Intervention of DeGroot model by soft control

Opinion dynamics study the consensus or disagreement of group opinions. Consensus can be achieved under some circumstances. But when the consensus opinion is not what we expect, how can we intervene the system? In this paper a mechanism named soft control is first introduced in opinion dynamics to guide the group's opinion when the population are given and play rules are not allowed to change. According to the idea of soft control, one or several agents called shills are added to the group, then they act and are treated as normal agents. It's proved that the change of collective opinion is affected by the initial opinion and influential value of the shill, as well as how the shill connect to normal agents. An interesting and surprising phenomenon is discovered: when the shill connects to an agent (or virtual agent) whose initial opinion is larger (or smaller) than the original collective opinion, even though the shill's opinion is smaller (or larger) than the original collective opinion, the new collective opinion value will be larger (or smaller) than the original one.