Network feature and the intervention performance

We apply soft control method on an opinion dynamic model, the weighted DeGroot model, to change the convergent opinion value x̅'. The interaction network plays an important role in the dynamics of system, and the soft control performance (Δx̅, the difference between the new convergent opinion value x and the original convergent opinion value x̅). In this paper, we define a new network feature Ω, called `network differential degree', to measure how node degrees couple with influential values in the network, i.e., large Ω indicates nodes with high degree is more likely to couple with large influential value. We investigate the relationship between the soft control performance and the network differential degree Ω in the following three intervention scenarios: (1) add one special agent (shill) to connect to one normal agent to achieve the best soft control performance which can maximize |Δx̅|; (2) add one edge between two normal agents which can maximize |Δx̅|; (3) add a number of edges among agents which can maximize |Δx̅|. Through simulations we find significant correlation between |Δx̅| (the maximum value of |Δx̅|) and Ω in all three scenarios: the smaller Ω is, the larger |Δx̅*| is. So Ω could be used to predict how difficult it is to intervene and change the convergent opinion value of the weighted Degroot model by soft control intervention method. Meanwhile, a theorem of adding one edge and an algorithm for adding optimal edges are given.