. 84, no. 3, pp. 442–448, 2008.
 R. Abdulghafor, F. Shahidi, A. Zeki, and S. Turaev, Dynamics classifications of extreme doubly stochastic quadratic operators on 2d simplex, in Advanced Computer and Communication Engineering Technology, pp. 323–335, Springer, 2016.
 F. A. Shahidi, Doubly stochastic operators on a finite-dimensional simplex, Siberian Mathematical Journal, vol. 50, no. 2, pp. 368–372, 2009.
 F. Shahidi and M. Abu Osman, The limit behaviour of trajectories of dissipative quadratic stochastic operators on finite-dimensional simplex, Journal of Difference Equations and Applications, vol. 19, no. 3, pp. 357–371, 2013.
 F. Shahidi, On infinite-dimensional dissipative quadratic stochastic operators, Advances in Difference Equations, vol. 2013, no. 1, pp. 1–13, 2013.
 R. Abdulghafor, F. Shahidi, A. Zeki, and S. Turaev, Dynamics of doubly stochastic quadratic operators on a finite-dimensional simplex, Open Mathematics, vol. 14, no. 1, pp. 509–519, 2016.
 F. Shahidi, R. Ganikhodzhaev, and R. Abdulghafor, The dynamics of some extreme doubly stochastic quadratic operators, Middle-East Journal of Scientific Research (Mathematical Applications in Engineering), vol. 13, pp. 59–63, 2013.
 R. Abdulghafor, S. Turaev, M. Tamrin, and M. Izzuddin, Nonlinear consensus for multi-agent systems using positive intractions of doubly stochastic quadratic operators, International Journal on Perceptive and Cognitive Computing (IJPCC), vol. 2, no. 1, pp. 19–22, 2016.
 R. Abdulghafor, S. Turaev, A. Zeki, and F. Shahidi, The convergence consensus of multi-agent systems controlled via doubly stochastic quadratic operators, in Agents, Multi-Agent Systems and Robotics (ISAMSR), 2015 International Symposium on, pp. 59–64, IEEE, 2015.
 L. Panait and S. Luke, Cooperative multi-agent learning: The state of the art, Autonomous Agents and Multi-Agent Systems, vol. 11, no. 3, pp. 387–434, 2005.
 A. Ajorlou, A. Momeni, and A. G. Aghdam, Sufficient conditions for the convergence of a class of nonlinear distributed consensus algorithms, Automatica, vol. 47, no. 3, pp. 625–629, 2011.
 A. N. Bishop and A. Doucet, Distributed nonlinear consensus in the space of probability measures, arXiv preprint arXiv:1404.0145, 2014.
 T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, and O. Shochet, Novel type of phase transition in a system of self-driven particles, Physical review letters, vol. 75, no. 6, p. 1226, 1995.
 J. N. Tsitsiklis, D. P. Bertsekas, M. Athans, et al., Distributed asynchronous deterministic and stochastic gradient optimization algorithms, IEEE transactions on automatic control, vol. 31, no. 9, pp. 803–812, 1986.
 A. Jadbabaie, J. Lin, and A. S. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, Automatic Control, IEEE Transactions on, vol. 48, no. 6, pp. 988–1001, 2003.
 F. Cucker, S. Smale, and D.-X. Zhou, Modeling language evolution, Foundations of Computational Mathematics, vol. 4, no. 3, pp. 315–343, 2004.
 N. A. Lynch, Distributed algorithms. Morgan Kaufmann, 1996.
 S. M. Ulam, A collection of mathematical problems, New York, vol. 29, 1960.
 Z.-H. Guan, Y. Wu, and G. Feng, Consensus analysis based on impulsive systems in multiagent networks, Circuits and Systems I: Regular Papers, IEEE Transactions on, vol. 59, no. 1, pp. 170–178, 2012.
 G. Cui, S. Xu, F. L. Lewis, B. Zhang, and Q. Ma, Distributed consensus tracking for non-linear multiagent systems with input saturation: a command filtered backstepping approach, IET Control Theory & Applications, vol. 10, no. 5, pp. 509–516, 2016.
 L. Yu-Mei and G. Xin-Ping, Nonlinear consensus protocols for multi-agent systems based on centre manifold reduction, Chinese Physics B, vol. 18, no. 8, p. 3355, 2009.
 F. Shahidi, On dissipative quadratic stochastic operators, arXiv preprint arXiv:0708.1813, 2007.