Algebraic criteria for consensus problem of discrete-time networked systems

This paper is mainly devoted to the algebraic criteria for consensus problems of discrete-time networked systems with the fixed and switching topology. A special eigenvector w of the Laplacian matrix is first correlated with the connectivity of a digraph, and then the relations between a class of Laplacian-type matrix and the stochastic matrix are established. Based on these tools, some necessary and/or sufficient algebraic conditions are proposed, which can directly determine whether the consensus problem can be solved or not. Furthermore, it is proved that only the agents corresponding to the positive elements of w contribute to the group decision value and decide the collective behavior of the system. Particularly for the fixed topology case, it is shown that not only the role of each agent is exactly proportional to the value of the corresponding element of w but also the group decision value can be calculated by such a vector and the initial states of all agents.