Dynamical models of social influence may present discontinuous rules of interactions: discontinuities are unavoidable when interactions are of topological nature, i.e. when the dynamics is the outcome of interactions with a limited number of nearest neighbors. Here, we prove that classical solutions are not sufficient to describe the dynamics that is produced by such interactions, but one needs to use non-classical concepts of solutions instead. 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 an algorithm defining an interaction graph.