Control Models on Social Networks in Marketing
A base model of impact in a social network is a weighted directed graph. Its vertices correspond to the members of a social group, and its arcs describe their mutual impact. A real value (an opinion of the group's member) as a function of time is ascribed to each vertex, and a real number (weight) is ascribed to each arc (a degree of impact of one member of the group to another or, what is the same, a degree of confidence of one member of the group to another). All strong subgroups (non-degenerated strong components of the digraph) develop their final opinions which depend only to the initial opinions of their members. The final opinions of other members of the group are linear convolutions of the final opininions of all strong subgroups. The most interesting question for applications arises when one or several decision makers are not satisfied by the received final opinions. To formalize such situation the base model is extended by an introduction of a set of the impact agents which can exert influence to the basic agents. The impact agents may change the initial opinions of the basic agents or the factors of their interaction. This leads to the optimal control problems or dynamic games on networks. The respective setups and their marketing interpretation are presented in the paper.