How effectively can the initial opinions influence the opinion formation in social networks with stubborn agents?

In this paper, we investigate an opinion formation model with stubborn agents, where each agent updates her opinion according to opinions of her neighbors as well as her own initial opinion with some degree of stubbornness. Generally, the presence of stubborn agents will preclude consensus and result in a steady distribution with different opinions. Here, we investigate how effectively the initial opinions can influence the opinion formation. And, we find that any steady distribution can be reached if the stubbornness of each agent is nonzero; otherwise, only the steady distribution located in a special space can be reached. Furthermore, we develop a simple criterion to judge whether or not a steady distribution is reachable. For a reachable steady distribution, a method for the choice of the corresponding initial opinions is presented; while for an unreachable distribution, an algorithm is designed to find a reachable distribution which is closest to the desired one.