Sarymsakov cubic stochastic matrices

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.