  We describe a model for polarization in multi-agent systems based on Esteban
and Ray's standard measure of polarization 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 show that polarization does not necessarily vanish in
weakly-connected graphs under confirmation bias. We illustrate our model with a
series of case studies and simulations, and show how it relates to the classic
DeGroot model for social learning.
