Disagreement, Limit Cycles and Zeno Solutions in Continuous Opinion Dynamics with Binary Actions

This paper studies a mathematical model of opinion dynamics on social networks, which features continuous opinions and binary actions. The binary actions are a suitable quantization of the opinions, which evolve in continuous time. The model thus takes the form of a differential equation with discontinuous right-hand side: we explore the asymptotic behavior of its Caratheodory solutions, which turns out to be unexpectedly rich. By considering specific classes of graphs, namely lines and rings, we not only find attractive extended equilibria where the opinions are not in agreement, but also limit cycles and solutions that exhibit the Zeno phenomenon, whereby switching points accumulate in finite time.