  We describe a model for polarization in multi-agent systems based on Esteban
and Ray's standard family of polarization measures from economics. Agents
evolve by updating their beliefs (opinions) based on an underlying influence
graph, as in the standard DeGroot model for social learning, but under a
confirmation bias; i.e., a discounting of opinions of agents with dissimilar
views. We show that even under this bias polarization eventually vanishes
(converges to zero) if the influence graph is strongly-connected. If the
influence graph is a regular symmetric circulation, we determine the unique
belief value to which all agents converge. Our more insightful result
establishes that, under some natural assumptions, if polarization does not
eventually vanish then either there is a disconnected subgroup of agents, or
some agent influences others more than she is influenced. We also prove that
polarization does not necessarily vanish in weakly-connected graphs under
confirmation bias. Furthermore, we show how our model relates to the classic
DeGroot model for social learning. We illustrate our model with several
simulations of a running example about polarization over vaccines and of other
case studies. The theoretical results and simulations will provide insight into
the phenomenon of polarization.

Logic in Computer Science;Social and Information Networks