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. 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.