  We propose a linear model of opinion dynamics that builds on, and extends,
the celebrated DeGroot model. Its primary innovation is that it concerns
relative opinions, in that two opinion vectors that differ only by a
multiplicative constant are considered equivalent. An agent's opinion is
represented by a number in R, and evolves due to positive and negative impacts
resulting from the other agents' opinions. As a consequence of the relative
interpretation, the model exhibits nonlinear dynamics despite its DeGroot-like
linear foundation. This means that, even when agents are "well-connected", the
dynamics of the relative opinion model can reproduce phenomena such as
polarization, consensus formation, and periodic behavior. An important
additional advantage of remaining in a DeGroot-type framework is that it allows
for extensive analytical investigation using elementary matrix algebra. A few
specific themes are covered: (i) We demonstrate how stable patterns in opinion
dynamics are identified which are hidden when only absolute opinions are
considered, such as stable fragmentation of a population despite a continuous
shift in absolute opinions. (ii) For the two-agent case, we provide an
exhaustive description of the model's asymptotic (long-term, that is) behavior.
(iii) We explore group dynamics, in particular providing a non-trivial
condition under which a group's asymptotic behavior carries over to the entire
population.
