  We investigate disagreement and polarization in a social network with two
polarizing sources of information. First, we define disagreement and
polarization indices in two-party leader-follower models of opinion dynamics.
We then give expressions for the indices in terms of a graph Laplacian. The
expressions show a relationship between these quantities and the concepts of
resistance distance and biharmonic distance. We next study the problem of
designing the network so as to minimize disagreement and polarization. We give
conditions for optimal disagreement and polarization, and further, we show that
a linear combination of disagreement and polarization of the follower nodes is
a convex function of the edge weights between followers. We propose algorithms
to address some related continuous and discrete optimization problems and also
present analytic results for some interesting examples.
