Reaching an Agreement Using Delayed Information

This paper studies a modified version of the Vicsek\´s problem, also known as the "consensus problem". Vicsek et al. (1995) consider a discrete-time model consisting of n autonomous agents all moving in the plane with the same speed but with different headings. Each agent\´s heading is updated using a local rule based on the average of the headings of its "neighbors". We consider a modified version of the Vicsek\´s problem in which integer valued delays occur in sensing the values of headings which are available to agents. By appealing to the concept of graph composition, we side-step most issues involving products of stochastic matrices and present a variety of graph theoretic results which explains how convergence to a common heading is achieved