Numerous studies are devoted to systems of connected agents and network control. Historically, the greatest interest of control theory has been in averaging systems, particularly the consensus problem. However, network interaction can be characterized by more specific functions reflecting the dependence on the actions of network neighbors, which is particularly evident in models of strategic interaction on a network, the subject of game theory. This paper surveys game-theoretic models of network interaction from the class of linear best-response games. The models are formally described, and formulations of control problems in this class of games are considered. Special attention is paid to the connection with consensus models and the well-known related control problems that can be stated in terms of a strategic interaction of agents. Despite the general similarity with the models of linear systems from control theory, this class of games allows capturing qualitative aspects of strategic interaction of connected agents and highlighting the role of structural characteristics.