  This paper studies the evolution of social power in influence networks with
stubborn individuals. Based on the Friedkin-Johnsen opinion dynamics and the
reflected appraisal mechanism, two models are proposed over issue sequences and
over a single issue, respectively. These models generalize the original
DeGroot-Friedkin (DF) model by including stubbornness. To the best of our
knowledge, this paper is the first attempt to investigate the social power
evolution of stubborn individuals basing on the reflected appraisal mechanism.
Properties of equilibria and convergence are provided. We show that the models
have same equilibrium social power and convergence property, where the
equilibrium social power depends only upon interpersonal influence and
individuals' stubbornness. Roughly speaking, more stubborn individual has more
equilibrium social power. Moreover, unlike the DF model without stubbornness,
we prove that for the models with stubbornness, autocracy can never be
achieved, while democracy can be achieved under any network topology.
