  The recently proposed DeGroot-Friedkin model describes the dynamical
evolution of individual social power in a social network that holds opinion
discussions on a sequence of different issues. This paper revisits that model,
and uses nonlinear contraction analysis, among other tools, to establish
several novel results. First, we show that for a social network with constant
topology, each individual's social power converges to its equilibrium value
exponentially fast, whereas previous results only concluded asymptotic
convergence. Second, when the network topology is dynamic (i.e., the relative
interaction matrix may change between any two successive issues), we show that
each individual exponentially forgets its initial social power. Specifically,
individual social power is dependent only on the dynamic network topology, and
initial (or perceived) social power is forgotten as a result of sequential
opinion discussion. Last, we provide an explicit upper bound on an individual's
social power as the number of issues discussed tends to infinity; this bound
depends only on the network topology. Simulations are provided to illustrate
our results.
