Persistent Flows in Deterministic Chains

Abstract

This paper studies, the role of persistent flows in the convergence of infinite backward products of stochastic matrices of deterministic chains over networks with nonreciprocal interactions between agents. An arc describing the interaction strength between two agents is said to be persistent if its weight function has an infinite $l_1$ norm; convergence of the infinite backward products to a rank-one matrix of a deterministic chain of stochastic matrices is equivalent to achieving consensus at the node states. We discuss two balance conditions on the interactions between agents, which generalize the arc-balance and cut-balance conditions in the literature, respectively. The proposed conditions require that such a balance should be satisfied over each time window of a fixed length instead of at each time instant. We prove that in both cases global consensus is reached if and only if the persistent graph, which consists of all the persistent arcs, contains a directed spanning tree. The convergence rates of the system to consensus are also provided in terms of the interactions between agents having taken place. The results are obtained under a weak condition without assuming the existence of a positive lower bound of all the nonzero weights of arcs and are compared with the existing results. Illustrative examples are provided to validate the results and show the critical importance of the nontrivial lower boundedness of the self-confidence of the agents.