Leader-Follower Consensus Modeling Representative Democracy

In this work, we study a leader-follower consensus protocol where both leaders and followers negotiate their states over a stochastically-switching network. The model incorporates the phenomenon of numerosity, which limits the perception of exact numbers. We derive a closed form expression for the asymptotic convergence factor that provides a necessary and sufficient condition for convergence and is used to study the expected decay rate of disagreement among agents. These results are validated with Monte Carlo simulations and we explore the dependence of the asymptotic convergence factor on model parameters using numerical simulations. This system can be used to model decision making in a representative democracy, where representatives negotiate among themselves and drive the opinion of the population.