On two approaches to analyzing consensus in complex networks

Recently there is a great interest in studying consensus and flocking problems by analyzing dynamics of linear interacting systems coupled via a complex network. Two approaches have emerged to study this problem. In one approach the system is analyzed via the theory of inhomogeneous Markov chains and consensus is related to weak ergodicity of Markov chains. More recently, consensus problems are studied with communication delays between the coupled systems and the approach used is based on paracontracting and pseudocontracting operators. The purpose of this paper is to illustrate the connections between these two approaches and show that under certain conditions they are in fact equivalent.