On the impact of agents with influenced opinions in the swarm social behavior

We consider a simplified version of the Taylor model, typically used in the collective dynamics of continuous exchange of opinions, to describe the properties of swarm formation in the presence of external sources of influence or prejudices affecting a number of agents in the network. Such external sources are responsible for the breakdown of the consensus equilibrium and directly influence certain other individuals in the network, which we denote as quasi-stubborn agents. These quasi-stubborn agents participate in consensus with other individuals, but are able to indirectly influence the opinions of the entire system. In particular, we show that the swarm in steady-state moves towards the convex hull of the opinions of the quasi-stubborn agents. This is an interesting result that allows a more accurate estimation of the final opinions in a social network. In the case of two prejudiced agents, an explicit expression of the stationary opinions is provided in terms of the Moore-Penrose inverse of the Laplacian of the graph. Numerical simulations are presented to illustrate the properties of the considered model.