This paper studies the evolution of self-appraisal and social power, for a
group of individuals who discuss and form opinions. We consider a modification
of the recently proposed DeGroot-Friedkin (DF) model, in which the opinion
formation process takes place on the same timescale as the reflected appraisal
process; we call this new model the single-timescale DF model. We provide a
comprehensive analysis of the equilibria and convergence properties of the
model for the settings of irreducible and reducible influence networks. For the
setting of irreducible influence networks, the single-timescale DF model has
the same behavior as the original DF model, that is, it predicts among other
things that the social power ranking among individuals is asymptotically equal
to their centrality ranking, that social power tends to accumulate at the top
of the centrality ranking hierarchy, and that an autocratic (resp., democratic)
power structure arises when the centrality scores are maximally nonuniform
(resp., uniform). For the setting of reducible influence networks, the
single-timescale DF model behaves differently from the original DF model in two
ways. First, an individual, who corresponds to a reducible node in a reducible
influence network, can keep all social power in the single-timescale DF model
if the initial condition does so, whereas its social power asymptotically
vanishes in the original DF model. Second, when the associated network has
multiple sinks, the two models behave very differently: the original DF model
has a single globally-attractive equilibrium, whereas any partition of social
power among the sinks is allowable at equilibrium in the single-timescale DF
model.
