We consider a nonlinear protocol for a structured time-invariant and synchronous multi-agent system. In the multi-agent system, we present opinion sharing dynamics as a trajectory of a cubic triple stochastic matrix. We provide a criterion for a uniform consensus of the multi-agent system. We show that the multi-agent system eventually reaches a consensus if either one of the following two conditions is satisfied: (i) every member of the group people has a positive subjective opinion on the given task after some revision steps or (ii) all entries of the given cubic triple stochastic matrix are positive.

We provide a general nonlinear protocol for a structured time-varying and synchronous multi-agent system. We present an opinion sharing dynamics of the multi-agent system as a trajectory of a polynomial stochastic operator associated with a multidimensional stochastic hypermatrix. We show that the multi-agent system eventually reaches to a consensus if either one of the following two conditions is satisfied: (i) every member of the group people has a positive subjective opinion on the given task after some revision steps or (ii) all entries of a multidimensional stochastic hypermatrix are positive. Numerical results are also presented.

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Abstract. This paper is a continuation of our previous studies on nonlinear consensus which unifies and generalizes all previous results. We consider a nonlinear protocol for a structured time-varying synchronous multi-agent system. We present an opinion sharing dynamics of the multi-agent system as a trajectory of non-autonomous polynomial stochastic operators associated with multidimensional stochastic hyper-matrices. We show that the multi-agent system eventually reaches to a nonlinear consensus if either one of the following two conditions is satisfied: (i) every member of the group people has a positive subjective distribution on the given task after some revision steps or (ii) all entries of some multidimensional stochastic hyper-matrix are positive. IntroductionThis paper is a continuation of our previous studies [12,13,14] on nonlinear consensus. Namely, we study a nonlinear consensus in a multi-agent system having opinion sharing dynamics as a trajectory of non-autonomous polynomial stochastic operators associated with stochastic multidimensional hyper-matrices. In the paper [12] the nonlinear consensus problem was studied for a single cubic triple stochastic hyper-matrix meanwhile in the paper [14] the nonlinear consensus problem was studied for a sequence of cubic triple stochastic hyper-matrices. The nonlinear consensus problem for a single k−dimensional k−tuple stochastic hyper-matrix was studied in [13]. In this paper, we are aiming to study the nonlinear consensus problem for a sequence of k−dimensional k−tuple stochastic hypermatrices which unifies and generalizes all previous results.The novelty of the paper is that a new nonlinear protocol for a structured time-varying and synchronous environment was presented as a trajectory of higher dimensional stochastic hyper-matrices. One of the classical results in the theory of Markov chains states that if every element of the square stochastic matrix is positive then its trajectory converges to its unique fixed distribution. To the best of our knowledge, the similar problem for the higher dimensional hyper-matrix was open. In this sense, our paper is the pioneering study for the higher dimensional hyper-matrices. One of our main results states that if every element of the k−dimensional k−tuple stochastic hyper-matrix is positive then its trajectory converges to its unique fixed distribution. Since the Markov chains and the consensus problems are dual to each other, we present the application of our results into the nonlinear consensus problem.

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