Leader–follower consensus over numerosity-constrained random networks

In this work, we study a discrete-time consensus protocol for a group of agents which communicate over a class of stochastically switching networks inspired by fish schooling. The network model incorporates the phenomenon of numerosity, that plays a prominent role in the collective behavior of animal groups, by defining the individuals’ perception of numbers. The agents comprise leaders, which share a common state, and followers, which update their states based on information exchange among neighboring agents. We establish a closed form expression for the asymptotic convergence factor of the protocol, that measures the decay rate of disagreement among the followers’ and the leaders’ states. Handleable forms of this expression are derived for the physically relevant cases of large networks whose agents are composed of primarily leaders or followers. Numerical simulations are conducted to validate analytical results and illustrate the consensus dynamics as a function of the number of leaders in the group, the agents’ persuasibility, and the agents’ numerosity. We find that the maximum speed of convergence for a given population can be enhanced by increasing the proportion of leaders in the group or the agents’ numerosity. On the other hand, we find that increasing the numerosity has also a negative effect as it reduces the range of agents’ persuasibility for which consensus is possible. Finally, we compare the main features of this leader–follower consensus protocol with its leaderless counterpart to elucidate the benefits and drawbacks of leadership in numerosity-constrained random networks.