  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 followers. Each person is assumed to maintain
a social state, which can be an emotional state or an opinion. Followers update
their social states based on the states of local neighbors, while social
leaders maintain a constant desired state. Social interactions are modeled as a
general 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.
