A Multi-Agent System (MAS) gives a complete description for large-scale systems consisting of small subunits, called agents. The behavior of MAS is particularly interesting because the agents may fulfill certain tasks as a group, even the individual agent does not know about the overall task. A lot of examples come from nature, such as schooling fishes or fireflies flashing in unison. A collective behavior is also interesting for engineers when solving problems such as flocking or synchronization. This is mainly due to its important applications, including the cooperative control of unmanned air vehicles, autonomous underwater vehicles, congestion control in communication networks, swarms of autonomous vehicles or robots, autonomous formation fight, etc. In all cases the goal is to control a group of agents connected through a communication network to reach an agreement on certain quantities of interest. This problem is called the consensus problem.

Many results have been achieved on this problem. Most researches in MAS consider a linear rule of an information exchanging. However, many systems, such as for instance the well-known Kuramoto oscillator exhibit nonlinear, locally passive dynamics.

In this work, we have considered a nonlinear protocol for a structured time-invariant multi-agent system. In the multi-agent system, we present an opinion sharing dynamics as a trajectory of a cubic triple stochastic matrix. We showed 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 the given cubic triple stochastic matrix are positive.

We know from the theory of Markov chains that if all entries of a doubly stochastic matrix are positive then its trajectory starting from any initial point taken from the simplex converges to the center of the simplex. The similar problem was open for cubic triple stochastic matrices. In this work, we gave an affirmative answer. To the best of our knowledge, this is a pioneering result for higher dimensional stochastic matrices.