Containment control for a directed social network with state-dependent connectivity

Social interactions influence our thoughts, opinions and actions. In this paper, social interactions are studied within a group of individuals composed of influential social leaders and follower groupies. Each person is assumed to maintain a social state, which can be an emotional state or an opinion on a social event. Followers update their own states based on the states of local neighbors, which are considered as reasonable, while the social leaders maintain a desired constant state. Social interactions are modeled as a directed graph where each directed edge represents an influence from one person to another. Motivated by the non-local property of fractional-order systems, the social response of individuals in the network are modeled by fractional-order dynamics whose states depend on influences from local neighbors and past experiences. A decentralized influence method is then developed to maintain existing social influence between individuals (i.e., without isolating peers in the group), and to influence the social group to a common desired state (i.e., within a convex hull spanned by social leaders). Mittag-Leffler stability methods are used to prove asymptotic stability of the networked fractional-order system.