Caratheodory Solutions and their Associated Graphs in Opinion Dynamics with Topological Interactions∗

Models of social influence may present discontinuous dynamical rules, which are unavoidable with topological interactions, i.e. when the dynamics is the outcome of interactions with a limited number of nearest neighbors. Here, we show that classical solutions are not sufficient to describe the resulting dynamics. We first describe the time evolution of the interaction graph associated to Caratheodory solutions, whose properties depend on the dimension of the state space and on the number of considered neighbors. We then prove the existence of Caratheodory solutions for 2-nearest neighbors, via a constructive algorithm.