A Continuous Opinions Discrete Actions Model on Line Graphs

The starting point of this work is a well-known class of linear systems on a graph that asymptotically converge to a consensus state. We consider a variation of this dynamics, by modifying some of the states through the nearest integer function: this change sets the dynamics apart from the consensus dynamics. We focus our study on the case in which the underlying graph is a line, which is particularly significant as it exhibits asymptotic behaviors that are far from consensus. In this case, we compute the equilibria of the system and prove convergence of solutions.