Leader-follower consensus and synchronization in numerosity-constrained networks with dynamic leadership

In this work, we study leader-follower consensus and synchronization protocols over a stochastically switching network. The agents representing the followers can communicate with any other agent, whereas the agents serving as leaders are restricted to interact only with the other leaders. The model incorporates the phenomenon of numerosity, which limits the perceptual capacity of the agents while allowing for shuffling with whom each individual interacts at each time step. We derive closed form expressions for necessary and sufficient conditions for consensus, the rate of convergence to consensus, and conditions for stochastic synchronization in terms of the asymptotic convergence factor. We provide simulation results to validate the theoretical findings and to illustrate the dependence of this factor on system parameters. The closed form results enable us to study the factors affecting the feasibility of consensus. We show that agents' traits can be chosen for an engineered system to maximize the convergence speed and that protocol speed is enhanced as the proportion of the leaders increases in certain cases. These results may find application in the design and control of an engineered leader-follower system, where consensus or synchronization at the fastest possible rate is desired.