The asymptotic consensus problem on convex metric spaces

We consider the consensus problem of a group of dynamic agents whose communication network is modeled by a directed time-varying graph. In this paper we generalize the asymptotic consensus problem to convex metric spaces. A convex metric space is a metric space on which we define a convex structure. Using this convex structure we define convex sets and in particular the convex hull of a (finite) set. Under minimal connectivity assumptions, we show that if at each iteration an agent updates its state by choosing a point from a particular subset of the convex hull generated by the agent’s current state and the states of his/her neighbors, then the asymptotic agreement is achieved. In addition, we give bounds on the distance between the consensus point(s) and the initial values of the agents. As application example, we use this framework to introduce an iterative algorithm for reaching consensus of opinion. In this example, the agents take values in the space of discrete random variable on which we define an appropriate metric and convex structure. For this particular convex metric space we provide a more detail analysis of the convex hull generated by a finite set points. In addition we give some numerical simulations of the consensus of opinion algorithm.